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Prime ideals in rings of continuous functions
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Kimber, Chawne Monique, 1971-
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Integers ( jstor )
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Thesis (Ph.D.)--University of Florida, 1999.
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Includes bibliographical references (leaves 110-112).
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Typescript.
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Vita.
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by Chawne Monique Kimber.

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Full Text

PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS

By

CHAWNE MONIQUE KIMBER

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1999

This work is dedicated to those women who preceded me and to those who
are yet to follow.

ACKNOWLEDGMENTS

First and foremost, I express my wholehearted gratitude to my advisor, Jorge

Martinez. In the past few years, his humane guidance has helped me to achieve so

very much, in fact, more than I would ever have hoped. I follow his example both in

becoming a mathematician and a caring teacher, and in the enjoyment of the finer

things in life like wine, cheese, and chocolate.

Also, sincere thanks go to my committee members: Richard Crew, for show-

ing me some algebra; Alexander Dranishnikov, for teaching me a heap of topology;

Scott McCullough, for introducing me to real analysis (back when we were both

much younger); and Mildred Hill-Lubin, for expanding my world-view through lit-

erature.

Cheers and warm hugs to my friends, neighbors, and family, especially to the

immediate: Johnnie, Charles, Frances, Maribell, Chinene, Jean, and the inimitable

Poopy-girl, Cei.

ACKNOWLEDGMENTS ................................. iii

ABSTRACT .................................... v

CHAPTERS

1 PRELIMINARIES ................... ............ 1
1.1 History ................... ............. 1
1.2 Lattice-Ordered Groups .. ..................... 3
1.3 f-Rings .. .. .. .. ... .. .... .. ... .. .. .. 7
1.4 Rings of Continuous Functions ................. ... 10
1.5 Approaches ..... ............ ............ 15

2 CHARACTERS .................. ............... 18
2.1 Hahn Groups ............................. 18
2.2 Lex Kernels and Ramification ................... 20
2.3 Rank . .. . . . . . .. 26
2.4 Rank via Z#-Irreducible Surjections ............... 32
2.5 Prime Character .............................. 36
2.6 Filet Character ....................... ..... 43

3 GENERALIZED SEMIGROUP RINGS .................... 46
3.1 Specially Multiplicative f-Rings ................. 46
3.2 r-Systems and l-Systems ...................... 52
3.3 f-Systems ................................ 58
3.4 Survaluation Ring and nth-Root Closed Conditions ........ 66

4 RAMIFIED PRIME IDEALS .......................... 75
4.1 Ramified Points .................... ....... .. 75
4.2 Ramified Ga-points .. ..................... 79
4.3 Ramification via C-Embedded Subspaces ............ 84

5 tm-QUASINORMAL f-RINGS .......................... 88
5.1 Definitions ................................. 88
5.2 (B, m)-Boundary Conditions ................... 94
5.3 #X, m-Quasinormal and SV Conditions .............. 107

REFERENCES ................................... 110

BIOGRAPHICAL SKETCH .............................. 113

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS

By

Chawne Monique Kimber

May 1999

Chairman: Jorge Martinez
Major Department: Mathematics
Given a completely regular topological space X, we wish to determine the
order structure of < Spec(C(X)), C>, the root system of prime ideals of the ring of
real-valued continuous functions on X; and vice versa.
We present four approaches which give partial solutions to these problems.
First, we define three measures on < Spec+(G), C>, the set of prime subgroups of a
lattice-ordered group, which determine some arithmetic properties of the group, and
vice versa. Second, given any root system, we construct a generalized semigroup
ring R which is a commutative semiprime f-ring such that < r(%), C>, its root
system of values, is order-isomorphic to the given root system. Then we characterize
those non-isolated G6-points whose corresponding maximal ideal is the sum of the
minimal prime ideals it contains. Finally, we characterize those spaces X for which
C(X) has the property that the sum of any m minimal prime ideals is a maximal
ideal or the entire ring.

CHAPTER 1
PRELIMINARIES

The focus of this dissertation is the order structure of < Spec(C(X)), C>, the
spectrum of prime ideals of the ring C(X) of real-valued continuous functions on a
topological space, X. To start, we informally present the history and give motivation
for the discussion herein. We then review some essentials about lattice-ordered
groups, f-rings, and rings of continuous functions in detail and then formally indicate
the manner in which this thesis proceeds.

1.1 History

Our history begins with the independent research by Cech and Stone in 1937
(see the papers [Ce] and [St]), in which they describe a compactification 3X of a
topological space X which has the property that every real-valued continuous func-
tion on X extends to a continuous function on #X. Further, via #X, they establish
correspondences between the topological structure of X and certain algebraic prop-
erties of its ring C(X) of real-valued continuous functions under pointwise addition
and multiplication. For instance, Stone shows that the maximal ideals of the sub-
ring C*(X) of bounded functions are in one-to-one correspondence with points of

IX. The map p '-+ M*P = {f E C*(X) : f(p) = 0} witnesses this correspon-
dence and is a homeomorphism of topological spaces when the set of maximal ideals
of the ring is endowed with the hull-kernel (Zariski) topology. In particular, this
shows that for compact Hausdorff spaces X, Y, we have that X L" Y if and only if
C(X) 9 C(Y). The next significant result came in 1939 when Gelfand and Kolo-
mogoroff proved in [GK] that the maximal ideals of C(X) are exactly those of the

form MP = {f E C(X) : p E d#xZ(f)}, where Z(f) = {p E X : f(p) = 0} and
pe #X.
In the 1950's, Gillman, Henriksen, Jerison and Kohls began a formal in-
vestigation of topological/algebraic correspondences of this form. The elementary
techniques and results are recorded in the text Rings of Continuous functions, [GJ].
Concerning prime ideals, in [GJ, 14.3c] we learn that the prime ideals in C(X), con-
taining a given prime ideal, form a chain and in [GJ, 4J], it is shown that the
topology on X is closed under countable intersections (i.e., X is a P-space) if and
only if every prime ideal of C(X) is maximal. More generally, in [GJ, 14.25] we find
that X has the property that every bounded continuous function on set of the form

X \ Z(f), for some f E C(X), extends to a continuous function on X (that is, X is
an F-space) if and only if every maximal ideal of C(X) contains a unique minimal
prime ideal. Knowing these three facts, we can describe the graph of the prime ideal
spectrum of C(X) in each case, where vertices are prime ideals and edges indicate
set-inclusion. In the F-space situation, the graph is a disjoint set of strands (one
for each point of #X) with no branching; a P-space yields a graph consisting solely
of vertices (one for each point of #X).
It is from these topological characterizations of the graphical structure of
the spectrum of prime ideals of C(X) that we formulate our questions. Roughly
speaking, we wish to know:

Is it possible to determine the order structure (under inclusion) of the prime
ideal spectrum of the ring of continuous functions of a given topological space?

Conversely, given a graph, is it possible to construct a topological space such
that the given graph, in some sense, determines the structure of the spectrum
of its ring of continuous functions?

It turns out that both of these are rather ambitious pursuits and the ques-
tions must be refined before we can approach them. Extending the knowledge of
properties of the prime ideals of C(X), Kohls published a series of papers ([K1],
[K2], and [K3]) in 1957. In [K2], he addresses the properties of chains of prime
ideals of C(X). First he shows that the quotient ring C(X)/P is totally- ordered for
every prime ideal P and concludes that the prime ideals of the quotient ring form a
chain. Second, it is demonstrated that if P is nonmaximal, then the chain of prime
ideals in C(X)/P contains an 7rl-set (that is, a totally ordered set E such that for
every pair of disjoint countable subsets A, B C E such that A < B, there exists
c E E such that a < c and c < b for every a E A and every b E B). Hence, the chain
of prime ideals contains at least 2Ns primes. We may thus immediately reduce the
second of our questions to only consider those graphs for which each nontrivial edge
passes through an r77-set of vertices. The facts presented in the next three sections
show us that the class of graphs to consider can be further reduced.
In this dissertation, we continue to refine the questions and present four
perspectives-ranging from the very general to the very specific-which give partial
results. In order to properly introduce these approaches, we must recall some facts
and constructs which are fundamental to the ensuing investigation.

1.2 Lattice-Ordered Groups

Let (L, <) be a partially ordered set. If a, b E L are incomparable, then we
write a (I b. L is totally ordered if any two elements are comparable. We say that
L is a lattice if any two elements a, b E L have a least upper bound and a greatest
lower bound, denoted a V b and a A b, respectively. A lattice L is distributive if
a A (bV c) = (aV b) A (aV c), and dually for all a, b, cE L.
A group (G, +, 0, <) with partial order < is a lattice-ordered group (hence-
forth, -group) if it is a lattice and ifg < h implies that c+g < c+h and g+c < h+c

for all c E G. The majority of the groups we consider are abelian, so the additive
notation here is for convenience. It is important to note that any L-group is torsion-
free [D, 3.5] and its lattice is distributive [D, 3.17]. A (real) vector lattice is an
f-group G which is also an R-vector space such that rg > 0 for all positive g E G
and for all positive r E RB
By G+ we mean the set of elements g E G such that g > 0. Each element of G
may be written as a difference of elements of G+ : let g+ = g V 0 and g- = (-g) V 0,
then g = g+ g-. This follows from the fact that g+ A g- = 0. The absolute value
of an element is given by Igl = g+ + g-. In general, we say that a pair of elements
g, h E G are disjoint if g A h = 0. We will write g < h if ng < h for all n E N.
An I-homomorphism is a group homomorphism that also preserves the lattice
structure. An I-subgroup H of an I-group G is a subgroup which is also a sublattice
of (G, <). We call an I-subgroup convex if 0 < g h E H implies that g E H. G(S)
denotes the convex I-subgroup of G generated by the set S C G. When S = {g},
we write G(g). In fact, G(g) = {h E G : 3n E N, IhI| < nlgl}. In the special
case that G(g) = G, we call g a strong (order) unit. Let ((G) denote the set of
all convex I-subgroups of G ordered by inclusion. This set is a distributive lattice
under the operations of arbitrary intersection and Vie Hi = G(UIeHi), where

{Hi}iAr C C(G) and I is any indexing set; see [D, 7.10] for details.
Let S C G. Then the polar of S is given by

SI = {g E G: IgI A Is = 0 for all s E S}.

If S is a singleton, say S = {g}, then we write g- for the polar of S. Such a polar
is called principal. If g = 0, then g is termed a weak (order) unit. Note that for
any S C G, we have that S- E C(G) and (S')"- = S-. Let T(G) represent the
set of polar subgroups of G ordered under set-inclusion. Then T(G) 9C (G), but
in general this is not as a sublattice. Under the operations of arbitrary intersection,

I, and V., Hi = (Uie Hi))", where {Hi)3}i, C (G) and I is any indexing set,
we have that 3(G) is a Boolean lattice, by [D, 13.7].
The convex e-subgroups of greatest interest to us are the prime subgroups of
G. These are the subgroups H E C(G) for which any one of the following equivalent
conditions is satisfied (see [D, 9.1],[AF, 1.2.10], or [BKW, 2.4.1]):

1. IfgAh=OthengEHor h H.

2. Ifg,h> O and gAhEH then g H or h E H.

3. The right costs of H are totally ordered.

4. The convex I-subgroups of G containing H form a chain.

As suggested by the terminology and the second condition listed above, the
concept of a prime subgroup is related to that of a prime ideal in a ring. The
difference becomes apparent in considering the final equivalent condition which in-
dicates that the prime subgroups form a root system. That is, the graph of the
prime subgroups of G, in which nodes indicate prime subgroups and edges repre-
sent containment going up, has the property that incomparable elements have no
common lower bound. Illustrating that prime ideals differ from prime subgroups,
we note that the zero subgroup is the only prime subgroup of the totally ordered
group of integers, Z; whereas, the zero ideal and the ideals which are generated by
a prime integer comprise the set of prime ideals of Z. The structure of the graph of
the prime subgroups is the subject of our investigation. Note that we use Spec+ (G)
to denote the set, or spectrum, of all prime subgroups of G and to stand for the
associated graph.
By Zorn's Lemma, minimal prime subgroups exist. Let Min+(G) denote the
set of all minimal prime subgroups of G. If P E Min+(G) then by [AF, 1.2.11] we

have P = U{g : g V P}. This implies 0(Q) de= n{P E Min+(G) : P C Q} is the
set U{g : g V Q}, for a prime subgroup Q C G, by [BKW, 3.4.12].
A basis for an e-group G is a maximal pairwise disjoint set {gi}i4W C G+ such
that for each i E I, the set {g E G+ : g : g,} is totally ordered. The following is
Conrad's Finite Basis Theorem presented as [D, 46.12] and [C, 2.47]. It will figure
in our discussion in the next chapter.

Theorem 1.2.1. Let G be an t-group. The following are equivalent:

1. G has a finite basis.

2. Min+(G) is finite.

3. q(G) is finite.

4. There is a finite upper bound on the number of pairwise disjoint elements of
G.

5. There is a finite upper bound on the number of elements of strictly increasing
chains of proper polars.

Another application of Zorn's Lemma establishes the existence of convex I-
subgroups which are maximal with respect to not containing a fixed element g E G.
Any such subgroup is generally termed a regular subgroup and specifically called
a value of g. The set of all regular subgroups of G is usually represented by r(G).
Regular subgroups are prime, by [D, 10.4], and a prime subgroup is precisely a convex
I-subgroup which is an intersection of a chain of regular subgroups, [D, 10.8]. In
particular, the minimal prime subgroups of G correspond to the maximal chains
in r(G). For these reasons, we call the root system given by P(G) the skeleton of
Spec+(G). By convention, we view F(G) as a partially-ordered indexing set F whose

elements are denoted by lower case Greek letters and then represent the regular
subgroups by V, for 7 E r.
Topologize Spec+ (G) using the hull-kernel (or Zariski) topology whose open
base is given by U(g) = {P E Spec+(G) : g V P} for all g E G. In this topology,
Spec+(G) is Hausdorff if and only if Spec+ (G)=Min+(G) by [CM, 1.4]; on the other
hand, in the subspace topology, Min+(G) is always Hausdorff and U(g) n Min+(G)
is both open and closed for every g E G. The space Spec (G) is compact if and only
if G has a strong unit, by [CM, 1.3]; it is demonstrated in [CM, 2.2] that Min+(G)
is compact if and only if G is complemented, that is, if and only if for every g E G+
there is an h E G+ such that g A h = 0 and g V h is a weak unit.

1.3 f-Rings

Let (R, +, ., <) be a ring whose underlying group is an e-group and satisfies
the relations rc < sc and cr < cs whenever r < s and c > 0. Such a ring is a lattice-
ordered ring (abbreviated I-ring). If an i-ring R also satisfies ca A b = ac A b = 0
whenever a A b = 0 and c > 0, then R is called an f-ring. The following is found in
[BKW, 9.1.2]:

Theorem 1.3.1. Let R be an I-ring, then the following are equivalent:

1. R is an f-ring.

2. Every polar in R is an ideal.

3. Every minimal prime subgroup of R is an ideal.

It is not difficult to verify that every G-ring which is 1-isomorphic to a subdi-
rect product of totally-ordered rings (with coordinatewise operations) is an f-ring.
In [BP], it is shown that the converse of this statement holds when we assume the
Axiom of Choice (abbreviated, AC). Since we routinely apply AC, let us formally

state that we will work within the axioms of ZFC. Then we may use this equiv-
alent definition of an f-ring R in order to obtain this list of arithmetic properties
given in [BKW, 9.1.10], for a, b, c E R:

1. Ifc > 0 then c(aV b) = ca V cb and (aV b)c = acV bc.

2. Ifc > 0 then c(a A b) = ca A cb and (aA b)c = ac A be.

3. IallbI = jabl.

4. If aAb = 0 then ab = 0.

5. a2 > 0.

An f-ideal of an I-ring R is an ideal which is a convex I-subgroup of R. We

call an 1-ideal a prime -idealif it is also a prime ideal. Let Spec(R) denote the space
of all prime ideals of R in the hull-kernel toplogy. Let Max(R) and Min(R) denote
the subspaces of maximal and minimal prime ideals, respectively. By property (4)
above, we see that prime I-ideals of an f-ring are prime subgroups, hence, as in the
case of I-groups, the subset Spect(R) of prime I-ideals forms a root system. Denote
the subspaces of maximal and minimal prime I-ideals by Maxt(R) and Mint(R),
respectively.
We call a commutative ring semiprime if it contains no nonzero nilpotent
elements. In the case of commutative f-rings, we have [BKW, 9.3.1]:

Theorem 1.3.2. Let R be a commutative f-ring, then the following are equivalent:

1. R is semiprime.

2. For any a, b E R, we have that [al A lbI = 0 if and only if ab = 0.

3. Every polar of R is an I-ideal which is an intersection of prime ideals.

4. Min(R) = Min(R).

5. R is e-isomorphic to a subdirect product of totally-ordered integral domains.

We say that an I-ring R with multiplicative identity, 1, has the bounded
inversion property if a > 1 implies that a is a multiplicative unit. By [HIJo, 1.1],
a commutative f-ring R with 1 has the bounded inversion property if and only if
Max*(R) = Max(R).
Let A be any commutative ring. Then, in the hull-kernel topology, Min(A) is
a Hausdorff space with a base of clopen sets. If A is a semiprime ring, then Spec(A)
is Hausdorff if and only if Min(A) =Max(A); this occurs if and only if A is von
Neumann regular (or absolutely flat,) i.e., for every a E A there exists b E A such
that a = a2b), see [AM, p. 35]. In [HJ] it is demonstrated that if A is a semiprime
f-ring, then Min(A) is compact if and only if A is complemented (i.e., for every
a E A there exists b E A such that ab = 0 and a + b is not a zero-divisor). Max(A)
is compact for any commutative ring A with identity, and if A is a commutative
f-ring with identity which has the bounded inversion property, then the subspace
is Hausdorff, see [HJo, 2.3].
Let A be a commutative ring with identity and P E Spec(A). Define

O(P) = {a E A: 3b P,ab = 0}.

If A is also a semiprime f-ring, then this is the same as the I-subgroup O(P) defined
in the previous section. Recall that the localization of A at P is the subring, Ap, of
the classical ring of quotients of A/O(P) consisting of the elements whose denomi-
nator is not in P/O(P). (For a review of this construction and general facts about
localizations, see [AM] or [G]). It is the case that Ap is a local ring whose unique
maximal ideal is generated by P/O(P) and there is a one-to-one correspondence
between prime ideals of Ap and the prime ideals Q of A such that O(P) C Q C P.

Thus, if A is also an f-ring with bounded inversion, then by the root system struc-
ture of Spece(A), we have that AM "I A/O(M) since the quotient ring is already
local with unique maximal ideal M/O(M).

1.4 Rings of Continuous Functions

Let X be a Hausdorff topological space. X is called completely regular (or
Tychonoff) if for every closed set A C X and z E X \ A, there exists a real-valued
continuous function on X such that f(x) = 1 and f(A) = {0}. Unless otherwise
stated, we assume that all spaces are completely regular. Let C(X) denote
the set of real-valued continuous functions on a space X. Under the operations of
pointwise addition and multiplication, C(X) is a semiprime ring. Order the ring
via: f < g if and only if f(x) < g(x) for all x E X. This ordering gives an f-ring
structure such that C(X) has the bounded inversion property. Let C*(X) denote
the convex I-subring of bounded functions.
The zeroset of f, is the set Z(f) = {x E X : f(x) = 0}. The complement,
coz(f) = X \ Z(f), is the cozeroset of f. By [GJ, 3.6], a Hausdorff space X is
completely regular if and only if its topology is the same as the weak topology
generated by C(X). Equivalently, the set of all zerosets, Z(X), is a base for the
closed sets of such a space, [GJ, 3.2].
Sets A, B C X are completely separated if there exists f E C(X) such that
f(A) = {0} and f(B) = {1}. If for every f E C(A) there exists f E C(X) such that
fAA = f, then we say that A is C-embedded in X. Likewise, A is C*-embeddedin X if
bounded continuous functions on A extend to bounded continuous functions on X.
These embedding properties are characterized by complete separation of particular
subsets, as follows:

1. Urysohn Extension Theorem [GJ, 1.17]: A C X is C*-embedded in X if and
only if any two completely separated sets in A are completely separated in X.

2. A C*-embedded set is C-embedded if and only if it is completely separated
from every zeroset disjoint from it, [GJ, 1.18].

Recall that a Hausdorff topological space X is normal if any two disjoint
closed sets are separated by disjoint open sets. Assuming this stronger separation
axiom, the results listed above give rise to the theorem stated in [GJ, 3D], in which
the equivalence of the first two statements is known as Urysohn's Lemma.

Theorem 1.4.1. Let X be Hausdorff. The following are equivalent:

1. X is normal.

2. Any two disjoint closed sets of X are completely separated.

3. Every closed set of X is C*-embedded in X.

4. Every closed set of X is C-embedded in X.

For many reasons, it is often preferable to work with compact spaces. The
Stone-Cech compactification 3X is our compactification of choice, since fX is char-
acterized by the property that it is (up to homeomorphism) the unique compact
space in which X is dense and C*-embedded. There are at least three different ways
to construct iX, we begin with the one based on ultrafilters, described in detail in
Chapter 6 of [GJ], which we now summarize.
Let X be a completely regular space, let C be a subset of the power set of
X and Y C C. is a C-filter if 0 V ., it is closed under finite intersections and if
for every F E 7, the fact that F C F' E C implies that F' e F. If 7 is a Z(X)-
filter, then T is also called a z-filter. A maximal filter is an ultrafilter; similarly, a
z-ultrafilter is a maximal z-filter. Let fX be the set of all z-ultrafilters on X which
we index by {A' : p E X}. A closed base for the topology on fX is given by sets
of the form Z = {p E XX : Z e AP}, for Z E Z(X). Let p E [X and define

MP = {f E C(X): p E clxZ(f)}.

The theorem of Gelfand and Kolmogoroff [GK] is stated simply as:

Theorem 1.4.2. For a completely regular space X, the set Max(C(X)) is given by
{MP : p E X}.

In fact, this result gives rise to a homeomorphism of fX with Max(C(X)).
That is, since the sets Z[MP] = {Z(f) : f E MP} are precisely the z-ultrafilters on
X, by [GJ, 2.5], and Theorem 1.4.2 shows that the map p 1-+ Z[MP] is the desired
correspondence.
If p E X, then we will write Mp and, in this case, the maximal ideal and
corresponding z-ultrafilter are called fixed. Otherwise, a maximal ideal and its
corresponding z-ultrafilter is called free. It is evident that X is compact if and only
if every maximal ideal of C(X) is fixed. Maximal ideals are also classified by the
residue field C(X)/MP. Identifying the constant functions with their constant, we
see that these fields always contain a copy of R We call a maximal ideal real if the
field is exactly R; otherwise, the maximal ideal is called hyper-real. This concept is
the basis for considering the Hewitt realcompactification of X. Denoted vX, it is the
smallest subspace of (X in which X is dense and such that every maximal ideal of
C(vX) is real. In fact, by [GJ, 8.5], vX is the largest subspace of fX in which X
is C-embedded.
With these facts about the maximal ideals firmly in place, we now proceed
to consider the nonmaximal prime ideals. We know that every prime ideal of C(X)
is convex, by [GJ, 5.5]; so we deduce that Spec(C(X)) is a root system. In order
to understand this root system, we are required to consider the properties of other
ideals. For instance, for p E #X, the ideals of the form

0" = O(M0) = {f E C(X) : claxZ(f) is a neighborhood of p}

are of paramount interest when examining the prime ideals of C(X). One reason is
given in [GJ, 7.15]:

Theorem 1.4.3. Every prime ideal P in C(X) contains OP for a unique p E #X
and MP is the unique maximal ideal containing P.

If OP is prime, then we call p an F-point. If X has the property that OP is
a prime ideal for every p E X, then we call X an F-space. We see that in this case,
the graph of Spec(C(X)) consists of a set of strands with no branches. Note [GJ,
14.25]:

Theorem 1.4.4. Let X be completely regular. The following are equivalent:

1. X is an F-space.

2. /X is an F-space.

3. The prime ideals contained in any given maximal ideal form a chain.

4. Every cozeroset of X is C*-embedded.

5. Any two disjoint cozerosets of X are completely separated.

6. Every ideal of C(X) is convex.

7. Every finitely generated ideal of C(X) is principal (i.e., C(X) is BEzout).

A special case of an F-point is when Op = Mp and we call p a P-point if
this occurs. Call X a P-space if every point of X is a P-point. In this case, the
spectrum of C(X) consists only of vertices. Equivalent definitions of P-space are
presented in [GJ, 14.29] and are recorded below. First, recall that an ideal I of C(X)
is called a z-ideal if f E I and Z(f) = Z(g) implies that g E I. It is immediate
from the definitions that MP and OP are z-ideals for all p E 8iX. Note that not all
prime ideals are z-ideals; however, the following says that this is the case in a von
Neumann regular ring.

Theorem 1.4.5. Let X be completely regular. The following are equivalent:

1. X is an P-space.

2. vX is an P-space.

3. Every prime ideal of C(X) is maximal.

4. Every cozeroset of X is C-embedded.

5. For each f e C(X), the zeroset Z(f) is open.

6. Every ideal of C(X) is a z-ideal.

7. For every f e C(X), there exists g E C(X) such that f = gf2 (that is, C(X)
is von Neumann regular).

We now recall the definitions of other types of spaces which are useful to us.
X is basically disconnected if the closure of any cozeroset is clopen. X is extremally
disconnected if any open set has open closure. Discrete spaces are extremally dis-
connected; extremally disconnected spaces are basically disconnected and all such
spaces are F-spaces by [GJ, 14N.4]. Every P-space is basically disconnected by [GJ,
4K.7]. A space is a quasi-F space if every dense cozeroset is C*-embedded. Clearly,
from [GJ, 14.25], we see that every F- space is quasi-F. The converses of the pre-
ceding statements do not hold. That is, these are distinct classes of spaces, as we
now illustrate.

Example 1.4.6. Consider the following spaces:

1. Let U be a free ultrafilter on N. Let E = N U {a}, in which points of N
are isolated and neighborhoods of a are of the form U U {1}, where U E U.
Then E is an extremally disconnected subspace of #iN, but not a P-space. In
particular, O is a prime ideal which is not maximal; see [GJ, 4M]. Therefore,
E is an F-space.

2. Let D be an uncountable set. Let AD = D U {A}, where points of D are
isolated and a neighborhood of A is given by any cocountable set containing
it. Then AD is basically disconnected, but not extremally disconnected by [GJ,
4N.3]. Moreover, the topological sum X = AD II E is basically disconnected,
but neither extremally disconnected nor a P-space, by [GJ, 4N.4].

3. The corona, fN \ N is a quasi-F space which is an F-space, yet not basically
disconnected; see [GJ, 6W.3, 140].

1.5 Approaches

Starting as generally as possible in Chapter 2, we define three cardinal-valued
characters on the spectrum of prime subgroups of an f-group. The value of each
measure determines a portion of the arithmetic and/or polar structure of the I-
group, and vice versa. For instance, we define the prime character, ir(G) of an
-group, G to be the least cardinal K such that for any family {Q0a}a<, Min+(G),
of distinct minimal prime subgroups, we have that Va<, Qa is the smallest convex
I-subgroup of G containing all the elements of Min+(G). Roughly speaking, it is
a measure of the complexity of minimal paths in the graph of Spec+(G) between
minimal prime subgroups. We will show that the measure being finite satisfies the
following, where lex(G) denotes the smallest convex (-subgroup of G containing all
the elements of Min+(G) :

Proposition 1.5.1. Let G be an (-group and m a positive integer. The following
are equivalent:

1. r(G) = m < oo.

2. m is minimal with respect to the property that lex(G) = G(Uji= a -) for any
m pairwise disjoint positive elements, {aj}>Ti C lex(G)+.

8. m is minimal such that for any prime P C lex(G), the chains of proper polars
in P have length at most m 1.

Chapter 3 is devoted to a discussion of the properties of F(A, R), the gener-
alized semigroup ring of real-valued maps on a root system A (which has a partially
defined associative operation, +) each of whose support is the join of finitely many
inversely well-ordered sets. The ring structure on this group is introduced in [Cl]
and [C2]; we endow this ring with an f-ring structure. In particular, we show that
if (A, +) is a root system such that each of the following holds:

1. + is associative (when it makes sense);

2. if a, / E A are comparable, then a + 3, / + a are defined;

3. if a < p and a + 7, / + 7 are defined, then a + 7 < P + 7 and if 7 + a, 7 +
are defined then 7 + a < 7 + 3;

4. and if j is maximal, then 6 + p + 6, p + p are defined and 6 + = ~+6 = 6
for every 6 < IA,

then F(A, R) is an f-ring if and only if 6 = a + P implies a,3 > 6. And when
this occurs, the f-ring is semiprime and satisfies the bounded inversion property.
Moreover, by [CHH, 6.1], given any root system A, one of these f-rings has A
order-isomorphic to its root system of values. Thus, the second of our questions is
answered in the class of f-rings on the level of skeletons.
However, the solution to the second problem remains unclear in the smaller
class of rings of continuous functions. To gain a modicum of clarity on the situation,
we look to the work of Attilio LeDonne, published in 1977 in [Le], in which he
addresses the incidence of branching in the graph of Spec(C(X)). He shows, for
instance, that the root system branches at every prime z-ideal when X is a metric
space. In [Le, 2], LeDonne includes a result of DeMarco which states that there

is branching at each Mp when X is a first-countable space and p is non-isolated.
In Chapter 4, we show that, for a non-isolated Ga-point of a completely regular
space, there is branching at Mp if and only if X \p is not C*-embedded in X. This
result is then used to examine branching in Spec(C(X)) when X is not necessarily
first-countable.
Both of our questions are addressed in Chapter 5, in which we generalize a
few of the results of Suzanne Larson on quasinormal f-rings that are found in the
series of papers [Lal], [La2], and [La3]. The semiprime commutative quasinormal
f-rings with identity are the ones having the property that the graph of the root
system of prime -ideals does not contain a subgraph of the form:

A (1.1)

By [La3, 3.5], a normal space X has the property that C(X) is quasinormal if
and only if cl(U) n cd(V) is a P-space for any disjoint cozerosets U, V C X. Our
generalizations similarly describe those normal spaces X for which Spec(C(X)) does
not contain a subgraph of any of the following forms:

a2 an k k k k k (1.2)

where n, k, a1,... aO are positive integers satisfying some specified conditions.

CHAPTER 2
CHARACTERS

We seek a collection of measures on root systems whose values will determine some
portion of the structure of a lattice-ordered group. In this chapter we describe three
such measures: rank, prime character, and filet character. The rank measures the

width of a connected component of the spectrum, the prime character determines,
roughly speaking, the complexity of minimal paths between minimal primes, and
the filet character counts the maximum length of a chain of branching incidences.
The first sections of this chapter are a review of two constructs essential to the
discussion to follow.

2.1 Hahn Groups

To begin, we recall a method of constructing examples of e-groups having a
specified root system as the skeleton of its prime spectrum. Let A be a root system
and define

V(A,R) = {v : A -4 R : supp(v) has ACC},

where supp(v) = 16 E A : v(6) # 0}. V(A, R) is an e-group under pointwise addition
ordered by the relation: v > 0 if and only if v(6) > 0 for every maximal element

6 E supp(v). This f-group is called a Hahn group. In the paper of Conrad, Harvey
and Holland [CHH], it is demonstrated that any abelian f-group can be embedded
in a Hahn group of a more general description than we give here. Of interest to
us is the f-subgroup of maps with finite support denoted by E(A, R) and the -
subgroup of maps whose support is the join of finitely many inversely well-ordered
sets, denoted by F(A,R). Clearly, E(A,R) C F(A,R).

The proof of the first statement of Proposition 2.1.1 is analogous to that
of Theorem 6.1 in [CHH]. This establishes that r(E(A, R)), r(F(A, R)) and A are
isomorphic as partially-ordered sets. For the sake of completeness, we present an
elementary proof of this fact for the case of F(A, R), although the result is easily
obtained from the theory of finite-valued f-groups. The proof is identical in the case
of E(A, R).
Recall that an -group is finite-valued if each element has only a finite number
of values. A special value is a prime subgroup which is the unique value of an element.
An -group G is finite-valued if and only if every value of G is special and if and
only if every element of G is a finite sum of pairwise disjoint special elements; for
details, see [AF, 10.10]. If G has a set S of special values such that S is a filter and
AS = {0}, then G is called special-valued.

Proposition 2.1.1. Let A be a root system. For each 6 E A define

V = {f E F(A, R) : v(7) = 0 when 7 > 6}.

Each V6 is a special value. Further, every value of an element of F(A, R) is of
the form V6 for some 6. Thus, A is the skeleton of Spec+(F(A,R)) and F(A,R) is
finite-valued.

Proof: Let 6 E A and let Xs E F(A, R) be the characteristic function on {6}. Then

Xs 4 Vs and we will show that V6 is the unique value of XS. Let V be a value of Xg
and let v E V+ \ V6. Then there exists 7 > 6 such that 7 is maximal in supp(v), and,
hence, v(7) > 0. If 6 < 7, then 0 < Xs < v, a contradiction. If 6 = 7, then there
exists a positive integer n such that 0 < Xa < nv and hence X; E V by convexity,
which is a contradiction. Thus V = V6.
Let v E F(A,R)+ and let V be a value of v. Let D be the finite set of
maximal elements of supp(v). Then the characteristic function XD is not in V; else,

there exists an integer n such that 0 < v < nXD, a contradiction. Since V is prime
and the set {xa : 6 E D} is pairwise disjoint, there exists a unique element 6 E D
such that Xj 4 V. By the above, we know that V C Vi. Finally, since v V Va, we
have that V = V5, as desired.
The final statement follows from [AF, 10.101 since we have shown that every
value is special. *

2.2 Lex Kernels and Ramification

Throughout, we will describe the location of a prime subgroup in the graph
in reference to a designated convex f-subgroup, called the lex kernel of an -group G
and denoted by lex(G). It is the least convex -subgroup containing all the minimal
prime subgroups of G. It is always the case that lex(G) is a prime subgroup [D,
27.2] which is normal in G [D, 27.13]. The following is a summary of a part of the
discussion of lex kernels in [D, 27] and gives a description of the -subgroup in
terms of its generators.

Proposition 2.2.1. Let G be an t-group and let C be a convex -subgroup. The
following are equivalent:

1. C = lex(G).

2. C is the least prime subgroup such that if 0 < g V C then g > h for every
hE C.

3. C is the convex f-subgroup of G generated by {g E G : g (I 0}.

4. C = {0} U {g E G: 3g1,g2,...g n E G,g II 1 g2 II ." II II 0}.

5. C is the convex I-subgroup of G generated by the nonunits of G.

6. C is prime and is the smallest among all convex t-subgroups of G which are
comparable with every convex t-subgroup of G.

7. C is the maximal convex f-subgroup of G such that lex(C) = C.

8. C is the supremum of the proper polars of G in the lattice of convex f-subgroups
of G.

It is natural to now introduce a concept which we will discuss in more detail in
Chapter 4. This is a generalization of a concept from [Le]. Let A be a commutative
ring with identity and for each a E A, let Max(a) = {M e Max(A) : a E M}. Recall
that an ideal I of A is a z-ideal if a E I and Max(a) = Max(b) imply that b E I.

Definition 2.2.2. Let A be a commutative f-ring with identity. A prime t-ideal P
is ramified if it is the sum of the minimal prime ideals that it contains. A maximal
t-ideal M is totally ramified if every prime z-ideal contained in M is ramified. A
completely ramified ring is one in which every prime z-ideal is ramified.

Graphically, a prime f-ideal P < A is ramified if and only if it is minimal or
if the root system of prime G-ideals of A branches at P. We begin with the f-group
characterization of ramification. It is the case that a ramified maximal f-ideal M
of A is the lex kernel of the local f-ring A/O(M). In order to discuss a proper lex
kernel in an f-ring, A, we must operate inside a localization. Henceforth, we will
obtain results for local rings and tacitly extend to the general case by referring to
localizations.
The following characterization of ramified maximal f-ideals is immediate from
Proposition 2.2.1.

Corollary 2.2.3. Let A be a commutative semiprime local f-ring with identity and
bounded inversion and let M be the maximal ideal. The following are equivalent:

1. M is ramified.

2. M is the convex t-subgroup of A generated by {f E A: f || 0}.

s. M= {O}U{f EA:3fi,f2,... fn E A,f II fi f2. l f IIf 0}.

4. M is the convex I-subgroup of A generated by the set

{f E A:3g A,g> O,gA f =0}.

5. M is the convex I-subgroup generated by the elements of A which are not order
units.

6. M is the smallest among all convex e-subgroups of A which are comparable
with every convex l-subgroup of A.

7. M is the supremum of the proper polars of A in the lattice of convex f-subgroups
of A.

It is well-known that the lex kernel of an f-group is a prime subgroup (see [D,
27.2]). We now show that the lex kernel of a commutative local semiprime f-ring
with identity is an ideal and then give conditions which guarantee that the lex kernel
is a prime ideal.
Let A be a commutative f-ring with identity. Recall that an ideal I < A is
pseudoprime if ab = 0 implies a E I or b I. An ideal J < A is semiprime if a E J
whenever a2 E J. A is square-root closed if for any 0 a E A, there exists 0 < b E A
such that a = b2. Let a, b E A, then A is n-convex if whenever 0 < a < b", there
exists u E A such that a = bu.

Proposition 2.2.4. Let A be a commutative semiprime local f-ring with identity.
Then lex(A) is a prime subgroup which is a pseudoprime f-ideal. If, in addition, A
is square-root closed, then lex(A) is a semiprime f-ideal.

Proof: Let f E lex(A). Then there exists g > 0 such that f A g = 0. If af = 0
then af E lex(A); else, af A g = 0 and we conclude again that af E lex(A). Hence
lex(A) is an ideal.

Let N be the set of nonunits of A and recall that lex(A) = A(N). Let ab = 0.
If a or b is 0, then there isThen by convexity, we see that a+, a- E A(N). Hence
a E A(N) and we have that the lex kernel is pseudoprime. Since any prime ideal is
semiprime and the lex kernel is the sum of the minimal prime ideals, the lex kernel
is semiprime, if A is also square-root closed; see [HLMW, 2.12(d)]. n

Corollary 2.2.5. Let A be a commutative semiprime local f-ring with identity and
bounded inversion and let M be the maximal -ideal. M is ramified if and only if
lex(A) is a z-ideal.

Proof: Since the maximal 1-ideal is the only z-ideal of a local f-ring, this is
immediate. *

Corollary 2.2.6. If A is a commutative local 2-convex semiprime f-ring with iden-
tity which is square-root closed, then the lex kernel of A is a prime -ideal.

Proof: By the remark after [La4, 4.2], under these hypotheses, we have that an

-ideal is a prime ideal if and only if it is pseudoprime and semiprime. n

For the remainder of this section, let G be an abelian lattice-ordered group.
Recall the following for H an -subgroup of G. H is rigid in G if for every h E H
there is g E G such that h" = g". It is shown in [CM, 2.3] that if H is rigid in G
then the contraction of minimal prime subgroups of G to minimal prime subgroups
of H is a homeomorphism of minimal prime spaces. If H E I(G), then H is very
large in G if it is not contained in any minimal prime subgroup of G. It is shown in
[CM] that if H E (G) then H is very large in G if and only if H is rigid in G. It
turns out that ramification in a rigid subring indicates global ramification and vice
versa. This is a direct consequence of the lex kernel correspondence demonstrated
below.

We will also need the following facts (see [BKW, 2.4.7, 2.5.8]):

Proposition 2.2.7. Let H E C(G).

1. The contraction map from the set of prime e-subgroups of G not containing H
to the set of prime I-subgroups of H is an order-preserving bijection.

2. If V is a value ofh E H in G, then V V n H is a bijection between the set
of values of h in G and the set of values of h in H.

Proposition 2.2.8. Let H < G be a convex i-subgroup. Assume that u E H is a
weak unit of H and a weak unit of G. Let V be a value of u in G. Then we have
that V n H = lex(H) if and only if V = lex(G).

Proof: Assume that H C P E Min (G) then u E P and we have that u1D 1 P by
[AF, 1.2.11]. This is a contradiction since u'G = 0 e P. Thus H is rigid in G since
it is a convex I-subgroup which is very large in G.
Assume that V n H is the lex kernel of H. Then V n H is the least convex
I-subgroup of H containing all the minimal prime subgroups of H. Since H is rigid
in G, by the bijection given in the first part of Proposition 2.2.7, all the minimal
prime subgroups of G are contained in V, and V is the least such convex I-subgroup
of G. That is, if W c V also contains the minimal prime subgroups of G, then
W n H is a convex -subgroup of H containing all the minimal prime subgroups of
H and hence W n H = V n H. But this says that W n H E F(H) is a value of u
and hence, W E r(G) is a value of u Therefore V = W.
If V is the lex kernel of G then VnH contains all the minimal prime subgroups
of H. Thus the lex kernel of H is contained in V n H. Let P C V f H be a
prime I-subgroup of H containing all the minimal prime subgroups of H. Then by
Proposition 2.2.7, there exists a prime convex I-subgroup Q < G not containing
H such that P = Q n H and since we have a rigid embedding, Q contains all the

minimal prime I-subgroups of G. Hence, Q = V, P = V nH and V n H is the lex
kernel of H. w

Let A be a commutative semiprime f-ring with identity and bounded in-
version. If M E Max(A) then AM is semiprime with bounded inversion. This is
a result of the well-known facts that the e-homomorphic image of an f-ring with
bounded inversion has bounded inversion and that AM -- A/O(M); see the proof
of [La3, 2.7]. Since we must localize an f-ring in order to have a proper lex kernel,
the following allows application of Proposition 2.2.8 to f-rings.

Proposition 2.2.9. Let B be a commutative semiprime f-ring with identity and
let A be a rigid convex f-subring of B. If M E Max,(A) is such that M = N n A
for some N E Maxe(B), then AM is a rigid convex f-subring of BN.

Proof: Recall that AM c A/O(M) and BN 9 B/O(N). Define a map 4 : A -+ BN
by a -+ a+O(N). This map is an f-ring homomorphism. We show that the image is
convex in BN and that the kernel is O(M). For a E A, let 0 < b+O(N) < a+O(N).
Then there exists n E O(N)+ such that 0 < b < a+n. If b-n < 0 then 0 < b < n and
hence b E O(N) since O(N) is convex in B. Thus we may assume that 0 < b-n < a.
Then b n E A and b + O(N) = b n + O(N) Elm(n). Therefore the image of 4 is
convex in BN.
The kernel of 4 is O(N) n A. It is easy to show that O(N) nA C O(M) since
M = N n A. For the reverse inclusion, assume that a E A and a O(N). Since
O(N) is the intersection of the minimal prime ideals of B contained in N, there exists
P EMin(B) such that P C N and a V P. By the rigidity of A in B, Pn A EMin(A),
and therefore a V O(M). We now have that Ker(4) = O(N) n A = O(M) and
therefore AM is a convex f-subring of BN.
Since AM contains the identity element of BN, AM is very large in BN. For
rigidity, we need only recall that very large convex embeddings are rigid, [CM]. m

Corollary 2.2.10. Let B be a commutative semiprime f-ring with identity. Let A
be a rigid convex f-subring of B. Let M E Maxi(A) be such that M = N n A for
some N E Maxt(B). Then M is ramified in A if and only if N is ramified in B.

Let A* denote the f-subring of bounded elements of the commutative semiprime
f-ring A with identity. Note that A* is convex and rigid in A. In his dissertation
[Wo], Woodward proves the following fact:

Theorem 2.2.11. Let A be a semiprime f-ring with identity and bounded inver-
sion. Let M be a maximal ideal of A and let M be the unique value of A* containing
Mn A*. The map M I-+ gives a homeomorphism between Max(A) and Max(A*).
That is, Max(A*) is the subspace consisting of values of 1 in A*. In particular, if M
is a real maximal ideal of A, then M n A* E Max(A*).

Corollary 2.2.12. Let A be a commutative semiprime f-ring with identity satisfy-
ing the bounded inversion property. Let M E Max(A) be real. Then M is ramified
if and only if I = M n A* is ramified in A*.

2.3 Rank

The first character on Spec+ (G) that we consider is simply one which counts
the minimal prime subgroups contained in a convex f-subgroup.

Definition 2.3.1. The rank, rkG(H) of a convex I-subgroup H < G is the cardi-
nality of the set of minimal prime subgroups of G contained in H. If that cardinal
is not finite, then we will say that H has infinite rank; we may choose to specify the
cardinal when its value is of significance in a discussion. If H is a minimal prime
subgroup of G, then we define rkG(H) = 0.

This is a variation of the following definition given in [HLMW]: Let A be a
commutative f-ring with identity and M a maximal G-ideal of A. The rank of M,
denoted rkA(M), is the cardinality of the subspace of minimal prime ideals of A

contained in M. By convention, if the rank of M is infinite and we don't necessarily
care about the exact cardinality, we write rkA(M) = oo. The rank of a point p E X,
rkx(p), is the rank of Mp. The rank of the f-ring A is the supremum of the ranks
of the maximal e-ideals of A, when it exists; the rank of a space X is the rank of
C(X).
We begin with illustrations of the extremal values of ranks. An e-group is
semiprojectable if for any g, h E G+, (g A h)- = G(gL U b-). In [BKW, 7.5.1], it
is proved that G is semiprojectable if and only if each prime subgroup contains
a unique minimal prime subgroup, which is equivalent to rkG(P) < 1 for every
P E Spec+(G). Thus, it is evident that a space X is an F-space if and only if C(X)
is semiprojectable which is equivalent to rk(C(X)) < 1. In particular, X is a P-space
if and only if C(X) is von Neumann regular, which is equivalent to rk(C(X)) = 0.
The one-point compactification of the natural numbers, aN, is an example of a space
for which C(X) has infinite rank, [GJ, 14G]. In fact, if a is the point at infinity,
then the maximal ideal corresponding to a contains 2C minimal prime ideals one
for each free ultrafilter on N. Moreover, by [HJ, 4.8], this subspace of minimal prime
ideals is homeomorphic to the corona, /N \ N. Proposition 2.3.3 describes a general
situation in which we have infinite rank.
We recall some definitions. From [LZ, 39.1]: let G be a vector lattice, v E G+,
and let {g,}=1 C G be a sequence. We say that the sequence converges relatively
uniformly to g E G along the regulator v, and write gn -14 g, if for every e > 0 there
exists Ne > 0 such that for all n > N,, we have that Ig gnl ev. The sequence is
relatively uniformly Cauchy with respect to v if for every e > 0 there exists Ne > 0
such that for all n,m > Ne, we have that Ig, gn < ev. G is called uniformly
complete if for every v E G+, every sequence which is relatively uniformly Cauchy
with respect to v relatively uniformly converges along the regulator v.

Lemma 2.3.2. Let G be a uniformly complete vector lattice with weak order unit
u E G+. For any set {gj}jl, there exists g E G such that g = nfEW gf.

Proof: Let g = E', 2-(lgjl A u). a

Proposition 2.3.3. Let G be a uniformly complete complemented vector lattice with
weak order unit u E G+. If for some Q E Spec+(G) we have rko(Q) > w, then Q
contains at least 2C minimal prime subgroups.

Proof: Note that, by [CM, 2.2], Min+(G) is compact since G is complemented. Let
P = {Pn)nEN be a countably infinite set of minimal prime subgroups which are
contained in Q which is discrete in the hull-kernel topology on Min+(G). We first
show that P is C*-embedded in Min+(G) and conclude that the minimal prime space
contains an homeomorphic copy of #N. Then we describe the elements of Min+(G)
that correspond to the points in this copy of #N \ N.
Let A, B C P be completely separated in P and index them by I, J C N
as A = {A : i E I} and B = {Bj : j J}. Fix i E I and let A, E A. For each
Bj E B, Let ai, E A+ \ Bj and bj, E Bj \ Ai. Then Ai E njJ U(bj) V' Ki and
B C UjEJ U(oj) = U(EjEJ 2-i(4 A u)) V Li. Then Ki, Li are disjoint closed sets
in Min+(G).
For each i E I, generate the disjoint pair Ki, Li. Then A C cl(USIK,) V1 K
and B C niElLi = l L. By Urysohn's Lemma, the disjoint closed sets K,L are
completely separated in Min+(G) since Min+(G) is normal. Consequently, A, B are
completely separated in Min+(G) and therefore P is C*-embedded in Min+(G) by
the Urysohn Extension Theorem. Finally, by [GJ, 6.5], the closed subset of Min+(G)
of minimal prime subgroups in Q contains an homeomorphic copy of #N.
Let U be a free ultrafilter on N. For g E G, let N(g) = {n : g E P,}. Define
a new prime subgroup P by g E P if and only if N(g) E U. We show that P is a

minimal prime subgroup. The following proof is the same as that for [HJ, 4.8] and
for [HLMW, 4.1].
Let g, h E P. P is a subgroup since N(g h) D N(g) n N(h) E U implies
g h E P by filter properties. By convexity and since the P, are prime subgroups,
N(gVh) D N(g) E U and N(gAh) D N(g) E U, we have that P is a sublattice of G.
Thus P is an I-subgroup of G. Let 0 < g < h E P. Then N(g) D N(h) E U since each
P, is convex, and thus P is convex. Let gAh E P, then N(g) UN(h) N(gAh) E U
implies that N(g) or N(h) is in U since U. Thus P is a prime subgroup of G.
Let g E P. Since P, is a minimal prime subgroup for each n, we have that for
each n E N(g), there exists an hn E G \ Pn such that g A hn = 0. By Lemma 2.3.2,
there exists h E G such that h- = nEN(,, ) hn. Then g A h = 0 and h Pn for all
n N(g) since h'- C hI C Pn for each n E N(g). Thus N(g)n H(g) V U and hence
h V P and P is a minimal prime subgroup of G. a

Recall that a space X is cozero-complemented if for any cozeroset U C X
there exists a cozeroset V C X such that Un V = 0 and U U V is dense in X.
A concrete example of a maximal ideal of infinite rank is found in C(X) where
X is cozero-complemented and first countable. DeMarco shows in [Le, 2] that
rk(Mp) 2 2, for any nonisolated point of a first countable space (the result actually
says more than this, and we will discuss this in Chapter 4). By modifying DeMarco's
proof, we show that Mp contains infinitely many minimal prime ideals and hence
has rank at least 2'.

Proposition 2.3.4. Let X be first countable and let p E X be nonisolated. For
every m E N there exists a family of m distinct prime ideals which sum to M,.
Moreover, if X is also cozero-complemented, then there exist at least 2C minimal
prime ideals contained in M,.

Proof: Since p has a countable base, there exists g E C(X)+ such that Z(g) = {p}.
Let {Vi,}i be a neighborhood base at p and define a sequence of real numbers {a,}
recursively as follows: for each i, let ai E g(Vi) such that 0 < < a3 < a2 < al
and limn-+oo an = 0. Let {xn} C X be a sequence of distinct preimages under g such
that xi E Vi. Then the sequence {xn} converges to p and may be considered as a
discrete set in X.
Let m > 1 be given and let Ui,U2,... ,Um be distinct free ultrafilters on the
sequence {xn}. Define Pi = {f E C(X) : 3A E Ui,A C Z(f)}, for each i. DeMarco
shows that each of these sets is a prime z-ideal of C(X) and that My is the sum
of any two. Thus these prime ideals are noncomparable. We will demonstrate, as
DeMarco has done for m = 2, that Mp is the sum of these m noncomparable prime
ideals.
Let {Ai,}1 be a collection of m pairwise disjoint subsets of {xn} such that
Ai e U, for each 1 < i < m. If B, = g(Ai), then Bi U {0} is a closed subset of R
Thus B, U {0} is a zero-set of K Choose Wi E C(R) for each 1 < i < m such that

Z(p,) = B, U {0} and E', i = a.
Let ui = pig. Then A, c Z(g) = Z(ui), hence u, E Pi. Finally, we have

g = ul + u2 +- -. + Urn. If h E Mp, then Z(h- + g) = {p} and Z(h + g) = {p} By
the above, h+ + g, h- + g E -i Pi and hence h+, h-, h E !, P.
The final statement of the proposition follow from the previous proposition,
since the cardinality of /N is 2C, by [GJ, 9.2]. *

Now that we have illustrated the extreme cases of 0,1 and infinite rank, we
present a result of [HLMW, 3.1], which gives a test for finite rank of a point of a
compact space.

Proposition 2.3.5. Let X be a compact space. Then p E X has rank n < oo if
and only if there exist n pairwise disjoint cozerosets {Uj} 1 with p E njl cl(Uj),
and no larger family of pairwise disjoint cozerosets has this property.

An f-ring A is called an SV-ring if A/P is a valuation domain (i.e. principal
ideals are totally-ordered) for every prime ideal P. A space X is an SV-space if
C(X) is an SV-ring. We will discuss this class of rings in more detail in Chapter 3.
However, using the above, it is shown in [HLMW, 4.1] that any compact SV-space
has finite rank. The validity of the converse of this result is unknown. Presently,
our objective is to show that the result in Proposition 2.3.5 does not hold for infinite
rank. To demonstrate this, we define a cardinal function on compact spaces and
compare its value with a known cardinal invariant.

Definition 2.3.6. Let K be a cardinal and let X be a compact space with p E X. Let

{U,,}a< be a family of pairwise disjoint cozerosets of X and call the set fl<, cl(UA)
a K-boundary. Define p(p, X) to be the infimum over all (infinite) cardinals n such
that p is not contained in a K-boundary and let p(X) be the supremum over all the
points p E X of the cardinals p(p, X).

Recall that the cellularity of a space X, denoted c(X) is the infimum over
all (infinite) cardinals K such that every family of pairwise disjoint open sets of X
contains at most rK many sets.

Proposition 2.3.7. Let X be compact. Then p(X) < c(X)+.

Proof: If p(X) > c(X)+, then there exists a s-boundary of cardinality greater
than the cellularity of the space, which is nonsense. m

Example 2.3.8. Let 7 > Ro. The product space, 21, where 2 is the two-point
discrete space is called the Cantor space of weight 7. We show in Example 4.3.4

that every point of 2' has infinite rank, thus rk(C(2')) > 2' by Proposition 2.3.3.
By [E, 3.12.12(a)], we have that c+(2T) = Ri. Thus, Proposition 2.3.7 gives us that
Ro < p(2T) < R1. Therefore, since Ri \$ 2c, we see that p(2T) \$ rk(C(27)). O

2.4 Rank via Z#-Irreducible Surjections

We must first recall a few definitions from [Ha] and [HVW2]. Let X, Y be
Tychonoff Hausdorff spaces. Let f : Y -+ X be a subjective continuous map. Then
f is perfect if it is a closed map such that the inverse image of any point is compact.
A perfect map is irreducible if proper closed sets of Y map to proper closed sets
of X. The pair (Y, f) is a cover of X if f is a perfect irreducible surjection from
Y to X. Let (Y1,fl) and (Y2,f2) be covers of X. We define (Y1, f) ~ (Y2,f2) if
there exists a homeomorphism g : Yi -+ Y2 such that f2g = fi. Order the set of
,-equivalence classes of covers via: (Y1, fi) ( (Y2, f2) if and only if there exists a
continuous map g : Y1 -+ Y2 such that f2g = fi. A class of spaces C is a covering
class if for any space X there exists a least cover (Y, f) of X such that Y E C.
The minimal extremally disconnected, basically disconnected and quasi-F covers of
compact spaces are described in [PW], [V], and, respectively, in [DHH], [HudP], and
[HVW1].
Certain covering maps allow us to compute the rank of a space externally.
A perfect irreducible surjection 4 : Y -+ X is Z#-irreducible if for each cozeroset
U C Y, there is a cozeroset V C X such that cly(U) = cly(q-'(V)). This condition
on maps is also known as sequential irreducibility and wl-irreducibility. It turns out
that a map 4) is Z#-irreducible if and only if C(4) is a rigid embedding of C(X) inside
C(Y), by [HaM, 2.2]. Hence we have a homeomorphism Min(C(Y)) ^ Min(C(Y))
via contraction, by [CM, 2.3]. It is therefore not surprising that these maps are
useful for calculating rank.

Example 2.4.1. Let X be a compact space. The quasi-F cover of X, (QFX, Ox),
constructed in [HVW1] has the property that 4x is Z#-irreducible. We summarize
this construction.
Let Z#(X) = {clxintx(Z) : Z E Z(X)}. For A E Z#(X) denote the set
of ultrafilters on Z#(X) containing A by A. The authors of [HVW1] show that
T(Z#(X)), the compact space of ultrafilters on Z#(X) whose topology has a closed
base given by {A: A E Z#(X)}, is quasi-F and define a perfect irreducible surjection
x : T(Z#(X)) -- X by a E T(Z#(X)) maps to the unique point in n{A: A E a}.
Then QFX = T(Z#(X)) with the map bx is the quasi-F cover of X.
The map bx is Z#-irreducible: It is shown in [HVW1, 2.9] that if we have
A e Z#(QFX), then qbx(A) E Z#(X), which is equivalent to the property of Z#-
irreducibility. In fact, the quasi-F cover of X is characterized up to equivalence
in [HVW1, 2.13] as the only cover (Y, f) of X for which Y is quasi-F and f is
Z#-irreducible. 0

Before we continue, we discuss the question (now answered) which was our
motivation for considering this line of investigation. Let X be a compact space of
finite rank and W = (z E X : rkx(x) > 1}. In [La2], Suzanne Larson asks if W is
always closed in X. The answer is no. Her counterexample, presented at ORD98 (a
conference on 1-groups held in Gainesville, FL in 1998) follows:

Example 2.4.2. Let U be a free ultrafilter on N. Let E = N U {a} where points
of N are isolated and neighborhoods of a are of the form U U {a}, where U E U.
Let Ej = E for j = 1,2 and define Y = (E1 II E2)/(al ~ a2). Let Yr = Y for each
r E R, and let X = (UJe Yr) U {oo} where neighborhoods of oo contain all but
countably many copies of the Yr. Then oo is a P-point which is in the closure of the
set W= {x E X: rkx(x) > 1}. D

We provide a characterization of points of finite rank of a compact space X
via Z#-irreducible maps onto X. Let B be an f-subring of A and let 0 denote the
natural surjection Max(A) -+ Max(B). The following is proved in [HaM, 2.5].

Lemma 2.4.3. Let A and B be commutative f-rings with identity and bounded
inversion. Let B be an f-subring of A. Then if B is rigid in A, we have that
O(OM) = n{O(N) n B : ON = OM}, and if ON, = OM (for j=1,2) with N| II N2,
then (O(N) n B) 1 (O(N2) n B).

Let X and Y be compact spaces and q : Y -+ X a Z#-irreducible map. Then
S: Max(C(Y)) -- Max(C(X)) is given by M i-+ {f E C(X): fob E M}.

Lemma 2.4.4. Let X and Y be compact and 0 : Y -+ X a Z#-irreducible map.
Let p E X then O(Mq) = M, if and only if O(q) = p. Therefore, we have that
Op = n{o, n C(X) : q E -1p}}.

Proof: Let q E O-'{p}. If f E O(Mq) then foq E Mq and therefore we have that
0 = fof(q) = f(p) and f E M,. Let g E Mp; then (go)(q) = g(p) = 0, and hence
g4 E O(M,). Thus O(Mq) = Mp. Conversely, assume O(q) = r 9 p. By complete
regularity, there exists f E C(X) such that f(p) = 0 and f(r) = 1. We have
f E Mp, but foq(q) = f(r) = 1 0 0 and hence f O0(M,). The final statement then
follows from the first and Lemma 2.4.3. *

Proposition 2.4.5. Let X and Y be compact spaces and f : Y -+ X Z#-irreducible.

1. Ifp E X such that rkx(p) = n, then t-1{p}) = n.

2. IfY is an F-space and -l({p}l = n then rkx(p) = n.

3. If Y has finite rank and -{-'{p} = n, then rkx(p) < oo. Explicitly, we have
that rkx(p) = E=i rky(qi) where 4-l{p} = {fqi}l.

4. If p E X is an F-point of X, then q E 4- {p} is an F-point of Y.

Proof: (1) Let -'1{p} = {qi}iEI for some index set I. For each i E I, choose
Qj E Min(C(Y)) such that 0, C_ Qi. Then by Lemma 2.4.4, it follows that we
have Op = ni~(O, n C(X)) C Qi n C(X) C Mp, for every i E I. By the bijection
described in Proposition 2.2.7, the set of minimal prime ideals contained in M, is
given by {Qi n C(X)}SEI and III = n.
(2) Let -l1{p) = (qi}l=. If Y is an F-space, then Oq, is prime for each i and we have
that Op = n!=x(0, n C(X)) C Oq, n C(X) C M for each i. Thus, by the bijection
described in Proposition 2.2.7, the subspace of minimal prime ideals contained in
Mp is {Oq, n C(X)}L1 and rkx(p) = n.
(3) Let 4-r{p} = {q}?=1 and let the subspace of minimal prime ideals in Mq be
given by {Qi, : 1 ideals contained in Mp is given by {Qi n C(X) : 1 < j < rky(qi)}~1 and hence we
see that rkx(p) = Eni1 rky(qi) < oo, as desired.
(4) Let p E X be an F-point and 4b-{p} = {q}. Then O, = O, n C(X) C Mp is a
minimal prime. There exists a unique Q E Min(C(Y)) such that Op = Q n C(X).
Since Oq n C(X) = Q n C(X) and Q is unique, Oq = Q E Min(C(Y)). U

If Y = QFX in Proposition 2.4.5, then the third statement is a partial
converse of [HLMW, 5.1] which states that if X is compact and has finite rank then
QFX has finite rank. The final statement is an extension of [HVW1, 3.12] in which
it is shown that the preimage of a P-point of X is a P-point of QFX. The second
statement says that if QFX is an F-space, then the points of X of rank one are
precisely the points with unique preimage under the covering map Ox. In this light,
one should ask when a quasi-F cover of a space is an F-space.
Recall that a space X is fraction dense if the classical ring of quotients of
C(X) is rigid in the maximal (Utumi) ring of quotients of C(X). In [HVW1, 2.16],

it is demonstrated that QFX is basically disconnected if and only if X is cozero-

complemented. In fact, the basically disconnected cover is the quasi-F cover in this

case, see [HaM, 2.6.2]. The fact that QFX is realized by the extremally disconnected
cover if and only if X is fraction dense is proved in [HaM, 2.4]. By [HudP, 6.2], QFX
is an F-space if and only if for any two disjoint cozero-sets C1, C2 C X, there exist

Z, Z2 E Z(X) such that Ci C Zi for i=1,2 and int(Z n Z2) = 0.
We now provide an example of a space X such that QFX is an F-space which

is not basically disconnected. Recall that a space is a-compact if it is a countable
union of compact spaces.

Lemma 2.4.6. Let X be a noncompact a-compact locally compact F-space which
is not basically disconnected. Let X1 = X2 = #X and define Y be the quotient
space of the topological sum of X1 and X2 where pairs of corresponding points of

Xj \ X (j=1,2) are collapsed to a single point. Then Y is not quasi-F and QFY is
an F-space which is not basically disconnected.

Proof: The disjoint union U = X II X is a dense cozero set of Y which is not
C*-embedded in Y. Thus Y is not quasi-F. The quasi-F cover of Y is X1 II X2,

which is an F-space but not basically disconnected. *

Example 2.4.7. Let X be the disjoint union of a countable number of copies of the
corona [IN \ N. Then X is a a-compact noncompact F-space which is not basically
disconnected. Construct Y as defined in Lemma 2.4.6. Then QFY is an F-space
which is not basically disconnected. O

2.5 Prime Character

The second character we consider counts the minimum number of minimal
prime subgroups that we must sum in order to obtain the lex kernel of an G-group.

Definition 2.5.1. Let S be a family of minimal prime subgroups of G. We call S
ample if VS = lex(G). The prime character of G, denoted Ir(G), is the least cardinal
so that any family of distinct minimal prime subgroups of that cardinality is ample.
Note that iflex(G) = 0, i.e., if G is a totally-ordered group, then we say 7r(G) = 1.
A prime subgroup properly contained in lex(G) is called embedded.

Proposition 2.5.2. Let G be an i-group and m a positive integer. The following
are equivalent:

1. r(G) = m < oo.

2. m = 1 + sup{rk(P) : P E Spec+(G) is embedded}.

3. m is minimal with respect to the property that lex(G) = G(UjI aj-) for any
m pairwise disjoint positive elements, {a}T, C_ lex(G)+.

4. m is minimal such that for any embedded prime P, the chains of proper polars
in P have length at most m 1.

Proof: (1) =* (2) : Let P be embedded. If P contains the m minimal prime
subgroups {Qj}(g C Min+(G) then V Qij C P C lex(G). Hence r(G) > m. Thus
we have shown that 7r(G) < m implies rk(P) < m. Thus by (1), rk(P) < mn 1 and
sup{rk(P) : P embedded} < m 1. If sup{rk(P) : P embedded} < m 1, then for
any family S of m 1 minimal prime subgroups of G, VS is not embedded since
rk(VS) > m-1. Thus VS = lex(G) and 7r(G) 5 m- 1. Thus 7r(G) = m implies that
sup{rk(P) :P embedded} > m 1. Therefore sup{rk(P) : P embedded} = m + 1.
(2) =* (3): Let {ai,}im C lex(G)+ be pairwise disjoint. Let g E lex(G) \ G(UJ afj)
and let V be a value of g such that G(Ujt= aJ) C V C lex(G). Then by the polar
characterization of O(V), [BKW, 3.4.12], we have that aj 0 O(V) for each j such
that 1 < j < m. By (2), V contains at most m 1 minimal prime subgroups,

{Qi,}Y1. Since O(V) = (_1'- Qi, we have by the pigeonhole principle that there

must be a minimal prime subgroup, Q, contained in V which fails to contain two of
the elements of {aj}=1. But since these elements are pairwise disjoint, this contra-
dicts the fact that Q is prime. Therefore, lex(G) = G(U,_J a%).
For the minimality of m, let P be an embedded prime of rank n < m.
Then by the Finite Basis Theorem, P/O(P) contains n pairwise disjoint elements
of corresponding to elements of P which are not in O(P), say {bk}i=1 C P. Then
bC- C P for each k such that 1 < k < n and G(U = b6~) C P C lex(G).
(3) =: (1) : Assume that i(G) > m. Then there exists S = (Qj}im= C Min+(G)
which contains m elements and is not ample. Let Q = VS C lex(G). For each
j, let q, E Qf \ Qj. Then qi = Vjlqiy E Q \ (Uo# Qk). Disjointify by defining
qi- = Aiojqj Alfqk V Qi. Then W E Qi for every j 9 i and then we obtain that
G(U -i, i') c VS C lex(G). Hence, Ir(G) < m. By the minimality of m in (3) and
by (1) = (3), we must have 7(G) = m.
(3) 4 (4) : Follows directly from the Finite Basis Theorem [D, 46.12] applied to
P/O(P) for any embedded prime P. *

The following is immediate:

Corollary 2.5.3. Let G be an l-group. The following are equivalent:

1. 7(G) < oo.

2. sup{rk(P): P embedded} < oo.

3. There exists m E N such that lex(G) = G(Uj'Li af) for all families of m
pairwise disjoint elements {aj)}T1.

We now consider some -group-theoretic properties of the prime character.
Note that for I-groups A and B, AEBB denotes the I-group Ax B with componentwise
operations and is called the cardinal sum.

Proposition 2.5.4. Let G be an -group.

1. For any e-homomorphic image, H, we have r(H) < Ir(G).

2. For any C E (G), ir(C) < r(G). If r(G) < oo, then r(C) = r(G) if and only
if C contains every minimal prime subgroup of G.

3. If r(A),7r(B) < oo00 and G = A B, then r(G) < r(A) + r(B) 1.

Proof: (1) Let Vp : G -- H be an e-surjection with kernel K. Then by [D, 9.11],
the prime subgroups of H correspond to prime subgroups of G containing K. Thus,
by the characterization of prime character in terms of ranks of prime subgroups, we
have that ir(H) < 7(G).
(2) This result follows from [BKW, 2.4.7] and [D, 27.8].
(3) Let G = AffB and let P be a prime subgroup of G. Then P = (PnA)OE(PnB),

by [D, 27.8]. Hence P contains at most (m 1) + (n 1) = m + n 2 minimal
prime subgroups and therefore r (G) < m + n 1, as desired. m

Recall from [D, 36.1] that a class C of -groups is a radical class if G E C
implies the following:

1. ((G) C C,

2. every 1-isomorphic image of G is in C, and

3. if {AA}AA C ~ (G) n C, then VAEAA E C.

In view of Proposition 2.5.4, it is natural to ask if the class of all I-groups of finite
prime character is a radical class. The answer is no; the following is a counterexam-
ple.

Example 2.5.5. We construct an I-group with two convex I-subgroups, A and B,
each of finite prime character such that the supremum A V B has infinite prime

character.

FA ;------- -c------ ----- --3\------ IFBB

S(2.1)

Let F be the root system above, where the subgraphs FA and rF each have infinitely
many identical branches descending from its maximal vertex. Define G = E(F, R),
A = {v E G : supp(v) C rA} and B = {v G : supp(v) C Fa}. Then we have that
A G E(FA, R), B E(rB, R) and A V B E(FA U rB, R). Now, it is evident that
ir(G) = oo, ir(A) = 3 = 7r(B), and 7r(A V B) = oo. O

Recall that an t-group G has a finite basis if it contains a finite maximal set
of elements {bj}'=1 such that the set {g E G+ : g < bj} is totally ordered for each
j. The following indicates when we can expect 7r(A V B) < oo:

Proposition 2.5.6. Let G be an f-group and let A, B be convex e-subgroups such
that ir(A) = m < 00 and 7r(B) = n < 00. If lex(A V B) = lex(A) V lex(B), then
r(A V B) < oo. Otherwise, 7r(A V B) < o00 if and only if each of A and B has a
finite basis.

Proof: Assume that lex(AVB) = lex(A)Vlex(B) and let P C lex(AVB) be a prime
subgroup of G. Then either lex(A) Z P or lex(B) Z P, or both. Say lex(A) Z P.
Then P n lex(A) is an embedded prime subgroup of A, and since 7r(A) = m, we
have that rkA(P n lex(A)) < m 1. Then by [BKW, 2.4.7], rkAVB(P) < m 1.
Likewise, if lex(B) Z P then rkAvB(P) < n 1. Thus, for every prime subgroup
P C lex(A V B), we have that rkAvB(P) < max{m 1, n 1} < oo. Therefore, by
Proposition 2.5.2, 7r(A V B) < 00.

Note that we always have that lex(A) V lex(B) C lex(A V B). We assume
now that lex(A V B) : lex(A) V lex(B) and let P E Spec+(A V B) have the property
that lex(A) V lex(B) C P C lex(A V B). If each of A and B has a finite basis, then

rkAVB(P) = IMin+(A)I + IMin+(B)I < oo and for any embedded prime subgroup Q
of AVB, we have that rkAvB(Q) <: IMin+(A) + Min+(B) Thus, by Corollary 2.5.3,
we have that ?r(AVB) < oo. Conversely, 7r(AVB) < oo implies that rkAVB(P) < 00.
Hence, since IMin+(A)I, IMin+(B)| < rkAvB(P), we have that each of A and B has
a finite basis by the Finite Basis Theorem. U

The proof of the following is evident:

Proposition 2.5.7. Let G be an -group and let A, B E Q(G). If A C B = lex(B)
or if lex(A) = A and lex(B) = B, then lex(A V B) = lex(A) V lex(B).

We now compare the property of finite prime character to Conrad's Property
F: every element g E G+ exceeds at most a finite number of disjoint elements. The
following is compiled in Conrad's Tulane Notes, [C]:

Proposition 2.5.8. Let G be an t-group. The following are equivalent:

1. G has Property F.

2. Every bounded disjoint set in G is finite.

3. For every element g E G, the convex i-subgroup G(g) has a finite basis.

4. Every element of G is contained in all but a finite number of minimal prime
subgroups.

Corollary 2.5.9. Let F be any root system in which each maximal element lies
above a finite number of minimal elements. Then E(F, R) has Property F.

Proof: Let v E (F, R). Let C be a maximal chain in F and let the associated
minimal prime subgroup be He = {v E E(r,R) : 7 E C =* v(') = 0}. If v c He,

then there exists 7 E C such that v(7) / 0. Since supp(v) is fnite, v is in all
but finitely many minimal prime subgroups of E(r, R). Thus by Proposition 2.5.8,
E(P, R) has Property F. *

Proposition 2.5.10. If G is a finite-valued I-group of finite prime character, then
lex(G) has Property F.

Proof: Let g E lex(G). Then any minimal prime subgroup of lex(G) not containing
g is contained in a value of g. Since each value of g contained in lex(G) contains a
finite number of minimal prime subgroups and there are only finitely many values
of g, there are only finitely many minimal prime subgroups of lex(G) not containing
g. Thus lex(G) has Property F. a

Proposition 2.5.11. Let G be an i-group and let m be a positive integer. If m
is minimal such that every pairwise disjoint subset of G contains at most m 1
elements, then 7r(G) = m.

Proof: Any proper prime subgroup of G is contained in a value of G, hence proper
prime subgroups of G contain at most m 1 minimal prime subgroups. Thus
7r(G) < m. Let n < m and assume that every family of n minimal prime subgroups
of G is ample. Then every proper prime subgroup (in particular, every value) of G
contains at most n 1 minimal prime subgroups. This contradicts the minimality
of m. Thus 7r(G) = m. m

Example 2.5.12. The following is an example of an I-group of finite prime char-
acter such that lex(G) has Property F but G has pairwise disjoint sets of any size
m. Thus the converse of Proposition 2.5.11 does not hold.
Let F be the root system:

AAA (2.2)

and let G = E(r, R). Then ir(G) = 3 and G is finite-valued. Thus, by Proposi-
tion 2.5.10, lex(G) has Property F. We demonstrate that there are bounded disjoint
families of any given size.
Index the maximal elements of F by pj, where j is a positive integer, and let
vj E G be such that vi(pj) = 1 and supp(vi) C {1y E F : 7 <- pj} where vi(6) = 0 for
all 6 < pj. Then {vj}j>l is an infinite pairwise disjoint family in G. Let a positive
integer m be given. Choose any m elements from this set, {vji, 2j,,..., vi,} and let
S = Ul supp(vj,). Let v be the characteristic function on the finite set S, then
v E (F,R) and v > vj, for k = 1,2,...m. O

Example 2.5.13. The converse of Proposition 2.5.10 does not hold. That is, we
present an example of a finite-valued -group with Property F and infinite prime
character. Consider the following root system F which is indexed by the positive
integers:

A A (2.3)

Then each prime subgroup of the f-group G = E(F, R) contains a finite number of
minimal prime subgroups, yet there is no bound on the number of minimal prime
subgroups in each prime subgroup. Thus 7r(G) = oo. G has Property F by Corollary
2.5.9. 0

2.6 Filet Character

The third character that we define measures the length of a chain of incidents
of branching.

Definition 2.6.1. Let G be an f-group. C = {P,Q,j E Spec+(G) : i > 0,j > 1} is
called a filet chain of prime subgroups if Po D P1 D Pa..., for all i, Pi |1 Qi, and

Pi+F V Qi+l C Pi for all i > 0 (see below).

P,
Q

Q (2.4)

The length of a filet chain is given by l(C) = max{j E N : 3Qj E C}. If the
maximum does not exist, we write 1(C) = oo. The filet character O(G) is given by
O(G) = sup{l(C) : C is a filet chain}. If Spec+(G) has no filet chains, i.e., if G is
semiprojectable, then we say that O(G) = 0.

Proposition 2.6.2. Let G be an e-group. Then O(G) < 1 if and only if 7(G) < 2.

Proof: Suppose that 7r(G) < 2. If O(G) > 1 then there exists a filet chain C of
length 2 in which we may assume that Po = lex(G). Since rk(Pi) = 2, we have
that 7r(G) > 2, by Proposition 2.5.2. Conversely, assume that 7(G) > 2. Then
there exist minimal prime subgroups P2, Q2 such that P1 = Pa V Q2 9 lex(G). Thus
for any minimal prime subgroup Qi Z P1, the set C = {Po, P,P2,Q1, Q2}, where
Po = lex(G), is a filet chain of prime subgroups in G of length 2. Therefore O(G) > 1.

For larger filet character, the relationship between it and the prime charac-
teris more complicated, as the following example illustrates.

Example 2.6.3. Let I be the following root system:

(2.5)

Then G = E(r, R) has =(G) = 2 while 7r(G) = 4. O

The following relationships between the characters hold:

Proposition 2.6.4. Let G be an I-group. Then

1. Ob(G) < rk(G) 1.

I. O(G) < 7r(G) 1 < rk(G).

Proof: Let rk(G) = m. If G has a filet chain C = {Pi, Qj E Spec+(G) : i > 0,j >
1}, then rk(Po) < m and hence 1(C) < m 1. Therefore, f(G) < m 1 = rk(G) 1.
If 7r(G) = n and C = {Pi, Qj Spec+(G) : i > 0,j > 1} is a filet chain
in G then 1(C) < oo since rk(Pi) 5 n 1 by Proposition 2.5.2. Hence, in fact,
1(C) < n 1 and therefore, O(G) < n 1 = 7r(G) 1. Now, 7r(G) < rk(G) + 1 by
Proposition 2.5.2. Thus, finally, 4(G) _< r(G) 1 < rk(G). n

At this time, any statement that we make about the filet character of an
-group requires a restriction on the rank and prime character. Rather, we can
not say much more than what we establish earlier in this chapter. We leave this
investigation for a later date.

CHAPTER 3
GENERALIZED SEMIGROUP RINGS

Let A be a root system. Starting from the standard semigroup ring and Hahn group
constructions and the investigation of the following section, we build an f-ring R
having A as the skeleton of the graph of Specq(R). We follow up on ideas presented
in two papers of Conrad [C1],[C2], in the paper of Conrad and Dauns [CD], and
in the paper of Conrad and McCarthy [CMc]. The first two papers look at the
conditions on A which will yield a ring structure on V(A, R) and on the subgroup
F(A, R) of elements v for which supp(v) is the join of finitely many inversely well-
ordered sets in A. The paper [CD] focuses on the case when V(A, R) is a division
ring, while in [CMc] the conditions are established for the ring to be an I-ring its
properties are studied when A is finite. Note that F(A, R) is denoted by W(A, R)
in [C1],[C2], and [CD].

3.1 Specially Multiplicative f-Rings

Let r be a partially ordered group which is a root system. Suppose also that
F is torsion free, that the subgroup H of F generated by the positive cone is totally
ordered and that F/H is finite. In the paper of Conrad and Dauns [CD, 2.2], it is
shown that V(F, R) = F(P, R) and that V(T, R) is a lattice-ordered division ring
under the usual group-ring multiplication: for u, v E V(r, R), and y E F

u* (7)= E u(a)v(3).
a+0=7
It is easy to see that if an element of (V(F,R),+, ) is special (i.e., has only one
maximal component), then its multiplicative inverse is also special. Hence, the

special elements of the ring V(I, R) form a multiplicative group. Moreover, the

following holds in general, [CD, Theorem I]:

Theorem 3.1.1. Let R be a lattice-ordered division ring with identity. The follow-
ing are equivalent:

1. The special elements of R form a multiplicative group or the empty set.

2. If a E R+ is special, then a-1 > 0.

3. If a E R is special, then a-1 is special.

4. For all a E R+ special and x, y E R, a(x V y) = ax V ay.

Since the authors of the paper [CD] seek an embedding theorem for f-fields,
their investigation in this realm is restricted to division rings. In this section, we
consider a class of f-rings (which are not division rings) in which the special values
form a partially ordered semigroup. The pursuit of a characterization similar to
Theorem 3.1.1 of I-rings satisfying this condition is left for another time. For the f-
rings that concern us here, some particular assumptions are needed on the associated
semigroups. As a formality, we define:

Definition 3.1.2. Call an f-ring A specially multiplicative if the special values of

A, with an appropriately adjoined 0, form a partially ordered semigroup.

Recall that an f-ring A is called an SV-ring ifA/P is a valuation domain (i.e.,
the set of principal ideals is totally ordered) for every prime ideal P. Let 9 be the

class of commutative f-rings which are local, bounded (that is, A* = A), semiprime,
finite-valued, finite rank and square root closed SV-rings with identity and bounded
inversion. We demonstrate that the elements of G are specially multiplicative and
then investigate the properties of the associated semigroups. We must first remind
the reader of a couple of facts about special values and of the relatively deep theorem,
recorded as [HLMW, 2.14]:

Theorem 3.1.3. Let A be an f-ring of finite rank with identity and bounded inver-

sion which is local, semiprime and square root closed. Then A is an SV-ring if and
only if whenever 0 < a < b and b is special, there exists x E A such that a = xb.

The proof of the following lemma is well-known and routine, however, it is
instructive, so we include it here. We remind the reader that we denote the root
system of values of an t-group G by F(G) = {V, : y E F}, where F is a partially-

ordered index set that is order-isomorphic to F(G).

Lemma 3.1.4. Let G be an t-group and let a, b E G+ be distinct special elements
with values at Va, Vp, respectively.

1. Va i VO if and only if a A b = 0.

2. Va < Vf if and only if a < b.

Proof: First, assume that Va II Vp. We show that a A b is contained in each value

of G and conclude that a and b are disjoint. Let V be a value in G and assume
that b V V. Then V C V# and a E V since, else, V C Va, which contradicts the

assumption of incomparability. Thus, by convexity we obtain that a A b E V. Again
by convexity, if b E V then a A b E V. Thus a A b = 0, as desired. Conversely,
assume that a and b are disjoint and, by way of contradiction, assume (without loss
of generality) that Va < Vs. Since a V Va and Va is a prime subgroup, b E Va < V,

Second, assume that Va < Vs. Then a E Vs. If there exists an integer n such
that na > b, then since b V Vs, we must have that na V Vs, by convexity. But this
contradicts that a E Vs, and so we conclude that a < b. Conversely, assume that
a < b. We know that b V Va and hence Va < Vp. If Va = Vs, then by [D, 12.6], there
exists an integer n such that na > b, which is nonsense. Thus Va < VS. m

Combining Theorem 3.1.3 and Lemma 3.1.4 leads to the main result of this

section:

Theorem 3.1.5. Let A E g.

1. If a, b E A+ are special and not disjoint, then ab is special.

2. If a, a' are special with value Va, b, are special with value V, and Va, V, are

comparable, then the special value of ab is the same as that of a'V.

Proof: Let a, b E A+ be special with values at Va, Vp, respectively, and let M
be the maximal ideal in A. Since A is bounded, we may assume without loss of

generality that Va, Vp < M. This gives rise to the relations a, 6 < 1 and ab < a, by

Lemma 3.1.4(2).
Theorem 10.15 of [AF] states that an abelian I-group is finite-valued if and

only if each positive element is a finite disjoint sum of positive special elements.

Thus we assume, by way of contradiction, that ab = cl + c2 + + c, where each

ci E A+ is special and ci A c, = 0 for all i # j. Then 0 < c < ab < a for each i
and therefore Theorem 3.1.3 gives the existence of xi E A+ such that ci = arx for
each i = 1,..., k. Without loss of generality, we may assume that 0 < zi < b, by

replacement with xz A b. Assume that az = Eii xi is a decomposition of xi into

a sum of special elements. Then ci = Ej axij implies that axi, = 0 for all but

one j, by [AF, 10.15]. Thus we may choose each az to be special, without loss of
generality.

For each i, let V&i be the value of zi. Then b V Vi, and therefore for all i, we
have that V6Y < V#. If V,% < VO, then xz < b. This gives a contradiction, since it
implies that axz < ab. On the other hand, if V, = Vp, then there exist n, m E N
such that xi < nb and b < mai. Therefore, axi < nab and ab < maxi. Thus, by [D,
12.6] we have that the value of ci is the unique value of ab and ab = ar,. Hence, ab

is special.

For the second statement, let Vh be the value of the special element ab. We
show that V5 is the value of al and then a similar argument and transitivity gives
that Vs is also the value of a'b, as desired. Towards this end, let V, be the value of
al.
If V, 1i Vs, then 0 = ab A ab = a(b A b) and hence 0 = a A (b A b). On the
other hand, if V, < V#, then since b A Y V Va, we have that a E Va. If Vp < Va,
then since a V Vp, we have b A and hence b or V is in Vp. These contradictions
lead us to conclude that Vs < VT or V, < Vs. If Vj < V, then ab << al. However,

by [D, 12.6], there exists an integer n such that nb > V and therefore, nab > nab.
Likewise, Vy Vs. Thus V, = Vs, as desired. *

Let A E G. Then by [AF, 10.10], all the elements of r(A) are special since
A is finite-valued. Abusing notation, we now identify values with their indices and
define an operation on r. Let a, b E A be special elements with corresponding values
at a, #, respectively. Append 0, a generic symbol, such that II1 a for all a E r and
define a multiplication on r U {4} such that a = = = a *- and

a = if a I3,
the value of ab otherwise.

By Theorem 3.1.5, this operation is well-defined. Some properties of this operation
are recorded in the following proposition. Define a ~ P if and only if a and # are
contained in the same maximal chains of F. Then ~ is an equivalence relation on r.
Let rf denote the --equivalence class of a.

Proposition 3.1.6. Let A E G and let a, #, 6 E r correspond to the values of the
special elements a, b, c, d E A+, respectively. Then

1. The operation is associative, for every a, #,7 : (a ) = a ( 7y)

2. If M is the maximal ideal of A, then M is a value of the identity. Let p = M.
Then pt a a = a p= a for every a < p.

3. If a #0 4 then a / < a, P.

4. If a < P and 7 is comparable with a, then a 7 < 3 7 and 7 a < 7 3.

5. If a \$p, then a a < a.

6. Maximal chains in r are closed under the operation.

7. For each a E F, the equivalence class rF is closed under the operation.

8. If < a, then there exists / E r such that 7 = a P.

Proof: (1) Note that the products under consideration are all 4 if any of the factors
multiplied are actually 4. If a j\ P then aAb = 0. Thus aA bc = 0 and hence a I /-7
which implies that (a /) 7 = 4 7 = a = a (/3 7). Likewise, if /- 7 = 4 then
both products are equal to 4.
Assume that a p 964 and /37 / 4). Then either (a /) 7 = 4, or it is the
value of the special element (ab)c = a(bc). Thus, it suffices to show that a -3 P 7 if
and only if aI / '7. If a I P3 -7 then aA bc = 0 and hence 0 = acAbc = (aA b)c and
so 0 = (a A b) A c. Therefore, for any value ir of a A b, we have 77 11 7. But a -3 < rl,
so a p | 7. The converse follows similarly.
(2) Since A is bounded and has bounded inversion, the maximal ideal is a value
of the identity, by [Wo, 2.3.4]. Hence, it follows that p a = a p is the value of
la = al = a, namely a.
(3) Assume, without loss of generality, that / < a < p. Since ab << b, we obtain
that a-/3 < 3 (4) If a < p then a < b and thus ac < be and ca < cb. By Lemma 3.1.4(2), this
just says that a. 7 < p/ 7 and 7 a < 7 /, as desired.
(5) Since a2 < a, we know that a a < a.
(6) Assume, without loss of generality, that / < a. Let 6 < P be such that 6 I a /.
Then abAd = 0 and 0 < d < b. Thus by Theorem 3.1.3, there exists y E A such that

d = yb. Then 0 = ab A yb = (aA y)b gives that 0 = aA yAb and hence 0 = ayb = ad
and therefore aAd = 0. This says that a |[ 6, a contradiction. We now have that a -
is comparable with any 6 < P. It follows from this fact and from (3) that maximal
chains are closed under the operation.
(7) Let 7, 6 E r,. By (6), we have that 7 6 is in the same maximal chains as 7 and

6. Hence 7 6 E Fa.
(8) We know that 7 < a if and only if Vy < Va if and only if 0 < c < a. This implies
that there exists a positive element b such that c = ab, by Theorem 3.1.3. We have,
by [AF, 10.15], that b is special. Thus, if Vp is the value of b, then 7 = a P, as
desired. *

Corollary 3.1.7. Let A E .. Then A is specially multiplicative. Moreover, the
operation is a surjection onto F(A) U {\$}.

3.2 r-Systems and L-Systems

Let A be a given root system. The discussion of the previous section and
of the papers [Cl], [C2], [CD], and [CMc] prepare us to construct an f-ring having
A order-isomorphic to its root system of values. However, under the assumptions

placed on A in these papers, these rings sometimes are e-rings, and rarely are f-rings;
in fact, as mentioned before, the paper [CD] focuses on the case when V(A, R) is a
division ring. Also, contrary to our intentions, [CMc] concentrates on the properties
of the ring when the root system is finite. We modify all these conditions on the
root system in order to get a ring multiplication, *, yielding an f-ring structure
on R = (F(A, R),+, *) such that the subring R = (E(A, R), +,*), as defined in
Chapter 2, is an f-subring. We start with [C1,Theorem I] which establishes the ring
structure.

Proposition 3.2.1. Assume that A is endowed with a subjective partial binary op-
eration + : A -- A defined on A C A x A. Let u,v E V(A, R), and define for

6 e A, u v(6) = Ia+-- u(a)v(/). Then both R and R are closed under *. In fact,
R and R are rings if and only if the operation on A is (Baer-Conrad) associative:

(a, 3), (a +3, 7) E A if and only if (3, 7), (a, 3 +7) E A, and if (a, /), (a+/, 7) hA
then (a + #) + 7 = a + (# + 7).

Certain properties of these rings, R and R, are completely determined by the
operation + on A; commutativity is one such attribute.

Definition 3.2.2. Let (A, +) be a root system with a partial binary operation, +.
We call (A, +) an r-system if the operation is surjective and (Baer-Conrad) asso-
ciative. An r-system (A, +) is commutative if (a, #) E A if and only if (/, a) E A
and a +/ = P + a for all such pairs.

Proposition 3.2.3. Let (A, +) be an r-system. (A, +) is commutative if and only
if 7 is a commutative ring.

Proof: If (A, +) is commutative, then for u, v E R we have

u*v(6)= u(a)v() = E v(#)u(a)= v(8)u(a)= v *u(6).
a+0=6 a+-=6 5+a=-
Thus 1Z is a commutative ring.
Conversely, if a + /3 /3 + a, then Xa+# = Xa X,= + X8 Xa = X#c+a.

In order to obtain an I-ring, we must ask that the operation on A preserve
the order of the root system and restrict the domain of the operation. As we will
indicate, the various strengths of order preservation and restrictions of domain yield
varying richness of order structure on the rings.

Definition 3.2.4. Let A be a root system. An r-system (A, +) is an i-system
if it satisfies: a < / and (a, 7) E A implies that (#,7) E A and, in this case,
a+,7 < /#+7; and if (7, a) E A implies that (7, #) E A and, in this case, 7+a < 7+/3.
If every connected component of A has a maximal element, we call A bounded above.

Note that the I-system condition gives that no nonmaximal element of such

a bounded above root system is idempotent. The following theorem is in [CMc, 2].

Theorem 3.2.5. Let (A, +) be an r-system. Then 7' is an I-ring if and only if A

is an I-system.

In the next section, we describe the conditions on an I-system which induce

an f-ring structure on %. As mentioned, there are some restrictions that we need

to place on the domain and range of the operation. Before we discuss this situation,

we present an example of an I-system that gives rise to an I-ring which is not an

f-ring. Probably, this is the simplest example of such an I-system.

Example 3.2.6. Let A > a > 2a > 3a... and p > 8 > 2/ > 3# .... Then

we let Aa = {A} U {na}O1, let A# = {p} U {n/p}01 and totally-order A4, x Ap

lexicographically such that (61, 71) < (62, 72) if and only if S6 < 62 or we have 61 = 62

and 71 < 72. Identify pairs with sums and let Aa+, = {6+7 : (6, 7) E Aa x A#}. We

define an associative and commutative addition on the root system given by disjoint

union A = An II A, II Aa+ as shown in the following table, where v = A + p and

k, 1, m, n are positive integers (note that the table is completed by reflection across

the diagonal):
+ a A p v A+ms ka+p noa+I,
a 2a a+6 a a+-- a+p a+m (k + )a+p (n+l)a +1
0 2- A+ 0 A+ A+(m+ 1) k-+# na+(TI+1)
A ___ A v v A+m ka+p na+I1
p _V + m ka + + nao +
v V' A+mO ka+ p noa+I
A +mS A+ 2m# ka +m na + (m -+)
ka+ p 2ka+p + (k + na + I
na + 10 2na + 21P
Let u, v E R+. We need to show that u v E R+ in order to conclude that R

is an I-ring. Since u and v are positive, it is evident that u*v(A), u*v(p), u*v(v) > 0.

If 0 = u v(A) = u(A)v(A), then assume that u(A) = 0. For an integer n,

u v(na) = u(A)v(na) + u(na)v(A) + u(ia)v(ja).
i+j--n
ij0

If u(na) = 0 for all n, then u v(na) = 0. Conversely, if n 0 0 is minimal such that
na E supp(u), then u v(na) = u(na)v(A) > 0. Thus, if v(A) > 0 then na is the
maximal element of supp(u v) below A.
If v(A) = 0 and v(ma) = 0 for all m, then u v vanishes on A,. Otherwise,
let m be minimal such that ma E supp(v). Let k = n + m then ka is maximal in
supp(u v) and u v(ka) = u(na)v(ma) > 0. A similar computation demonstrates
that u v > 0 at the maximal element of its support in A# also.
Assume that 0 = u v(v) and, when they exist let
Au = max(supp(u) n A.) Av = maz(supp(v) n A,)

= max(supp(u) nL A) p = maz(supp(v) n A#)

6u = max(supp(u) n Aa+#) 6v = maz(supp(v) n A,+O) (3.1)
Then the maximal element of supp(u v) lying below v is given by

7 = TmaX{A, + p ~,A+ 64, A, + Alu, \a + 64, pu + 4, p, + 6 + 6t}

and so we are left to show that u v(7) > 0. We have three cases to check here,
namely 7 = A + m/, ka + p or noa + I. Upon consideration of the formulae below, it
quickly becomes dear that the only nonzero summands of u v(7) are those of the
form u(a)v(r), where a and r are one of the six maximal support elements listed
above in (3.1), which implies that the summands are all positive. For the sake of a
certain degree of completeness of exposition, we list the possible summands in the
case that 7 = A + mp. The analysis in the other cases is similar.
u v(A + mp) = u(A)v(m#) + u(mn)v(A) + u(A)v(A + mp) + u(A + m/)v(A)

+ u(/A)v(A + mp) + u(A + m8p)v(() + u(v)v(A + mr) + u(A + mr)v(v)

+ E u(A+ip)v( ) + E u(i/)v(A+jp) + E u(A+i i)v(A+j j)
i+j=m i+j-m i+j=m
i-jo ijso ijsA

To compute u v(A + mp), consider all the possible combinations of the

1. If u(A) # 0 then if v(mn) # 0, we have that m6 < p,. If mr < ,, then
7 = A + m/ < Au + i which is nonsense. Thus m/ = pu, in this case and
u(A,)v()p,) is a summand of u v(7).

2. By an argument similar to the above, if v(A) # 0 and u(mp) # 0, then
mfn = p and u(p,)v(A,) is a summand.

3. If u(A) # 0, u(/) / 0 or u(v) # 0 then if v(A + m/7) # 0, we may conclude
that 6, = A + mp and u(Au)v(6J,), u(pu)v(,,) or, respectively, u(6,)v(6,) is a
summand.

4. Similarly, if v(A) # 0, v(/i) # 0, or v(v) # 0 then 6, = A +mpf and u(6,)v(A,),
u(6u)v(p,), or u(6,)v(6,) is a summand.

5. If, for some i + j = m, we have that u(A + iP)v(jP) # 0, then we also have
6, = A+iPif,, p = jp and u(6,)v()(t,) is the only nonzero contribution from this
large sum.

6. Likewise, if u(iP)v(A+j#/) # 0 for some i+j = m, then u(p,)v(6,) is the only
nonzero component of this summation.

7. Finally, if u(A+i)p)v(A+j/) # 0 then u(6u)v(6,) is the only nonzero summand
coming from this summation.

As stated before, the analysis in the other two cases is similar. We provide
the formulae below:
u v(ka + p) = u(ka)v(p) + u(p)v(ka) + u(A)v(ka + p) + u(ka + p )v(A)

+ u(s)v(ka + p) + u(ka + p)v(p) + u(i)v(ka + p) + u(ka + p)v(v)

+ E u(ia + p)v(ja) + E u(ia)v(ja + I) + u(ia + J)v(ja + p)
i+j=k i+j=k i+j=0
is3o idj# ij#o

u v(na + I1) = u(na)v(l1) + u(l/3)v(na) + u(A)v(na + I1) + u(na + 1ft)v(A)

+ u(pi)v(na + 10/) + u(na + l13)v(p) + u(v)v(na + 1/) + u(na + lIf)v(v)

+ u(na)v(A +1 0) + u(A + l/)v(na) + u(na + it)v(l1) + u(l/)v(na + ip)

+ u(ia + lf)v(ja) + E u(ia)v(jU + l) + u(na + iP)v(j#)
i+j=n i+j=n i+j=l
ij#O ijAo ijAo
+ u(ip/)v(na +j) + C u(ia + i)v(ja + 1) + E u(ia + 1)v(ja + p)
i+j=l i+j=n i+j=n
ijZo iJ9o idjo
+ u (A + ip)v(na + j#) + u(na + i/)v(A + ji)
i+j=L i+j=-
ij#o ij4o
+ E u(ia + kpl)v(ja + mni)
i+j=n
k+m=I
ijk,mv0

R is not an f-ring: Let HI+ = {v E R : 6 E Aa+6 =, v(6) = 0} be the
minimal prime subgroup. Then X,, X# E HA+,, yet Xa *X# = Xa+6 V HA.+ Thus,

HA+o is not an ideal and R is not an f-ring by [BKW, 9.1.2]. ]

The following two propositions record some consequences of the I-system
condition. Note that none of the excluded conditions occur in Example 3.2.6.

Proposition 3.2.7. Let (A, +) be a bounded above -system in which maximal ele-

ments act as an additive identity on elements below it. Then the following can not
occur for nonmaximal elements a, 3 E A,

1. 8 + a < a, where P, a are below the same maximal element, and / II P + a

2. 3
3. 53<0+a

4. If A, pI A are maximal and (A, M) e A, then A + p A1 it.

In particular, the second and third properties imply that if a, P, 3+a are comparable,
then 3+ a < a, 3.

Proof: If P + a < a,/ || + a, and p > a, 3 is maximal, then X,, Xu Xa are
positive. Yet, X# (X, Xa) = X6 X#+a is negative, contradicting that R is an
I-ring.
Assume that / < a < 3 + a. Let i > p + a be maximal. Then as above,

X# (Xi, Xa) < 0. The same contradiction is obtained in the case that we assume
13+ 3+a If A, p are maximal and A + p < p then Xx (X, 2X\+,) = -Xa+p < 0. U

Proposition 3.2.8. Let (A, +) be an -system.

1. There is no nonmaximal 6 E A such that 6 is idempotent and 6 + a = 6 for
all nonmaximal a > 6.

2. If 6 A is nonmaximal and idempotent then there does not exist a maximal
element p > 6.

Proof: If 6 E A such that 6 is idempotent and 6+ a = 6 for all nonmaximal a > 6,
Then we contradict the assumption that R is an -ring since Xj (Xa 2X6) = -X6.
If p > 6 is maximal and 6 is idempotent, then X ( (X, 2Xs) = -XS. U

3.3 f-Systems

We are much more interested in the f-ring situation. In [CMc, 2.1], the
authors demonstrate a condition on A which will give rise to an f-ring.

Theorem 3.3.1. Let (A, +) be an t-system. Then R is an f-ring if and only if the
root system also satisfies: if a | p and (a,'y) E A, then a+7 II 3; and if(7,a) E A,
then 7+a II p.

In this section we consider a condition on A which is equivalent to the one
stated above and proceed to investigate certain properties of the associated f-rings.
Recall that r(F(A,R)) is order-isomorphic to A by Proposition 2.1.1, where the
values are of the following form, for 6 E A :

V = {u E R: > 6 = u(y7) = 0}.

Definition 3.3.2. Let (A, +) be an e-system such that 6 = a + # implies a, f > 6.
Then we say that (A, +) is an f0-system.

Proposition 3.3.3. If (A, +) is an fo-system, then % is an I-ring and for every
6 E A, the subgroup V6 is an ideal. Hence, in particular, R is an f-ring and R is
an f-subring of R.

Proof: Assume that (A, +) is an f0-system. Let 6 E A, v E Vs, u E R, and 7 6.
Let 7 = a+#f, then by the f0-system assumption, # 7 6 and hence v(0) = 0, for
all such / since v E Vs. Then u* v(7) = E +=,y u(a)v(P) = 0 and we conclude that
u*v E VF. Therefore, the values of R are ideals. Moreover, since each minimal prime
subgroup is an intersection of a chain of values by [D,10.8], each is an intersection
of a chain of ideals; thus each minimal prime subgroup itself is an ideal. Finally, we
have shown that R and R are f-rings by [BKW, 9.1.2]. m

Proposition 3.3.4. An -system (A, +) is an f0-system if and only if % is an
f-ring.

Proof: Sufficiency is shown in Proposition 3.3.3. Conversely, assume that there
exist a, / E A such that a + /3 a or a + / /t Then by Proposition 3.2.7, we
have that a, /, a + / are not all comparable. If a I /, then we have that a |I a + /
by Theorem 3.3.1. Assume that / |j a + P and let C be a maximal chain in A
containing a + /. Let He = {v E % : 6 E C = v(6) = 0} be the associated minimal

prime subgroup. Then XB E He, yet Xa X, = Xa+ Hc. Thus He is not an ideal
and hence 17 is not an f-ring by [BKW,9.1.2]. *

Definition 3.3.5. An f0-system satisfying the following is called an f-system:

1. If a and # are comparable, then (a, /), (0, a) E A.

2. If p is maximal, then (6, p), (p, 6), (/s, If) E A and 6 + p = p + 6 for every

6 < p. In particular, p. + = p.

We will shortly see that these additional assumptions on a bounded above fo-
system make maximal chains in A into monoids. This is quite useful in our setting.
For instance, it is not difficult to figure out when the f-rings have a multiplicative
identity, under the f-system assumption.

Proposition 3.3.6. Let (A, +) be a bounded above f-system. R and 1Z each have
a two-sided multiplicative identity if and only if A has a finite number of connected
components.

Proof: Let (A, +) contain only a finite number of connected components with
maximal elements {i, #2, ..., p,}. Let Xj be the characteristic function on the set

{pj} and let e = E = X. Then for v E R~ and 6 E A,

v e(6) = E v(a)e() = v(6)e(k) = v(6),
a+O=6
where pk > 6 is maximal. Likewise,

e v(6) = e(a)v() = e(k)v(6) = v(6).
a+B=6
Thus e is a two-sided multiplicative identity.
Conversely, assume that e E 7I is an identity and let p E A be maximal with
characteristic function X,. Then 1 = X,(p) = e xm,() = e(p). Thus, supp(e) has

as many maximal elements as there are maximal elements in A. Since e E r?, we
must conclude that there are only finitely many connected components in A. U

It is handy to have the following definition:

Definition 3.3.7. Call an r-system (A, +) unital if the ring 7 has a multiplicative
identity.

The rings R and R are semiprime, as we will now show. Thus the minimal
prime subgroups are also prime ideals, by [BKW, 9.3.1].

Proposition 3.3.8. Let (A, +) be an f0-system. Then % is semiprime.

Proof: Let u E 7, where u = EjEi ajXj for an index set I of supp(u) and for
a, E R\0 for all j E I. Then u*u(6) = 6=,+6j aiaj. Let 6i be maximal in supp(u).
We show that 6i + 5i is maximal in supp(u u). Let 6 = 6j + 8k E supp(u u).
If 6j 1I Si, then by the f0-system condition, 6 |1 6 + 65. On the other hand, if

Jj, 6k < 6, then 6 < Si + i, as desired. Thus u* u(6i + 6) = ai > 0 and we conclude
that 7I is semiprime. *

Corollary 3.3.9. Let (A, +) be an f0-system. Then maximal chains in A are closed
under the operation +.

Proof: Let C C A be a maximal chain and let He = {v E : 6 E C =- v(6) = 0}
be the associated minimal prime subgroup. Let a, ~ E C and assume by way of
contradiction that a + p f C. Then X, X# = Xa+# E Hc but Xa, X# 0 He. Thus
He is not a prime ideal. This contradicts Proposition 3.3.8, by [BKW, 9.3.1]. m

One should ask if the maximal e-ideals of 7 are actually the maximal ideals;
or equivalently, one should ask if R has the bounded inversion property. The answer
is yes, if (A, +) is a unital f-system.

Proposition 3.3.10. Let (A, +) be a bounded above unital f-system. An element
u E R is a multiplicative unit if and only if u 0 V, for all maximal p E A. Thus 1
satisfies the bounded inversion property.

Proof: Let S C A be the set of maximal elements of A and let e = E-Es Xi be
the multiplicative identity. If u E R is a multiplicative unit, then there exists v E R
such that u v = e. Hence 1 = e(p) = u(L)v(/p) for every i E S. Thus u(p) 0 0 for
all 1 E S and hence u V, for all maximal p E A, as desired.
Assume that u V, for all maximal p E A. We define the multiplicative
inverse v of u recursively on each maximal chain in the support of u. Let v(6) = 0
for all 6 V supp(u) and let v(p) = 1/u(ip) for each maximal p E A. Then if 61 < / is
maximal in supp(u), we just solve the equation 0 = u*v(61) = u(61)v(pL)+u(p)v(61)
to get that v(61) = -u(61)/u(p)2. Proceed with the definition of v accordingly. That
is, let p > 6 E supp(u) and assume that v(7) is defined for all 7 > 6. Then u(a)v(r)
is a summand of u v(6) only if 6 < a, 7 E supp(u). Thus v(6) is the only unknown
in and is the unique solution of the equation 0 = u v(6).
Hence, since if e is the multiplicative identity of R and u > e then u 0 V,
for all maximal p, we have that u is a multiplicative unit. Therefore 7 satisfies the
bounded inversion property. *

Let (L, <) be a partially ordered set. Recall that A C L is called closed if

{ai}ier and Aiai or Viai exists in L then Aiai, Viai E A. It is the case that, if (A, +)
is an f-system, then all of the closed convex e-subgroups of 7 are ring ideals. We
will use the following special case of [D, 45.26].

Theorem 3.3.11. Let G be a finite-valued e-group. Then there is an order-preserving
correspondence between the closed convex e-subgroups of G and (order) ideals \$ of
the root system r(G) given by

K 1-4 (K = {G6 E F(G) : 3k E K such that Gs is a value of k}

4~ = {=g G : all values of g are in <}.

Proposition 3.3.12. Let (A, +) be a bounded above f-system. Any closed convex
-subgroup of R is an ideal of R.

Proof: Let K be a closed convex I-subgroup of R; let u e K+ and v E R. If u is
a unit, then 4)K = r(G) and by the preceding theorem, K = G. So assume that u
is a nonunit.
Let p E A be maximal and assume that u v(7) = 0 for all Jt > 7 > 6 and
u*v(6) # 0 for some 6. By the preceding theorem, we need to show that 6 E IK C A.
Let 6 = a + p such that u(a)v(3) # 0. Then there is a maximal a' > a such that
u(d) # 0. Then a' = a'+ a + > a+/ = 6 anda' E K Since u K.
Therefore, we conclude that 6 E by the preceding theorem. *

We now seek the prime -ideals and z-ideals among the prime subgroups Vs
and their associated value covers. The cover of V8 is the set

P = {v ER: v(7) = for all 7>6}

and is the smallest convex I-subgroup properly containing V, and Xs. First let us
recall the most general definition of z-ideal. Let G be a vector lattice, v e G+,
and let {g,}n=L C G be a sequence. Recall that the sequence converges relatively
uniformly to g E G along the regulator v and write gn -_-+ g, if for every e > 0
there exists N, > 0 such that for all n > N6, we have that Ig gI < ev. Let
H be a convex I-subgroup (sub-vector lattice) of G. The pseudo-closure of H is
H' = {g E G : 3{gn,},1 C H, gn --+ g for some v E H+}. Then H is relatively
uniformly closed if H = H'; let H denote the relative uniform closure of H. Then
if G(g) denotes the convex I-subgroup (sub-vector lattice) of G generated by g,
we define a convex f-subgroup (sub-vector lattice) H to be an abstract z-ideal if

h E H,g E G and G(g) = G(h) imply that g E H. In fact, [HudPI, 3.4] says that
this definition is equivalent to G(h) C H for all h E H.

Proposition 3.3.13. Let (A, +) be an f-system. If p E A is mazimal, then Vc is
a prime ideal of R which is an abstract z-ideal.

Proof: Let u v E V,, and u V V,. Assume without loss of generality that
u+, + E V, and u- f V,. We show that v- E V,. Since u- f V,, we have that
u-(p) 0. Thus since u+,v+ e V,,

S= u *(p) = u(a)v() = (+ u-)(a)(v+ v)(0),
a+0=0- a+#=I
and since p is maximal, we have that 0 = u-(Is)v-(I) and therefore, v~(p) = 0.
This gives that v- E V, and v E V,. Hence V, is a prime ideal.
Since I/V, is isomorphic to R via the evaluation map u "-* u(t), we have
that V, is uniformly closed by [HudPI, 2.1]. Thus by [HudPI, 3.4], V, is an abstract
z-ideal. n

Corollary 3.3.14. If (A, +) is a bounded above unital f-system, then the mazimal
ideals are given by {V, : pE A is maximal}.

Proposition 3.3.15. Let (A, +) be an f-system and let 6 E A be nonmaximal.
Define PS = {v E R : v(7) = 0 for all 7 > 6}. Then Ps is an abstract z-ideal and
it is a prime ideal if and only if a + P > 6 for all a, > 6.

Proof: Let v E P1. Then there exists {vn}il C Pj such that v, -4 v, for some
w E R+. Let 7 > 6, then for every e > 0 we have that lvl(7) < ew(y). Thus v(7) = 0
and Pa is relatively uniformly closed. Therefore, PS is a z-ideal by [HudPI, 3.4].
Assume that Pg is a prime ideal and that there exist a, / > 6 such that
a + P < 6. Then Xa+, E P6 and Xa+6 = Xa XB. But Xa, XB 1 PS, which is a

Conversely, assume that a+# > 6 for all nonmaximal a, # > 6. Let u*v E PS
and assume, by way of contradiction, that u, v Pg. Then u v(7y) = 0 for all
7 > 6 and there exist elements a E supp(u) and # E supp(v), each maximal in the
support set and such that a,3 > 6. Assume, without loss of generality that a > P. If
a' < a, ff 6 / and at least one of the inequalities is strict, then a' + ff < a + / and
we conclude that u*v(a+P) = u(a)v(p) +u(P)v(a). If a = /, then since a+ / > 6,
we have that 0 = u v(a + f) = 2u(a)v(/) t 0, a contradiction. If a > f, then
v(a) = 0 and hence 0 = u v(a + /) = u(a)v((/) 6 0, another contradiction. Thus
we conclude that either u or v is in P8 and therefore P6 is a prime ideal. *

Let S be a totally-ordered set. Recall that S is an ri -set if whenever A, B C S
are countable and A < B, then there exists c E S such that A < c < B. Since R
is not an i1-set, the ring R is never an 771-set. To see this, let 6 E A and consider
the sets {xj} > {(1 )x6 : n E N}. But, R can satisfy a related, slightly weaker
condition.

Definition 3.3.16. We call a totally-ordered set S an almost l1-set if A, B C S
are countable and ifA < B, then there exists c E S such that A < c < B. Note that
R is such a set.

Proposition 3.3.17. Let (A, +) be a totally-ordered f-system. R is an almost
r71-set if and only if A is an Ir -set.

Proof: Assume that R is an almost 71-set. We first note that A contains no suc-
cessor pair. Let a < / be a successor pair. Then the sets {nXa}neN and {1/nX#}nEN
Let A = {aI}}N, B = {lj}jEN C A, where a 2
XaY < Xa2 < ... < X& < X#i and there exists f E R such that X, < f < Xj for
all i,j. Let 7 be the maximal element of supp(f). Then f Xa, 0 implies that
7 = ai and f(7) > 1 or 7 > a. and f(7) > 0. If 7 = a,, then f X,, < 0, which

is a contradiction. Thus 7 > ai for all i. Similarly, 7 < p# for all j. Therefore, A is
an rh-set.
Conversely, assume that A is an ril-set and let fi < f2 < .* < g2 < gl E 1.
Let 1 < 2 < < 72 < 71 E A be the corresponding maximal elements of the
support sets. Let 4 = {}j)jEN and r = {7j}iNs. We have a few cases to consider: \$
and r are the same constant sequence, one of the sequences is eventually constant,
or neither sequence is eventually constant.
If there exists n E N such that Oj = a = 7, for i,j > n, then we get the
following string of inequalities in R : fi(41) < f2('2) < -* < 2(72) < 9g1(,). Since
R is an almost r7-set, there is r E R such that fj(41b) < r < gi(i,) for all i,j. Then

fj < rxa < gi for all i,j. If \$ is eventually constant and r is not, say Oj = a for
all i > n, for some n, then, by hypothesis, there exists P E A such that a < 0 < r.
Then fji X# < gi for all i, j. If neither sequence is eventually constant, then by the

ri-set hypothesis, there exists P such that 1j < f < 7y for all i and fj < XO < gi for
all i, j. M

3.4 Survaluation Ring and nf-Root Closed Conditions

Recall that a commutative ring A is a survaluation ring (or SV-ring) if A/P
is a valuation ring for every prime ideal P. In this section, we set down a character-
ization of those f-systems which give SV-rings.
Let (A, <, +, ) be a totally ordered and cancellative abelian monoid with
identity element p. We define the group of differences, qA, of A as it is done in
[Fu, X.4]. Define an equivalence relation on A x A by (i6, 62) ~ (7, 72) if and only
if 81 + 72 = 71 + 62. It is clear that the relation is reflexive and symmetric; the
transitivity follows from the cancellation property of the monoid. We let qA be the
quotient A x A/ ~, denote the class of the element (61, 62) by [61 62], and define
an operation + as is usual. That is, [J6 62] + [71 72] = [(6 + 71) (62 + 72)]. The

cancellation in the monoid ensures that the operation is well-defined. The element

[p p] is an identity and [62 61] is an additive inverse of [61 62]. We define
[61 621 5 [71 72] if and only if 61 + 72 < 71 + 62. By [Fu, X.4.4], this is the
unique order on qA which extends the order on A. Finally, o-embed A in qA via
8 -4 [6 p].
Let H be a partially ordered groupoid. Then h E H is called negative if
hx < x or xh < x, or both for all x E H. The groupoid H is called negatively
ordered if every element is negative.

Definition 3.4.1. Let A be a partially ordered semigroup. A is called inversely
naturally ordered if it is negatively ordered and 6 < a implies that there exists

Se A such that 6 = a + 3.

Example 3.4.2. Let A = {1 }11, U [1,oo) C R be inversely ordered with the
usual addition in the reals. Then A is an f-system which is not inversely naturally
ordered. To see this, note that (1 1) + c = 1 if and only if c = n = 2 or
c= 1,n= 1. [

Let (A, +, p) be a totally-ordered abelian cancellative monoid with maximal
element p, such that (A,+) is an f-system. Let X = {X6 : 6 E A}. Then (X, *, X,)
is a totally-ordered abelian monoid which is I-isomorphic to (A, +, p). Since X is
written multiplicatively, the elements of the group qX are quotients and we denote
them as such in the following proof.

Theorem 3.4.3. Let (A, +, p) be a totally-ordered abelian cancellative monoid with
maximal element p. Let X = {xs : 6 E A} and R = F(A,R). The following are
equivalent:

1. R is a valuation ring.

2. R is 1-convex.

3. I is Bizout.

4. 1 is convex in qR.

5. X is convex in qX.

6. A is convex in qA.

7. A is inversely naturally ordered.

Proof: The equivalence of (1), (2), and (3) is [MW, Theorem 1]; the equivalence
of (2) and (4) is [ChDi2, Lemma 2].
(2) =- (5) : Let X, <- Xa./X < Xa. Then 0 5 Xa < Xs+p and by (2), there exists
f E R such that Xa = f X+f-. Iff = EC j fj3Xo, where J is some index set,
fj E R and q dj E A for all j E J, then

Xa(e)= f X+b(e)={f ife= +6+

Thus, for some j E J, we have that a = 3 + 6 + j, fj = 1 and fA = 0 for all
k 5 j. Hence f = x~ and Xa/Xp = Xs+ (5) (6) : Follows since A L X.
(5) => (7) : Let a < p E A. Then a +/ < a and therefore Xa+, < Xa < X# which
implies that Xa < XaI/X < X,. Thus, by (5), we have that Xa/Xy# = X6 E X for
some 6 E A. Therefore, Xa = X8+6 and a = P + 6.
(7) = (2) : Let 0 < u < v E 7, and assume that u = YjE~aijX,, and that
v = EkeK bkxpk for index sets J, K, and where aj, bE E R for all j E J, k E K.
Also assume that a, < 61 are maximal elements in the respective support sets.
Then by (7), for every j E J and every k E K, there exists 6i, 7y E A such that
aO = A + b and P3 = #I + 7. Note that 71 = p. Then u = XA, E* jJ aixa and
v = XA EkEK b4Xk

Let w = ZEK bhX7k. Then w(p) = bl > 0 and hence -w is invertible in
R. Let z = Eij ajXs,. Then by cancellation in A, we have that 0 < x < w and
0 x< -w. If we let f = (-w)-* (Ix), then x = -w *f and = w f.
Finally, u = X, x = X, w f = v f, as desired. a

The following lemma is well-known and routine to verify.

Lemma 3.4.4. Let (A, +) be an l-system and C C A, a maximal chain. Denote the
associated minimal prime subgroup by He = {u E 7 : C n supp(u) = 0}. The map
cp : F(A, R) -+ F(C, R) given by restriction is a subjective I-ring homomorphism
with kernel He. Thus, %/Hc F(C, R.)

Corollary 3.4.5. Let (A, +) be a unital f-system. Then 7 is an SV-ring if and
only if each maximal chain in A is inversely naturally ordered.

Example 3.4.6. 1. If A1 = [0, oo) C R is inversely ordered with the usual
addition of real numbers, then F(A1, R) is an SV-ring.

2. If A2 = {1 1}=1 U [1, oo) C R is inversely ordered with the usual addition
in the reals, then F(A2, R) is not an SV-ring.

3. Let A = R[[x, y]] be the ring of formal power series in the indeterminates z, y.
Order the monomials lexicographically via 1 > x, y and x'yi < zxyl if and
only if k < i or k = i and I < j. The ring A is not an SV-ring since it is
not 1-convex: note that 0 < y < x and the equation y = xf has no solution
fEA.

Let Z,, = Z = {n E Z : n > 0} be inversely ordered. In the lex-order
described above, if A = Z, x Zy, then A c F(A, R). We convexify A in qA by
convexifying A in qA. That is, if Ac = A U {(n, m) E Z x Z : n > 0, m < 0}
then F(Ac, R) ^- A({xayi : i > 0 or i = 0 and j > 0}) UA is an SV-ring which
is the convexification of A in qA. O

Recall that an f-ring A is nth-root closed if for every a E A+ there exists
b E A such that a = b". This property arises in R if there is a certain amount of
divisibility in the arithmetic structure of A.

Definition 3.4.7. Let (A, +) be an r-system. A is called n-divisible if for every
6 E A there exists a E A such that na = 6. We say that the system is divisible if it
is n-divisible for all n E N.

Proposition 3.4.8. Let (A, +) be a totally ordered f-system such that R is nth-root
closed. Then A is n-divisible.

Proof: Let 6 E A. Then xs = v" for some v = E ij ajXa. If 6 = p is maximal
in A, then 6 = nip, so we assume that 6 # p. If ac E supp(v) is maximal in the
support set, then nal E supp(v") is maximal. Therefore 6 = nal, as desired. w

Proposition 3.4.9. Let (A, +) be a totally ordered inversely naturally ordered f-
system with maximal element p. If A is n-divisible, then R is nt-root closed.

Proof: We begin with square-roots. Let u E R+ be given by u = EjEj ajX", for
some index set J, aj E R, and j E A for all j E J.
If u(p) 6 0, then we define a square-root v recursively on A. To begin, let
v(p) = i) and assume that a, is maximal in supp(u) \ {p}. Let v(6) = 0 for
all al < 8 < p. Then we define v(al) = u(al)/2v(p,). Let S be the N-linear span
of supp(u) and define v(6) = 0 for all 6 V S. If v(7) is defined for all 7 > 6, then
we define v(6) to be the unique solution of the equation u(6) = ;-+r=~ v(a)v(r),
where, necessarily, a, r E {a E S : a > 6}. Then u = v v, by construction.
Now, assume that u(/p) = 0 and let al be the maximal element in the support
of u. Since A is inversely naturally ordered, for every j E J, there exists ij E A such
that aj = al +6i. Note that 61 = p. Then u = Xa1 *EjE, aXj. Let w = jEJJ aXj ,,
then w(/p) # 0. Thus w = vi vl by the above construction. Since A is 2-divisible,

al = 271 for some 71 E A. Therefore Xai = X, X-n. Letting v = X, vl, we then
have that v v = Xyi vl Xy, v = Xai w = u. Thus nth-roots exist when n is a
power of 2.
Let n be odd and let v E R be given by v = 'keK bkX#, for some index
set K, bk E R, and flP E A for all k E K. As with square roots we consider two
cases. First assume that v(p) # 0 and define an na-root recursively. Let w(p) be a
real nt-root of v(/i) and let v(6) = 0 for all A < 6 < /, where a is the maximal
element of supp(v) \ {p}. Then the n-fold convolution product equation we must
solve reduces to v(,8) = w"(A) = n(w(p))n-lw(1). In order to see this, we proceed
by induction on n. If n = 2, then w*w(/31) = w(=/)+w)+w(#)w(/) = 2w(p)w(A).
Assume that the statement holds for n = m. Then
wm'(Pi) = (w w')(1) = w(u)w'm(I) + w(#i)w'm()

= w(p)m(w(/p))m-1w(fi) + w(fi)(w(/A))m = (m + 1)(w(p))mw(/O)

as desired. We may then define w(/1i) = v(j1)/(n(w(p))n-1).
In general, we let w(7) = 0 if 7 is not in the N-linear span of the support of v.
Assume that w(7) is defined for all 7 > J. We show that the equation v(6) = wn(6) is
linear in w(6); hence, we may define w(6) to be the unique solution of this equation.
If n =2, then w2(6) = ,+r=, w(a)w(r) + 2w(5)w(p). Assume that wm(6) is linear
in w(6). Then

wm+1(6) = w (a)wm(7.) + w(w),m'(6) + w(6)w'(t)
or+-Tr=

is linear in w(6), by induction, since w(6) will not appear in wm(r), as 6 < 7 and A
is an f-system. Thus, in this case, v has an nt-root.
Second, assume that v(it) = 0 and proceed as in the square-root case. Let 61
be the maximal element in the support of v. Since A is inversely naturally ordered,
for every k E K, there exists 6k e A such that /3k = f/ + 6k. Note that 6~ = p.. Then

v = X EYeK b xk Let x = EkEKbkx6,, then x(p) 0. Thus x = w:, for some

wl, by the above construction. Since A is n-divisible, #I = nai for some 71 E A.
Therefore Xp, = (X,,) Let w = X, wi; then wn = (X, Wl)" = XI z = v. a

Corollary 3.4.10. Let (A,+) be a totally ordered inversely naturally ordered f-
system with maximal element p. A is divisible if and only if R is nh-root closed for
all n.

Example 3.4.11. 1. Let Ai = [0, oo) C R be inversely ordered with the usual
addition of the real numbers. Then F(A1, R) is nt-root closed for all n.

2. Let A2 = {n E Z : n > 0} be inversely ordered. Then A2 is not 2-divisible
and Xi > 0 has no square-root. That is, if Xi = v2, then v(0) = 0 and we then
conclude that 1 = Xi(1) = 2v(0)v(1) = 0, a contradiction.

3. Let A3 {1 } U [1, oo) C R be inversely ordered with the usual addition
in the reals, then A3 is not inversely naturally ordered and similarly, X1 has
no square-root since 1 has no nonzero summand. ]

Proposition 3.4.12. Let (A, +) be a totally ordered f-system with maximal ele-
ment p. Then if p has an immediate predecessor ir, then X, has no square-root. If
there exists p > 6 E A such that 6 has no nonmaximal summand other than itself,
then Xa has no square-root.

Based on the preceding examples and results on nt-roots, we formulate the
following:

Conjecture 3.4.13. Let (A, +) be a totally ordered f-system with maximal element
p. F(A,R) is square-root closed if and only if A is 2-divisible and every nonmaximal
element of A has a nonmaximal summand other than itself.

Recall that a field K is real-closed if every positive element is a square and
every polynomial p E K[z] of odd degree has a root in K. An integral domain
R is called real-closed if qR is a real-closed field. Let (A, +) be a totally ordered
inversely naturally ordered f-system with maximal element p. Assume that A is
also 2-divisible. Then R is a 1-convex and square-root closed f-domain. By [ChDil,
Theorem 1], under these conditions, R is real-closed if and only if every odd degree
polynomial over R has a root. What additional assumptions on A are necessary to
guarantee the real-closed property?

Conjecture 3.4.14. Let (A, +) be a totally ordered commutative f-system with
maximal element p. If A is divisible and inversely naturally ordered, then R is real-
closed.

We end this section by shedding a little light on this conjecture. Recall from
[HLM] that a commutative f-ring A with 1 satisfies the Intermediate Value Theorem
for polynomials (or is an IVT-ring, for short), if for every p(t) E A[t], and pair of
distinct elements u, v E A such that p(u) > 0 and p(v) < 0, there exists w E A such
that p(w) = 0 and uA v < w < u V v. We show that a totally ordered commutative
semiprime valuation f-domain with identity is real-closed if and only if it is an IVT-
ring. It is necessary to record the following unpublished theorem of Suzanne Larson,
which was communicated via electronic mail on April 17, 1997. Her proof follows.

Theorem 3.4.15. Let A be a commutative semiprime IVT-ring with identity. If
S is a multiplicatively closed subset of regular elements of A+, then the ring of
quotients, S-1A is an IVT-ring.

Proof: Let p(t) E S-'A[t] be given by p(t) = aoboW + albi't + + anbnt" and
assume that p(ulv-') > 0 and p(u2v-) 5< 0. Let d = v'vb2bobi .. b. Then d E S is
regular. Define a new polynomial q(t) E A[t] by

aovvb ** "bn+alvn-'v-bob ... bnt+a2vn-2v2n-2boblb3 .... bt2+.. +abob, ".

Then q(ulv2) = dp(uiv ') > 0 and q(u2vi) = dp(u2v1) < 0. Since A is an IVT-ring,
there exists w E A such that ulv2 A U2V1 < w < u1v2 V u2v1 and q(w) = 0. Then

u1vp1 A u2v21 < wv-'v1 < u1v1 V uO21 and dp(wv'v1') = q(w) = 0. Since d is
regular, p(wv lv2') = 0 and we conclude that the quotient ring is an IVT-ring. m

Proposition 3.4.16. Let A be a totally ordered commutative semiprime valuation
f-domain with identity. Then A is real-closed if and only if it is an IVT-ring.

Proof: If A is real-closed, then qA is a real-closed field and, by [ChDi2], qA is an
IVT-field. Let p(t) E A[t] be such that p(u) > 0 and p(v) < 0, for some u, v E A.
Then there exists w E qA such that p(w) = 0 and u A v < w < u V v. Since A
is a valuation domain, A is convex in qA by [ChDi2, Lemma 2]. Hence, w E A
and we have that A is an IVT-ring. Conversely, if A is an IVT-ring, then qA is an
IVT-field, by the preceding theorem of Larson. Then, by [ChDi2], qA is real-closed
and therefore A is real-closed. *

CHAPTER 4
RAMIFIED PRIME IDEALS

In this chapter we expand on the notion of a ramified prime ideal, as defined in [Le],
which we introduced in Chapter 2. We first examine the concept in general and then
move to try to understand ramified maximal ideals which correspond to nonisolated
Ga-points. This result is then used to consider local versus global ramification
conditions.

4.1 Ramified Points

Definition 4.1.1. Let X be a completely regular space. A prime ideal of C(X) is
ramified if it is the sum of the minimal prime ideals that it contains. We define
p E X to be ramified if Mp is ramified. A point p E X is totally ramified if every
prime z-ideal contained in Mp is ramified. The space X is (totally) ramified if every
nonisolated point of X is (totally) ramified.

A ramified 1-ideal of C(X) is a prime ideal, by Corollary 2.2.6. LeDonne
proves that a ramified prime ideal of C(X) is necessarily a z-ideal. Let us consider
two extreme conditions. Recall that we say a point p E X is an F-point if Op is
prime. Ifp is an F-point, then since Op is the unique minimal prime ideal contained
in Mp, Mp is not ramified. Likewise, in this case, no prime z-ideal contained in Mp is
ramified. On the other hand, the condition of total ramification ensures branching
at every prime z-ideal. Analytically, LeDonne shows [Le, 3]:

Theorem 4.1.2. If X is a metric space then every maximal ideal of C(X) is totally
ramified.

Note that this result says that every maximal ideal of C(X) (fixed or free) is
ramified, if X is metric. We do not know of any weaker topological condition which
guarantees total ramification of C(X).

Definition 4.1.3. Let A be a commutative f-ring with identity. For any integer
n > 2, call a prime t-ideal P n-limbed if P is the sum of n noncomparable prime
e-ideals of A which are properly contained in P. A point p of X is n-limbed if Mp is
n-limbed. Note that any n-limbed -ideal P is necessarily ramified and rk(P) > n.

Example 4.1.4. We now present an example of an f-ring 1 in which a maximal
ideal is ramified but not n-limbed for any n. Let A0 = [0, oo) C R and define
A' = (1/n, oo) R, for i = 1,2, where each interval is inversely ordered. Let
An = A II A: be the disjoint union and then let A = Ao H (II,,,A). We obtain a
root system with the induced ordering which we describe in the following diagram:

(4.1)

We endow A with a partial commutative associative binary operation. Let
(*)i denote the sum in parentheses as the usual sum of real numbers residing in A~;

the mark "-" signifies that the sum is undefined. Note, to conserve space, the table
is completed by reflection across the diagonal.

+ rEAo rEA A rE6 I rE A rE
E o (r + s)o (r + a) (r + a)' (r + _s)__ (_+ _
sE A (r + a) (r + s). (r + s) if k < n;else (r + a)if I < n; else -
E A(r +s (r + 8) if k sEA, ________ (r + a)2

Let R = F(A,R) and V, = Ao II (IIm>nA); let Co = Ao II (IIne1A') and
C, = Dn) II A. Then the minimal prime ideals of R correspond to these maximal
chains and are given by Q, = {u E R : u(Cn) = {0}}, where n = 0,1,2,....
Any supremum VjE Qj over a finite set J C w is the prime ideal P, given by

{u E R : u(2)) = {0}}, where m is the maximum element of J. Hence, for all
n E w, the maximal ideal Vo is not n-limbed since it is not a finite supremum of
minimal prime ideals. However, Vo = Ve,, Qn, and so it is ramified. O

We show in Proposition 2.3.4 that for any nonisolated point p of a first
countable space, the maximal ideal Mp is n-limbed for every n. If the space is also
cozero-complemented, then rk(M,) > 2C and Mp is 2C-limbed. From this, we also
obtain the following, which is weaker than Theorem 4.1.2:

Corollary 4.1.5. Every metric space is ramified.

The following theorem means that if X is a metric space that is not pseudo-
compact, then there exist points of /X \ X such that MP is minimal. Hence, not
every maximal ideal of C(X) branches nontrivially in the root system Spec(C(X)).
Recall that we call a topological space X perfect if every closed set of X is a Gs-set.
Note that any metric space is perfect. A point of fX is remote if it is not in the
#X-closure of any nowhere dense subset of X. A point p E #X \ X is a C-point
if p inttx\x(cldxZ(f) n 8X \ X) for all f E C(X). A theorem similar to the
following appears in [W, 4.40]. All the proofs there carry through here, verbatim,
under our reduced hypotheses.

Theorem 4.1.6. Let X be a completely regular space and consider the following
conditions on a point p E #X \ X:

1. p is a C-point.

2. Z[MP] contains no nowhere dense set.

3. MP=OP.

4. p is a remote point.

Without additional assumptions, (3) =4 (1). Let X be perfect and assume the exis-
tence of a remote point, then (4) = (2) =* (3). If X is perfect and the set of isolated
points of X has compact closure in X, then (2) => (4). If X is realcompact and
C-points exist, then (1) =o (2).

It is not known if a remote point p always has the property that MP = OP.
We do know the following about the rank of a remote point:

Proposition 4.1.7. Let X be a completely regular space. Let p E fX be a remote
point. Then rkc(x)(MP) = 1.

Proof: In [vD, 5.2], it is demonstrated that no remote point is in the closure of two
disjoint open sets of #X. Thus, in particular, no remote point is in the closure of two
disjoint cozero-sets of #X. By [HLMW, 3.1], we have 1 = rkax(p) = rkc(ox)(M*P).
Finally, since C(fJX) is rigid in C(X), we have rkc(x)(MP) = 1. U

Finally, we ask: does ramification of a point in X indicate ramification in
#X, or vice versa?

Proposition 4.1.8. A point p E X is ramified in X if and only if it is ramified in
38X. Likewise, a point p E vX is ramified in vX if and only if it is ramified in 8X.

Proof: This is a corollary of Proposition 2.2.12, since we know C(#X) = C*(X)
and MfX = Mx n C(#X). a

4.2 Ramified Gs-points

The main theorem, Theorem 4.2.5, of this section provides a good method
for checking the ramification of Ga-points. We will use it to characterize ramified
Ga-points in normal countably tight spaces and to find some ramified points in
product spaces. We first discuss the following proposition, which we will obtain as
a corollary to Theorem 4.2.5.

Proposition 4.2.1. Let p E X be a G6-point. If X \ p is not C*-embedded in X,
then rk(p) > 2.

Since an F-point has rank 1, the preceding proposition, proved in [Le] and
(in greater generality) by van Douwen in [vD], shows that a Ga-point, p, is not
an F-point if it has the property that X \ p is not C*-embedded in X. We give a
counterexample for the converse if the G6-condition is lifted.

Example 4.2.2. Let X = B2(N, /N \ N) be as defined in Example 5.2.5. There,
we show that there exists a point p of the corona such that rk(p) = 2, X \ p is
C*-embedded in X and p is not ramified. But no point of the corona is a G6. [

The following two results are Theorems 2.1 and 2.2 of [K2]; we will use these
to prove our theorem on the ramification of Ga-points.

Theorem 4.2.3. Let p be a nonisolated Ga-point of X. If Z e Z[C(X \ p)] then
clx(Z) E Z[C(X)].

Define 7 : Z[C(X \ p)] --- Z[C(X)] by 7(Z) = cx(Z). Let 4 be the
extension of the identity map X \ p -- X to the largest subspace X1 C #(X \ p)
such that it is extendible as a continuous map into X.

Theorem 4.2.4. Let p be a nonisolated Ga-point of X then

1. The mapping 7 is one-to-one from the set of prime z-filters on X\p converging
to points of 0-'({p}) onto the set of prime z-fiters on X contained properly
in Z[Mp].

2. A prime z-filter W on X \p converging to a point of 1~'({p}) is a z-ultrafilter
if and only if 7(W) is maximal in the class of prime z-filters on X contained
properly in Z[Mp].

Theorem 4.2.5. Let p be a nonisolated Gj-point of X. The point p is ramified if
and only if X \p is not C*-embedded in X.

Proof: Let p be a nonisolated Gj-point of X. If p is not ramified then the prime
z-ideal P = oQEMin(M) Q is properly contained in Mp. We will show, in this case,
that every point of X is the limit of a unique z-ultrafilter on X \ p. Then by [GJ,
6.4], X \p is C*-embedded in X.
Let q E X \ p. Then Mq E Max(C(X \ p)) gives rise to the z-ultrafilter
14 = Z[Mq] on X \p. Clearly q E n{clx(U) : U E t4}. The uniqueness of U4 is a
standard result [GJ, 3.18].
By [GJ, 6.3(b)], there exists a z-ultrafilter U on X\p converging to p. Assume
that there exists another such z-ultrafilter, V. Let U = yU and V = yV. Then
Qu = ZxU and Qv = Z V are prime z-ideals of C(X) which are properly contained
in M,. If Qu C P then U = Zx[Qu] C Zx[P]. Hence U = 'r-U c r-Zx[P], which
contradicts that U is a z-ultrafilter on X \p. Likewise, Qv is not properly contained
in P. Thus P C Qu, P C Qv and by [GJ, 14.8(a)], we must have either Qu C Qv
or Qv C Qu. But Qu C Qv gives that U C V and therefore U C V. Since U is an
ultrafilter, U = V, as desired. In a similar manner, if Qv C Qu, then V = U.
Conversely, assume that p is ramified. Then Mp = E Min(Mp) and there
exists more than one prime z-ideal in C(X) which is maximal in the class of prime

z-ideals properly contained in M,. These give distinct prime z-filters on X which
are maximal in the class of prime z-filters on X properly contained in Z[Mp]. Hence,
via 7, we have distinct ultrafilters on X \ p converging to p. Again by [GJ, 6.4],
X \p is not C*-embedded in X. *

Corollary 4.2.6. If X is a metric space, then X is ramified.

Corollary 4.2.7. If X is first countable and p E X is nonisolated, then X \ p is
not C*-embedded in X.

Corollary 4.2.8. Ifp E X is a Gj-point and X \p is not C*-embedded in X, then
rk(p) > 2.

Let X and Y be completely regular spaces which are not P-spaces and let
W = X x Y. We conjecture that every nonisolated point of W is ramified. We use
Theorem 4.2.5 to deduce two partial answers to this question.

Proposition 4.2.9. Let X and Y be completely regular spaces and let W = X x Y.
Let x E X and y E Y be nonisolated Ga-points and let p = (x, y) E W. Then
W \ {p} is not C*-embedded in W.

Proof: Since X is completely regular and x is a Gj-point of X, {x} is a zero-set
of X. Say, {x} = Zx(f) for some f E C(X). Then we have for E, = {(} x Y,

E, = {z} x Y = 7r(Zx(f)) = Zw(f o 7x)

where lrx denotes the natural projection from W onto X. Likewise, we have that
Ey = X x {y} = Zw(g o ry) where g E C(Y) such that Z,(g) = {y} and ry is the
natural projection from W onto Y. Let and g denote the restrictions of fo7rx and
golry to W\{p}. Then we have that E,\{p} = Zw\{p}(f) and E,\{p} = Zw\ p(g)
are disjoint zero-sets of W\ {p}. Thus E \ {p} and E\ {p} are completely separated
in W\{p}. But p E clw(E \{p})nclw(E\ {p}) and therefore E,\{p} and E,\{p}

are not completely separated in W. By the Urysohn Extension Theorem, W \ {p}
is not C*-embedded in W. *

Corollary 4.2.10. Let X and Y be completely regular spaces and let W = X x Y.
Let x E X and y E Y be nonisolated Ga-points and let p = (x, y) E W. Then p is
ramified in W.

Proof: Follows from Theorem 4.2.5. U

Proposition 4.2.11. Let X and Y be completely regular spaces and let W = X x Y.
Let p = (x, y) E W be nonisolated. If W \ {p} is normal, then W \ {p} is not C*-
embedded in W.

Proof: Let E, = {x} xY and Ey = X x {y}. Then E,\{p} and E,\{p} are disjoint
closed sets in the normal space W \ {p}. Thus, they are completely separated in

W \ {p}. But pE clw(E, \ {p}) n clw(E, \ {p}), so E, \ {p} and E, \ {p} are not
completely separated in W. Therefore, by the Urysohn Extension Theorem, W\ {p}
is not C*-embedded in W. m

Corollary 4.2.12. Let X and Y be completely regular spaces and let W = X x Y.
Let p = (x, y) E W be a nonisolated Ga-point of W If W \ {p} is normal, then p is
ramified in W

Proof: Follows from Theorem 4.2.5.

We now investigate the ramification of Gs-points in a class of spaces more
general than metric or first countable spaces.
A topological space X is countably tight if for a subset U c X we have that
any p E cl(U) is in the closure of a countable set S c U. A Frechet- Urysohn space is
one in which every p E cl(U) is the limit of a sequence of distinct points {pn} C U.
It is evident that any Frchet-Urysohn space is countably tight.

Lemma 4.2.13. Let X be a normal topological space and let p e X be nonisolated.
Then X \ {p} is C*-embedded in X if and only ifp 4 clx(A) ncldx(B), whenever A
and B are completely separated in X \ {p}.

Proof: By the Urysohn Extension Theorem, X \ {p} is C*-embedded in X if and
only if A and B are completely separated in X, whenever A and B are completely
separated in X \ {p}. Assume that X \ {p} is C*-embedded in X and let A and B
be completely separated in X \ {p}. Then A and B are completely separated in X
and hence are contained in disjoint closed sets of X. Thus p dclx(A) clx(B).
Conversely, let A and B be completely separated in X \ {p}. We wish to show
that A and B are completely separated in X. By hypothesis, p clx(A)n clx(B).
Thus, clx(A) and clx(B) are disjoint closed sets of the normal space X. Hence, A
and B are completely separated in X. *

Proposition 4.2.14. If X is a normal countably tight topological space and p E X
is nonisolated, then X \ {p} is C*-embedded in X if and only if for every two
countable sets St and S2 which are completely separated in X \ {p}, we have that
p clx(S) n ldx(S2).

Proof: Assume that X \ {p} is C*-embedded in X and let S, and S2 be two count-
ably infinite sets which are completely separated in X\ {p}. Then by Lemma 4.2.13,
we have that p V dx(S) n clx(S2).
Conversely, let A and B be completely separated in X \ {p}. Assume that
p E clx(A) n clx(B). Then there exist countable sets S1 C A and S2 C B such that
p E clx(Si) n clx(S2). Since A and B are completely separated, so are S1 and S2.
This contradicts the hypothesis. Thus by Lemma 4.2.13, A and B are completely
separated in X and X \ {p} is C*-embedded in X. *

Corollary 4.2.15. Let X be normal Frichet- Urysohn space and let p E X be non-
isolated. Then X \ {p} is C*-embedded in X if and only if there do not exist two
sequences in X which are completely separated in X \ {p} and converge to p.

Finally, Proposition 4.2.14 and Theorem 4.2.5 imply the following.

Corollary 4.2.16. If X is a normal countably tight topological space and p E X is
a nonisolated Gs-point, then p is ramified in X if and only if for every two countable
sets Si and S2 which are completely separated in X \ {p}, p dclx(Si) dx(S2).

4.3 Ramification via C-Embedded Subspaces

Let A and B be commutative rings with identity. Assume that we have a
subjective ring homomorphism f : A B, with K = Ker(f). Recall that there is a
one-to-one order-preserving correspondence between Spec(B) and the set of prime
ideals P E Spec(A) such that K C P. Let M E Max(B) be such that M = P1 + P2
for some nonmaximal proper primes Pi, P2 E Spec(B). If N E Spec(A) corresponds
to M, then we have B/M (A/K)/(N/K) A/N. So N E Max(A). Let Q, Q2
be the prime ideals of A corresponding to P1 and P2. Then (Q1 + Q2)/K P1 + P2
via the surjective map given by a + b -+ f (a) + f(b) with kernel K. Thus, by the
correspondence, we have that N/K M = Pi+P2 ( (Q,+Q2)/K and N = Qi+Q2.
In fact, we have shown:

Proposition 4.3.1. Let A and B be commutative rings with identity such that there
exists a subjective ring homomorphism f : A -- B with K = Ker(f). If P < B is a
prime ideal which is a sum of two distinct prime ideals then there exists Q E Spec(A)
such that Q/K K P and Q is a sum of two distinct prime ideals of A.

Ramification in a C-embedded subspace implies global ramification.

Corollary 4.3.2. Let X be a completely regular space and let Y be a C-embedded
subspace. If a point p of Y is 2-limbed in Y, then p is 2-limbed in X.

Proof: Since Y is C-embedded in X, we have a surjective ring homomorphism
from C(X) onto C(Y), given by restriction with kernel {f E C(X): Y C Zx(f)}.
Hence this result follows from Proposition 4.3.1. 0

Proposition 4.3.1 also gives the following, since any compact subspace of a
completely regular space is C-embedded.

Corollary 4.3.3. 1. Let X be compact, Y C X a closed subspace. If a point of
Y is 2-limbed in Y then it is 2-limbed in X.

2. Let X be a compact space consisting of more than one point. If every noniso-
lated point in a proper zeroset of X is 2-limbed, then every nonisolated point
of X is 2-limbed.

Example 4.3.4. The Cantor Set is metric, hence every point is 2-limbed. By [E,
3.12.12c], every point of 2T, the Cantor space of weight r, is contained in a closed
set which is homeomorphic to the Cantor Set. Thus every nonisolated point of 2T
is 2-limbed. In fact, induction on Proposition 4.3.1 gives that every point of the
Cantor space is n-limbed for every n E N. []

Conversely, if a maximal ideal of A containing K is a sum of primes containing
K, then by the correspondence given above, its image is a maximal ideal which is a
sum of primes in B. That is, if N E Max(A), K C N, and N = Q1 + Q2 such that
K C Qi and K C Q2, then f(N) E Max(B) as B/f(N) G (A/K)/(N/K) G A/N.
And f(N) = f(Qi + Q2) = f(Q1) + f(Q2) Q1/K + Q2/K.
This gives a partial converse:

Proposition 4.3.5. Let A and B be commutative rings with identity such that there
exists a subjective ring homomorphism f : A -+ B with K = Ker(f). Let P be
a prime ideal of A containing K. Then P is a sum of two distinct prime ideals

containing K if and only if f(P)/K is a prime ideal which is a sum of two distinct

prime ideals of B.

Corollary 4.3.6. The ring-homomorphic image of a commutative ramified ring

with identity is ramified.

Corollary 4.3.7. Let Y be a C-embedded subspace of X and let p E Y have finite
rank in X. If p is ramified in Y, then p is ramified in X. If any function in C(X)
vanishing on Y also vanishes on a neighborhood of p, then p is ramified in Y if and
only if p is ramified in X.

Proof: The first statement is an application of Proposition 4.3.1 by induction. The
second statement follows from Proposition 4.3.5 by induction. a

Note that the hypothesis of the second statement of the above merely de-
mands that the kernel of the restriction map be contained in Opx. This is satisfied
if Y is open or if Y is a P-set. The preceding results indicate that ramification is a
local property.
Let A be a commutative ring with identity, let S be a multiplicative system in
A such that 1 E S. Then there exists a one-to-one correspondence from Spec(S-1A)
onto {P E Spec(A) : P S = 0}. The proofs of the following are routine:

Proposition 4.3.8. Let A be a commutative ring with identity, let S be a multi-
plicative system in A such that 1 E S.

1. If P E Spec(S-'A) is a sum of two proper primes, then the preimage of P, the
set {x E A : x/1 E A}, is a prime ideal which is a sum of two proper primes
in A.

2. If Q E Spec(A) is a sum of primes and Q n S = 0, then S-'Q is a sum of
primes in S-1A.

87

Corollary 4.3.9. Let Y be a subspace of X such that C(X) -+ C(Y) is a ring of
fractions map. That is, there is a multiplicative system S C C(X) such that 1 E S
and C(Y) = S-1(C(X)). Then:

1. Forp E Y, rky(p) < rkx(p).

2. Let p E Y. If Mf n S = 0, then there is P E Spec(C(X)) such that P C Mf
and rky(p) = rkx(P).

CHAPTER 5
m-QUASINORMAL f-RINGS

In [Lal-3], Suzanne Larson defines the notion of a quasinormal f-ring; one in which

the sum of any two distinct minimal prime ideals is a maximal t-ideal or the entire
f-ring. We generalize this definition and a few of her results.

5.1 Definitions

Definition 5.1.1. Let A be a commutative f-ring with identity and let m be a

positive integer. Call A m-quasinormal if the sum of any m distinct minimal prime

ideals is a maximal e-ideal or the entire f-ring A. If X is a topological space such
that C(X) is m-quasinormal, then we call X an F,-space.

Note that the "2-quasinormal" is Larson's "quasinormal" condition, the "1-

quasinormal" condition is equivalent to von Neumann regularity, and if A is m-
quasinormal then A is n-quasinormal for any n > m. Hence, the Fi-spaces are

exactly the P-spaces and any Fr-space is an Fn-space, when n > m.

Theorem 5.1.5 generalizes [Lal, 3.3] and characterizes the m-quasinormal
semiprime f-rings. Note that [Lal, 2.2], which we now state, gives necessary and

sufficient conditions for a semiprime f-ring to have the property that the sum of
any two distinct minimal prime ideals is a prime e-ideal. This condition is stronger
than the assumption we make in our theorem, but this result indicates when one

can expect to be able to apply it.

Theorem 5.1.2. Let A be a semiprime f-ring. The following are equivalent:

1. The sum of any two semiprime -ideals is semiprime.

S. The sum of any two minimal prime -ideals is prime.

3. The sum of any two prime -ideals is prime.

4. For any a, b E A+, the -ideal a1- + b'- is semiprime.

5. For any two disjoint elements a, b E A+, the I-ideal a + b- is semiprime.

6. For any a, b E A+, the t-ideal a + b' is semiprime.

7. When x, a, b, c, d E A+, a, b 0 are such that x2 = c +d and aAc = bAd = 0,
there ezist g, h E A such that x = g + h and gAa= hA b= 0.

The theorem above holds for C(X) since the sum of two prime ideals is prime
by [GJ, 14B]. We will use the following lemmas. The first follows from the fact that
a prime f-ideal P of a semiprime f-ring is minimal if and only if for every p E P
there exists q V P such that pq = 0. The second lemma shows the existence of
certain functions which we will take for granted in the proofs to follow.

Lemma 5.1.3. Let A be a semiprime f-ring in which the sum of any m distinct
minimal prime ideals is a prime i-ideal. Let P be a prime f-ideal. Then P is
minimal with respect to containing J~m=' aJ if and only if for every p E P there
exists q P such that pq E -=,1 al.

Lemma 5.1.4. Let A be a commutative f-ring. Let n > 2 and let Q, ...,Q,, be
distinct minimal prime ideals. Then there exists an element f E Qf \ Uj, Qj, for
each i = 1,...,n.

Proof: Let fk E Q+ \ Q for k = 1,..., n. Then, by convexity, we have that

f = .fi E Q+ \ Uj Qj, as desired. m

Theorem 5.1.5. Let m be a positive integer. Let A be a commutative semiprime
f-ring with identity in which the sum of any m distinct minimal prime ideals is a
prime L-ideal. The following are equivalent:

1. A is m-quasinormal.

2. For every nonmaximal prime f-ideal P, rk(P) < m 1.

3. Let {aj}'m1 be a family of positive pairwuise disjoint elements of A. Proper
prime I-ideals containing j1= a are maximal t-ideals.

4. Let {aj}im1 be a family of positive pairwise disjoint elements of A, let M be
a maximal f-ideal containing the -ideal ~1 afJ, and let p E M. Then there
exists z f M such that zp e 7iam= -a.

Proof: (1) =* (2) : Let P be a nonmaximal prime -ideal of A such that rk(P) > m
and let Qx,... Qm be m distinct minimal prime ideals that are contained in P. Then
m-L1 Qj C P is not maximal, hence A is not m-quasinormal.
(2) = (3) : Let P be a prime f-ideal containing Esm!_ af. Then ai C P for every
j. Therefore, aj V O(P) for all j, and hence P contains at least m minimal prime
ideals by the pigeonhole principle. Thus condition (2) gives us that P is a maximal
f-ideal.
(3) (4) : Follows from Lemma 5.1.3.
(3) =o (1) : Let M be a maximal f-ideal and let Q,..., Q C M be minimal prime
ideals. Then by hypothesis, E~j= Qj is a prime -ideal and we are left to show
that it is a maximal -ideal. For each j = 1,... m, let as E Q+ \ UiqQi and define
bj = Aisjai--A^=a E YEj Qj. Then {bj)} = is a pairwise disjoint set of m distinct
elements of E~ 1 Qj and by the choice of the aj's, we have that Allak E nfm=lQk
by convexity and Ai,6ai V Qj for each j. Hence bj V Qj and bf 9 Qj, for each j.
Thus ESm= Qj is a maximal f-ideal by condition (3), since E'=l bj C jE I gj

The quasinormal condition is a variation of the normal condition, which is
that the sum of any two minimal prime ideals of a semiprime f-ring with identity
is the entire f-ring. This is discussed in [Hu]. The expected generalized definition

follows, along with a theorem recording two equivalent conditions. The result is a

special case of Theorem 5.1.5 and the proof follows immediately from the one above
and from [Hu, Theorem 8].

Definition 5.1.6. Let m > 2 be a positive integer. An f-ring A is m-normal if for
any pairwise disjoint family {aj}m1, we have that A = E=J= a .

Theorem 5.1.7. Let A be a commutative semiprime f-ring with identity and let
m > 2 be a positive integer. The following are equivalent:

1. A is m-normal.

2. For any maximal -ideal M, we have that rk(M) < m 1.

3. The sum of any m distinct minimal prime ideals is A.

Before we move to describe Fm-spaces, we first discuss a special class of
m-quasinormal f-rings.

Definition 5.1.8. Let A be a local f-ring. An embedded prime e-ideal P is high
if for every minimal prime ideal N E Min(A), either N C P or N V P = lex(A).
Otherwise, P is low. Call an f-ring A a broom ring if for every maximal -ideal M
every prime t-ideal in AM is high.

The following is immediate from Proposition 2.5.2 and Lemma 5.1.3.

Proposition 5.1.9. Let A be a local commutative semiprime f-ring with identity
and maximal -ideal M. The following are equivalent:

1. A is a broom ring.

2. r(A) < 2.

3. If P lex(A) is a prime I-ideal, then rkA(P) < 1.

4. If a, b E A are disjoint and P is a proper prime -ideal containing a + bL,
then lex(A) C P C M.

5. If a, b E A are disjoint and a1 + bV C lex(A), then for every p E lex(A), there
exists z 4 lex(A) such that zp E a1 + b-.

Proof: (1) < (2) : Since every prime f-ideal of A is high, we have that for any
two distinct minimal prime ideals Q, Q2 that Q, V Q2 = lex(A). Hence 7r(A) < 2
by definition. Conversely, 7r(A) < 2 implies that every minimal prime ideal is high
and therefore that every prime f-ideal is high, as desired.
(2) 4 (3): Since 7r(A) < 2, we have that rk(P) < 1 for any embedded prime f-ideal
P, by Proposition 2.5.2; and vice versa.
(2) =- (4) : By Proposition 2.5.2, we have that al + b- = lex(A).
(4) > (5): Follows from Lemma 5.1.3.
(4) = (3) : Assume that (4) holds. Let P be an embedded prime -ideal and assume,
by way of contradiction, that rk(P) > 2. Let Qi, Q2 C P be minimal prime ideals.
Let qi E Qf \ Q2 and let q2 E Q+ \ Q1. Disjointify by defining f = qj q^ A q2
for j = 1,2. Then i,' C Q~ since ji f Q2 and q4- C Qi since V Q1. Then
'1 + 41 C P and hence lex(A) C P, which contradicts that P is embedded. *

Example 5.1.10. We now present an example of an f-ring which is 3-quasinormal
but is not a broom ring. Let Ao = A = [0, oo) C R, and A2 = A3 = (1,oo) C R,
where each is inversely ordered. Identifying the copies of 0 in the disjoint union
A1 = (Ao II Ai)/(0o ~ 01) and letting A2 be the disjoint union A2 II A, we obtain
a root system A = A1t A2 with the induced ordering which we now describe. That
is, r < s if and only if either r, s e Aj for j = 0, 1, 2 or 3 and r < s in the inversely
ordered real numbers; or if r E A2, s E A1; or if r E A3, s E At. Explicitly, r |I s if
r E A2 and s Ao I A3 or if r E At and s E Ao.

We endow A with a partial binary operation. To begin, note that we define

Oo + o0 = 01 + 01 = + 01 = Oo ~ 01. Let rj, sj E Aj be nonzero for j = 0,1,2,3.

Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in

Aj; the mark "-" signifies that the sum is undefined.

+ ro 8o ri a1 r2 82 rs3 s
ro (2roo o +o)o -
so (ro + ) (2o)o -
rl (2ri)i ( 1+rl)i (r2+rl)2 (a2+ r)2 (r3+r r) (3s+r)3
si (rl + i) (281) (r2 + 1) (82 + (r2 + +)s (8s+ )s3
r2 (rl +r2)2 (i +r2)2 (2r2)2 (2 + r2)2 -
2 (r + 2)2 (81 +82)2 (r2 + ) (282)2 -
rs (ri + ra)s (8 + rs3) (2rs)3 (83 + rs)
83 (ri + s)s (81 +3)3 (rs + 83)3 (2ss)

Let Co = Ao, Ci = Ai,C2 = A II A2, and C3 = A I A3. Then the minimal prime

ideals of F(A, R) are of the form Qj = {u E F(A, R) : supp(u) C A \ C} for

j = 0,2,3. The similarly defined Qi is a prime ideal by Proposition 3.3.15. Now,

it is evident that L = lex(F(A, R)) = Qo V Q2 V Q3 and Q2 V Q3 = Q1 # L so we

know that 7r(F(A, R)) = 3. Therefore, Proposition 5.1.9 shows that the ring is not

a broom ring. Since L is the maximal ideal of the ring, we have shown that F(A, R)

is 3-quasinormal. O

We present a similar example of an broom ring that is not quasinormal.

Example 5.1.11. Let Ao = [0, oo) C R, and A, = A2 = (, oo) C R, where each

is inversely ordered. Let A' = Ao and let A2 be the disjoint union A II A2 in

order to obtain a root system A = Al A2 with the induced ordering which we now

describe. That is, r < s if and only if either r, s E Ai for j = 0, 1 or 2 and r < s

in the inversely ordered real numbers; or if r E A,,s E Ao; or if r E A2,s E Ao.

Explicitly, r |I s if r E Ai and s E A2.

We endow A with a partial binary operation. Let rj, sj E A, for j = 0, 1, 2.

Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in

AI; the mark "-" signifies that the sum is undefined.

+ ro 80 1r s81 2 82
ro (2ro)o (0o+ro)o (ri +ro)i (a +ro)i (r2 + )2 (82 + r)2
so (ro + oo) (20o)o (r +o)1o (l + 8o) (12 +o)2 (s +so)2
rt (ro+rl)l (so+r1) (2r)i (I1 + rl)l -
1I (ro+sl) (so+Ish) (r1 +ai (2ai) -
r2 (ro+r-)2 8(o+r2)2 (2a2)2 (2 +,2)2
2 (tro+ 2)2 (0+8)2 (r2 +2)2 (202)2

Let Co = Ao, C = A0 II A1 and C2 = Ao I A2. Then the minimal prime ideals of

F(A,R) are of the form Qj = {u E F(A,R) : supp(u) C A \ Cj} for j = 1,2; the
ideal Qo, is a prime ideal by Proposition 3.3.15. Now, Qo = lex(F(A,R)) = Qi VQ2

and so we know that ir(F(A, R)) = 2. Therefore, Proposition 5.1.9 shows that the
ring is a broom ring. Since Qo is not the maximal ideal of the ring, we have shown

that F(A, R) is not quasinormal. ]

5.2 (B. m)-Boundarv Conditions

Definition 5.2.1. Let m be an integer greater than 1 and let {Uj} =1 be a family of
m pairwise disjoint cozerosets of a topological space X. The subspace fl= dlx(Uj)
is called an m-boundary in X. Let B be a topological property. We say that a space

X satisfies the (B, m)-boundary condition if every m-boundary in X has property
B.

In [Lal, 3.5], Larson proves that if X is completely regular, every point of X

is a Ga-point, and C(X) is quasinormal, then X satisfies the (discrete, 2)-boundary
condition. This result is improved in [La3, 3.5] to say that if X is normal and for
every p E jX \ vX, the I-ideal OP is prime then C(X) is quasinormal if and only if

X has the (finite, 2)-boundary condition. Here we refine this theorem by removing
the restriction on the points of the corona.
First, a lemma, extending [La3, 3.1], which we henceforth refer to as "Larson's
Lemma":

Lemma 5.2.2. Let X be normal and let {gj} l C C(X)+ be a family of pairwise
disjoint functions. Define Ym = nLm= clx(coz(gj)). Then Eti- gf = rE M,.

Proof: Each function in =1 gj- must vanish on Y,, hence im1 g- C nev M,.
For the reverse inclusion, we use recursion. The proof of the base case of m = 2 is
in [La3, 3.1].
Let f E nrYn Mv and define

f (x) = f A 1 if E cl(coz(gl)),
o0 if xE n7,=2l(coz(g)).
Since X is normal and fi is defined on a closed set, the function has a continuous
extension, J1 E C(X). Then fAl-f1 E gjL and ft E n{Mp: p n.=2 cd(coz(gj))}.
Recursively define a function fk to be the continuous extension of

f() = I^0-i A 1 if xE cl(coz(gk))
t0 if x E nfk+l (coz(g))
Then fk- A 1 e- E g- and f E n{M, : p E ni'k+1, c(coz(gj))}. In particular, by
the base case, we have that fm-2 E ({Mp : p E n m-,,l1 oz(g))} = g9m-1 + g.m
But then fn-3 A 1 fm-2 E 9gL-2 implies that -3 A 1 E j m-2 g and therefore
we have that m-3 = (7m-3 A 1)(7m-3 V 1) E r-,-2g -. Thus, by recursion, we
deduce that f1 E E-, gj Hence fAl = (fAl-f1 )+1, E m=gf and therefore,
f = (f A )(f V 1) E Em=! gi, as desired. *

Let X be normal and let Y,, as above, be given. The set Ym is C-embedded
in X by [GJ, 3D]. Thus we have a surjective ring homomorphism p : C(X) -+ C(Yn)
given by restriction of functions. The kernel of the homomorphism is
m
K = {f E C(X): Ym C Zx(f)} = N M= gf,
yEYm j=l
by Larson's Lemma. Thus by the First Isomorphism Theorem, it follows that

C(Ym) C(X)/K = C(X)/l gf.
j=1
We utilize the one-to-one correspondence between the prime ideals of C(Y,) and
prime ideals of C(X) which contain K, the kernel of p.

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PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS By CHAWNE MONIQUE KIMBER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999

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This work is dedicated to those women who preceded me and to those who are yet to follow.

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ACKNOWLEDGMENTS First and foremost, I express my wholehearted gratitude to my advisor, Jorge Martinez. In the past few years, his humane guidance has helped me to achieve so very much, in fact, more than I would ever have hoped. I follow his example both in becoming a mathematician and a caring teacher, and in the enjoyment of the finer things in life like wine, cheese, and chocolate. Also, sincere thanks go to my committee members: Richard Crew, for show ing me some algebra; Alexander Dranishnikov, for teaching me a heap of topology; Scott McCullough, for introducing me to real analysis (back when we were both much younger); and Mildred Hill-Lubin, for expanding my world-view through lit erature. Cheers and warm hugs to my friends, neighbors, and family, especially to the immediate: Johnnie, Charles, Frances, Maribell, Chinene, Jean, and the inimitable Poopy-girl, Cei lll

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TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT CHAPTERS 1 PRELIMINARJES . . . . . . . 1.1 History . . . . . . . . 1.2 Lattice-Ordered Groups . 1.3 /-llings .......... 1.4 llings of Continuous Functions 1.5 Approaches . 2 CHARACTERS .............. 2.1 Hahn Groups .......... 2.2 Lex Kernels and Ramification . 2.3 Rank .............. 2.4 Rank via z#-Irreducible Surjections 2.5 Prime Character . . . . 2.6 Filet Character . . . . . 3 GENERALIZED SEMIGROUP RJNGS 3.1 Specially Multiplicative /-llings 3.2 r-Systems and -Systems . . 3.3 /-Systems . . . . . .. 3.4 Survaluation lling and n th -Root Closed Conditions 4 RAMIFIED PRIME IDEALS ............ 4.1 Ramified Points . . . . . . . 4.2 Ramified G 6 -points ........... 4.3 Ramification via C-Embedded Subspaces 5 m-QUASINORMAL /-RINGS .......... 5.1 Definitions . . . . . . . . 5.2 (B, m) Boundary Conditions ..... 5.3 (3X, m-Quasinormal and SV Conditions REFERENCES ....... BIOGRAPHICAL SKETCH. iv iii V 1 1 3 7 10 15 18 18 20 26 32 36 43 46 46 52 58 66 75 75 79 84 88 88 94 107 110 113

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRJME IDEALS IN RJNGS OF CONTINUOUS FUNCTIONS By Chairman: Jorge Martinez Chawne Monique Kimber May 1999 Major Department: Mathematics Given a completely regular topological space X, we wish to determine the order structure of < Spec ( C ( X)), >, the root system of prime ideals of the ring of real-valued continuous functions on X; and vice versa. We present four approaches which give partial solutions to these problems. First, we define three measures on< Spec+(G), ~>, the set of prime subgroups of a lattice-ordered group, which determine some arithmetic properties of the group, and vice versa. Second, given any root system, we construct a generalized semigroup ring 'R which is a commutative semiprime f-ring such that < f('R), ~>, its root system of values, is order-isomorphic to the given root system. Then we characterize those non-isolated G 0 -points whose corresponding maximal ideal is the sum of the minimal prime ideals it contains. Finally, we characterize those spaces X for which C(X) has the property that the sum of any m minimal prime ideals is a maximal ideal or the entire ring. V

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CHAPTER 1 PRELIMINARIES The focus of this dissertation is the order structure of < Spec(C(X)), ~~;>, the spectrum of prime ideals of the ring C(X) of real-valued continuous functions on a topological space, X. To start, we informally present the history and give motivation for the discussion herein. We then review some essentials about lattice-ordered groups, /-rings, and rings of continuous functions in detail and then formally indicate the manner in which this thesis proceeds. 1.1 History Our history begins with the independent research by Cech and Stone in 1937 (see the papers [Ce) and [St]), in which they describe a compactification {JX of a topological space X which has the property that every real-valued continuous func tion on X extends to a continuous function on {JX. Further, via {JX, they establish correspondences between the topological structure of X and certain algebraic prop erties of its ring C(X) of real-valued continuous functions under pointwise addition and multiplication. For instance, Stone shows that the maximal ideals of the sub ring C*(X) of bounded functions are in one-to-one correspondence with points of {JX. The map p 1-t M*P = {! E C*(X) : /(p) = O} witnesses this correspon dence and is a homeomorphism of topological spaces when the set of maximal ideals of the ring is endowed with the hull-kernel (Zariski) topology. In particular, this shows that for compact Hausdorff spaces X, Y, we have that X ~ Y if and only if C(X) C(Y). The next significant result came in 1939 when Gelfand and Kolo mogoroff proved in [GK) that the maximal ideals of C(X) are exactly those of the 1

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2 form MP = {f E C(X) : p E cl13xZ(f)}, where Z(f) = {p E X : f(p) = O} and p E {3X. In the 1950's, Gillman, Henriksen, Jerison and Kohls began a formal in vestigation of topological/algebraic correspondences of this form. The elementary techniques and results are recorded in the text Rings of Continuous Functions, [GJ]. Concerning prime ideals, in [GJ, 14.3c] we learn that the prime ideals in C(X), con taining a given prime ideal, form a chain and in [GJ, 4J], it is shown that the topology on X is closed under countable intersections (i.e., X is a P-space) if and only if every prime ideal of C(X) is maximal. More generally, in [GJ, 14.25] we find that X has the property that every bounded continuous function on set of the form X \ Z(f), for some f E C(X), extends to a continuous function on X (that is, Xis an F-space) if and only if every maximal ideal of C(X) contains a unique minimal prime ideal. Knowing these three facts, we can describe the graph of the prime ideal spectrum of C ( X) in each case, where vertices are prime ideals and edges indicate set-inclusion. In the F-space situation, the graph is a disjoint set of strands (one for each point of {3X) with no branching; a P-space yields a graph consisting solely of vertices (one for each point of {3X). It is from these topological characterizations of the graphical structure of the spectrum of prime ideals of C(X) that we formulate our questions. Roughly speaking, we wish to know: Is it possible to determine the order structure (under inclusion) of the prime ideal spectrum of the ring of continuous functions of a given topological space? Conversely, given a graph, is it possible to construct a topological space such that the given graph, in some sense, determines the structure of the spectrum of its ring of continuous functions?

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3 It turns out that both of these are rather ambitious pursuits and the ques tions must be refined before we can approach them. Extending the knowledge of properties of the prime ideals of C(X), Kohls published a series of papers ([Kl), [K2], and [K3]) in 1957. In [K2], he addresses the properties of chains of prime ideals of C(X). First he shows that the quotient ring C(X)/ Pis totallyordered for every prime ideal P and concludes that the prime ideals of the quotient ring form a chain. Second, it is demonstrated that if Pis nonmaximal, then the chain of prime ideals in C ( X) / P contains an 1/l -set ( that is, a totally ordered set E such that for every pair of disjoint countable subsets A, B C E such that A < B, there exists c EE such that a< c and c < b for every a EA and every b EB). Hence, the chain of prime ideals contains at least 2N 1 primes. We may thus immediately reduce the second of our questions to only consider those graphs for which each nontrivial edge passes through an 1/l -set of vertices. The facts presented in the next three sections show us that the class of graphs to consider can be further reduced. In this dissertation, we continue to refine the questions and present four perspectives ranging from the very general to the very specific which give partial results. In order to properly introduce these approaches, we must recall some facts and constructs which are fundamental to the ensuing investigation. 1.2 Lattice-Ordered Groups Let ( L, ~) be a partially ordered set. If a, b E L are incomparable, then we write a II b. L is totally ordered if any two elements are comparable. We say that L is a lattice if any two elements a, b E L have a least upper bound and a greatest lower bound, denoted a V b and a I\ b, respectively. A lattice L is distributive if a I\ (b V c) = (a Vb) I\ (a V c), and dually for all a, b, c EL. A group (G, +, 0, ~) with partial order is a lattice-ordered group (hence forth, -group) if it is a lattice and if g h implies that c+g c+h and g+c h+c

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4 for all c E G. The majority of the groups we consider are abelian, so the additive notation here is for convenience. It is important to note that any -group is torsion free [D, 3.5] and its lattice is distributive [D, 3.17]. A {real) vector lattice is an -group G which is also an JR-vector space such that rg 0 for all positive g E G and for all positive r E JR. By a+ we mean the set of elements g E G such that g 0. Each element of G may be written as a difference of elements of a+ : let g+ = g VO and g= (-g) V 0, then g = g+ g -. This follows from the fact that g+ I\ g = 0. The absolute value of an element is given by jgj = g+ + g-. In general, we say that a pair of elements g, h E Gare disjoint if g I\ h = 0. We will write g h if ng < h for all n EN. An -homomorphism is a group homomorphism that also preserves the lattice structure. An -subgroup H of an -group G is a subgroup which is also a sublattice of (G, ~). We call an -subgroup convex if O g h EH implies that g EH G(S) denotes the convex -subgroup of G generated by the set S G. When S = {g}, we write G(g). In fact, G(g) = {h E G : 3n E N, lhl njgj}. In the special case that G (g) = G, we call g a strong (order) unit. Let
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5 ..l, and V~ 1 Hi = (UiEI Hi)1-1-, where { HihEI s.Jl( G) and J is any indexing set, we have that q:l(G) is a Boolean lattice, by (D, 13.7]. The convex -subgroups of greatest interest to us are the prime subgroups of G. These are the subgroups HE
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6 have P = U{g 1. : g (j. P}. This implies O(Q) def n{P E Min+(G) : P Q} is the set U{g1.: g (j. Q}, for a prime subgroup Q G, by [BKW, 3.4.12). A basis for an l-group G is a maximal pairwise disjoint set {9ihe1 a+ such that for each i E J the set {g E a+ : g 9i} is totally ordered The following is Conrad's Finite Basis Theorem presented as [D, 46.12) and [C, 2.47). It will figure in our discussion in the next chapter. Theorem 1.2.1. Let G be an l group. The following are equivalent: 1. G has a finite basis. 2. Min+ ( G) is finite. S. s.JJ( G) is finite. 4. There is a finite upper bound on the number of pairwise disjoint elements of G. 5. There is a finite upper bound on the number of elements of strictly increasing chains of proper polars. Another application of Zorn's Lemma establishes the existence of convex subgroups which are maximal with respect to not containing a fixed element g E G. Any such subgroup is generally termed a regular subgroup and specifically called a value of g. The set of all regular subgroups of G is usually represented by f(G). Regular subgroups are prime, by [D, 10.4], and a prime subgroup is precisely a convex -subgroup which is an intersection of a chain of regular subgroups, [D, 10.8]. In particular, the minimal prime subgroups of G correspond to the maximal chains in r(G). For these reasons, we call the root system given by r(G) the skeleton of Spec+(G). By convention, we view r(G) as a partially-ordered indexing set r whose

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7 elements are denoted by lower case Greek letters and then represent the regular subgroups by V-y for 'YE r. Topologize Spec+ ( G) using the hull-kernel (or Zariski) topology whose open base is given by U(g) = {P E Spec+(G) : g ff. P} for all g E G. In this topology, Spec+(G) is Hausdorff if and only if Spec+(G)=Min+(G) by [CM, 1.4); on the other hand, in the subspace topology, Min+(G) is always Hausdorff and U(g) n Min+(G) is both open and closed for every g E G. The space Spec+ ( G) is compact if and only if G has a strong unit, by [CM, 1.3); it is demonstrated in [CM, 2.2) that Min+(G) is compact if and only if G is complemented, that is, if and only if for every g E G+ there is an h E G+ such that g A h = 0 and g V h is a weak unit. 1.3 /-Rings Let (R, +, :::;) be a ring whose underlying group is an -group and satisfies the relations re :::; sc and er :::; cs whenever r :::; s and c 2:: 0. Such a ring is a lattice ordered ring (abbreviated l-ring). Han -ring R also satisfies ca Ab= ac Ab= 0 whenever a I\ b = 0 and c 2:: 0, then R is called an f-ring. The following is found in [BKW, 9.1.2): Theorem 1.3.1. Let R be an l-ring, then the following are equivalent: 1. R is an f-ring. 2. Every polar in R is an ideal. S. Every minimal prime subgroup of R is an ideal. It is not difficult to verify that every -ring which is -isomorphic to a subdi rect product of totally-ordered rings ( with coordinatewise operations) is an f-ring. In [BP), it is shown that the converse of this statement holds when we assume the Axiom of Choice (abbreviated, AC). Since we routinely apply AC, let us formally

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8 state that we will work within the axioms of ZFC. Then we may use this equiv alent definition of an f-ring R in order to obtain this list of arithmetic properties given in [BKW, 9.1.10], for a, b, c ER: 1. If c > 0 then c(a Vb)= ca V cb and (a V b)c = ac V be. 2. If c > 0 then c(a I\ b) = ca I\ cb and (a I\ b)c = ac I\ be. 3. lal lbl = labl 4. If a I\ b = 0 then ab = 0. 5. a 2 0. An -ideal of an -ring R is an ideal which is a convex -subgroup of R. We call an -ideal a prime -ideal if it is also a prime ideal. Let Spec( R) denote the space of all prime ideals of R in the hull-kernel toplogy. Let Max(R) and Min(R) denote the subspaces of maximal and minimal prime ideals, respectively. By property (4) above, we see that prime -ideals of an f-ring are prime subgroups, hence, as in the case of -groups, the subset Spect(R) of prime -ideals forms a root system. Denote the subspaces of maximal and minimal prime -ideals by Maxt(R) and Mint(R), respectively. We call a commutative ring semiprime if it contains no nonzero nilpotent elements. In the case of commutative !-rings, we have [BKW, 9.3.1]: Theorem 1.3.2. Let R be a commutative !-ring, then the following are equivalent: 1. R is semiprime. 2. For any a, b ER, we have that lal /\ lbl = 0 if and only if ab= 0. 3. Every polar of R is an -ideal which is an intersection of prime ideals.

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9 4Min,(R) = Min(R) 5. R is -isomorphic to a subdirect product of totally-ordered integral domains. We say that an -ring R with multiplicative identity, 1, has the bounded inversion property if a 1 implies that a is a multiplicative unit. By [HIJo, 1.1], a commutative /-ring R with 1 has the bounded inversion property if and only if Max,(R) = Max(R). Let A be any commutative ring. Then, in the hull-kernel topology, Min(A) is a Hausdorff space with a base of clopen sets. If A is a semiprime ring, then Spec(A) is Hausdorff if and only if Min(A) =Max(A); this occurs if and only if A is von Neumann regular (or absolutely flat,) i.e., for every a E A there exists b E A such that a= a 2 b), see [AM, p. 35]. In [HJ] it is demonstrated that if A is a semiprime /-ring, then Min(A) is compact if and only if A is complemented (i.e., for every a EA there exists b EA such that ab= 0 and a+ bis not a zero-divisor) Max(A) is compact for any commutative ring A with identity, and if A is a commutative f-ring with identity which has the bounded inversion property, then the subspace is Hausdorff, see [HJo, 2.3]. Let A be a commutative ring with identity and PE Spec(A). Define O(P) = {a EA: 3b (j. P, ab= O}. If A is also a semiprime f-ring, then this is the same as the -subgroup O(P) defined in the previous section. Recall that the localization of A at Pis the subring, Ap, of the classical ring of quotients of A/O(P) consisting of the elements whose denomi nator is not in P/O(P). (For a review of this construction and general facts about localizations, see [AM] or [G]). It is the case that Ap is a local ring whose unique maximal ideal is generated by P/O(P) and there is a one-to-one correspondence between prime ideals of Ap and the prime ideals Q of A such that O(P) Q P.

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10 Thus, if A is also an /-ring with bounded inversion, then by the root system struc ture of Spec,(A), we have that AM A/O(M) since the quotient ring is already local with unique maximal ideal M/O(M). 1.4 Rings of Continuous Functions Let X be a Hausdorff topological space. X is called completely regular (or Tychonoff) if for every closed set A~ X and x EX\ A, there exists a real-valued continuous function on X such that J(x) = 1 and J(A) = {O}. Unless otherwise stated, we assume that all spaces are completely regular. Let C(X) denote the set of real-valued continuous functions on a space X. Under the operations of pointwise addition and multiplication, C(X) is a semiprime ring. Order the ring via: / g if and only if f(x) g(x) for all x E X. This ordering gives an /-ring structure such that C(X) has the bounded inversion property. Let C*(X) denote the convex l-subring of bounded functions. The zeroset of/, is the set Z(J) = {x EX : f(x) = 0}. The complement, coz(J) = X \ Z(J), is the cozeroset off. By (GJ, 3.6], a Hausdorff space X is completely regular if and only if its topology is the same as the weak topology generated by C(X). Equivalently, the set of all zerosets, Z(X), is a base for the closed sets of such a space, (GJ, 3.2]. Sets A, B X are completely separated if there exists f E C(X) such that J(A) = {O} and J(B) = {l}. If for every f E C(A) there exists TE C(X) such that jjA = f, then we say that A is C-embeddedin X. Likewise, A is C*-embeddedin X if bounded continuous functions on A extend to bounded continuous functions on X. These embedding properties are characterized by complete separation of particular subsets, as follows: 1. Urysohn Extension Theorem (GJ, 1.17]: A~ Xis C*-embedded in X if and only if any two completely separated sets in A are completely separated in X.

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11 2. A C* -embedded set is C-embedded if and only if it is completely separated from every zeroset disjoint from it, [GJ, 1.18]. Recall that a Hausdorff topological space X is normal if any two disjoint closed sets are separated by disjoint open sets. Assuming this stronger separation axiom, the results listed above give rise to the theorem stated in [GJ, 3D], in which the equivalence of the first two statements is known as Urysohn's Lemma. Theorem 1.4.1. Let X be Hausdorff. The following are equivalent: 1. X is normal. 2. Any two disjoint closed sets of X are completely separated. 9. Every closed set of X is C*-embedded in X. 4. Every closed set of X is C-embedded in X. For many reasons, it is often preferable to work with compact spaces. The Stone-Cech compactification {3X is our compactification of choice, since {3X is char acterized by the property that it is (up to homeomorphism) the unique compact space in which X is dense and C* -embedded. There are at least three different ways to construct {3X, we begin with the one based on ultrafilters, described in detail in Chapter 6 of [GJ], which we now summarize. Let X be a completely regular space, let C be a subset of the power set of X and :F C. :Fis a C-filter if 0 rt :F, it is closed under finite intersections and if for every F E :F, the fact that F F' E C implies that F' E :F. If :Fis a Z(X) filter, then :F is also called a z-filter. A maximal filter is an ultrafilter; similarly, a z-ultrafilter is a maximal z-:6.lter. Let {3X be the set of all z-ultrafilters on X which we index by { AP : p E /3X}. A closed base for the topology on {3X is given by sets of the form Z = {p E {3X: Z E AP}, for Z E Z(X). Let p E {3X and define MP= {f E C(X) : p E clpxZ(f)}.

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12 The theorem of Gelfand and Kolmogoroff [GK] is stated simply as: Theorem 1.4.2. For a completely regular space X, the set Max(C(X)) is given by {MP: p E ,BX}. In fact, this result gives rise to a homeomorphism of ,BX with Max(C(X)). That is, since the sets Z[MP] = { Z(f) : f E MP} are precisely the z-ultrafilters on X, by [GJ, 2.5], and Theorem 1.4.2 shows that the map pi-+ Z[MP] is the desired correspondence. If p E X, then we will write Mp and, in this case, the maximal ideal and corresponding z-ultrafilter are called fixed. Otherwise, a maximal ideal and its corresponding z-ultrafilter is called free. It is evident that X is compact if and only if every maximal ideal of C(X) is fixed. Maximal ideals are also classified by the residue field C(X)/MP. Identifying the constant functions with their constant, we see that these fields always contain a copy of JR. We call a maximal ideal real if the field is exactly R; otherwise, the maximal ideal is called hyper-real. This concept is the basis for considering the Hewitt realcompactification of X. Denoted vX, it is the smallest subspace of ,BX in which X is dense and such that every maximal ideal of C(vX) is real. In fact, by [GJ, 8.5], vX is the largest subspace of ,BX in which X is C-embedded. With these facts about the maximal ideals firmly in place, we now proceed to consider the nonmaximal prime ideals. We know that every prime ideal of C(X) is convex, by [GJ, 5.5]; so we deduce that Spec(C(X)) is a root system. In order to understand this root system, we are required to consider the properties of other ideals. For instance, for p E ,BX, the ideals of the form QP = O(MP) = {J E C(X) : dpxZ(f) is a neighborhood of p} are of paramount interest when examining the prime ideals of C ( X). One reason is given in [GJ, 7.15]:

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13 Theorem 1.4.3. Every prime ideal P in C(X) contains OP for a unique p E {3X and MP is the unique maximal ideal containing P. If OP is prime, then we call pan F-point. If X has the property that OP is a prime ideal for every p E X, then we call X an F-space. We see that in this case, the graph of Spec(C(X)) consists of a set of strands with no branches. Note [GJ, 14.25]: Theorem 1.4.4. Let X be completely regular. The following are equivalent: 1. X is an F -space. 2. {3X is an F-space. 9. The prime ideals contained in any given maximal ideal form a chain. 4. Every cozeroset of X is C* -embedded. 5. Any two disjoint cozerosets of X are completely separated. 6. Every ideal of C(X) is convex. 7. Every finitely generated ideal of C(X) is principal {i.e., C(X) is Bezout) A special case of an F-point is when Op = Mp and we call pa P-point if this occurs. Call X a P-space if every point of X is a P-point. In this case, the spectrum of C(X) consists only of vertices. Equivalent definitions of P-space are presented in [GJ, 14.29] and are recorded below. First, recall that an ideal J of C(X) is called a z-ideal if f E J and Z(/) = Z(g) implies that g E J. It is immediate from the definitions that MP and OP are z-ideals for all p E {3X. Note that not all prime ideals are z-ideals; however, the following says that this is the case in a von Neumann regular ring.

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Theorem 1.4.5. Let X be completely regular. The following are equivalent: 1. X is an P-space. 2. vX is an P-space. S Every prime ideal of C(X) is maximal ,4.. Every cozeroset of X is C-embedded. 5. For each f E C(X), the zeroset Z(f) is open. 6. Every ideal of C(X) is a z-ideal. 14 7. For every f E C(X), there exists g E C(X) such that f = g/2 {that is, C(X) is von Neumann regular). We now recall the definitions of other types of spaces which are useful to us. X is basically disconnected if the closure of any cozeroset is clopen. X is extremally disconnected if any open set has open closure. Discrete spaces are extremally dis connected; extremally disconnected spaces are basically disconnected and all such spaces are F-spaces by [GJ, 14N.4). Every P-space is basically disconnected by [GJ, 4K.7). A space is a quasi-F space if every dense cozeroset is C*-embedded. Clearly, from [GJ, 14.25), we see that every Fspace is quasi-F. The converses of the pre ceding statements do not hold. That is, these are distinct classes of spaces, as we now illustrate. Example 1.4.6. Consider the following spaces: 1. Let U be a free ultrafilter on N. Let = N U {a}, in which points of N are isolated and neighborhoods of a are of the form U U {a}, where U EU. Then~ is an extremally disconnected subspace of (3N, but not a P-space. In particular, O
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15 2. Let D be an uncountable set. Let )..D = D U {)..}, where points of D are isolated and a neighborhood of ).. is given by any cocountable set containing it. Then )..Dis basically disconnected, but not extremally disconnected by [GJ, 4N.3]. Moreover, the topological sum X = >-.D II~ is basically disconnected, but neither extremally disconnected nor a P-space, by (GJ, 4N.4]. 3. The corona, {JN \ N is a quasi-F space which is an F-space, yet not basically disconnected; see [GJ, 6W.3, 140]. 1.5 Approaches Starting as generally as possible in Chapter 2, we define three cardinal-valued characters on the spectrum of prime subgroups of an -group. The value of each measure determines a portion of the arithmetic and/or polar structure of the group, and vice versa. For instance, we define the prime character, 1r( G) of an -group, G to be the least cardinal K such that for any family {Qa}a<" Min+(G), of distinct minimal prime subgroups, we have that Va<" Q 0 is the smallest convex -subgroup of G containing all the elements of Min+(G). Roughly speaking, it is a measure of the complexity of minimal paths in the graph of Spec+ ( G) between minimal prime subgroups. We will show that the measure being finite satisfies the following, where lex( G) denotes the smallest convex -subgroup of G containing all the elements of Min+ ( G) : Proposition 1.5.1. Let G be an l-group and m a positive integer. The following are equivalent: 1. 1r(G) = m < oo. 2. m is minimal with respect to the property that lex(G) = G(LJ; 1 a{) for any m pairwise disjoint positive elements, {a;}J!: 1 lex(G)+.

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16 S. m is minimal such that for any prime P lex(G), the chains of proper polars in P have length at most m 1. Chapter 3 is devoted to a discussion of the properties of F(fl, JR.), the gener alized semigroup ring of real-valued maps on a root system fl (which has a partially defined associative operation, +) each of whose support is the join of finitely many inversely well-ordered sets. The ring structure on this group is introduced in [Cl] and [C2]; we endow this ring with an f-ring structure. In particular, we show that if (fl,+) is a root system such that each of the following holds: 1. + is associative (when it makes sense); 2. if a, /3 E fl are comparable, then a+ /3, /3 + a are defined; 3. if a< /3 and a+ 'Y, /3 + 'Y are defined, then a+ 'Y < /3 + 'Y and if 'Y + a, 'Y + /3 are defined then 'Y +a< 'Y + /3; 4. and if is maximal, then 8 + + o, + are defined and 8 + = + o = o for every o :::; then F(fl, JR.) is an f-ring if and only if 8 = a + /3 implies a, /3 2'.: o. And when this occurs, the f-ring is semiprime and satisfies the bounded inversion property. Moreover, by [CHH, 6.1), given any root system fl, one of these !-rings has fl order-isomorphic to its root system of values. Thus, the second of our questions is answered in the class off-rings on the level of skeletons. However, the solution to the second problem remains unclear in the smaller class of rings of continuous functions. To gain a modicum of clarity on the situation, we look to the work of Attilio LeDonne, published in 1977 in [Le), in which he addresses the incidence of branching in the graph of Spec(C(X)). He shows, for instance, that the root system branches at every prime z-ideal when X is a metric space. In [Le, ), LeDonne includes a result of DeMarco which states that there

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17 is branching at each Mp when X is a first-countable space and pis non-isolated. In Chapter 4, we show that, for a non-isolated G 6 -point of a completely regular space, there is branching at Mp if and only if X \ p is not c -embedded in X. This result is then used to examine branching in Spec(C(X)) when Xis not necessarily first-countable. Both of our questions are addressed in Chapter 5, in which we generalize a few of the results of Suzanne Larson on quasinormal /-rings that are found in the series of papers [Lal], [La2], and [La3]. The semiprime commutative quasinormal /-rings with identity are the ones having the property that the graph of the root system of prime -ideals does not contain a subgraph of the form: (1.1) By (La3, 3.5] a normal space X has the property that C(X) is quasinormal if and only if cl(U) n cl(V) is a P-space for any disjoint cozerosets U, V X Our generalizations similarly describe those normal spaces X for which Spec(C(X)) does not contain a subgraph of any of the following forms: n n (1.2) where n, k, ai, ... an are positive integers satisfying some specified conditions.

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CHAPTER 2 CHARACTERS We seek a collection of measures on root systems whose values will determine some portion of the structure of a lattice-ordered group. In this chapter we describe three such measures: rank, prime character, and filet character. The rank measures the width of a connected component of the spectrum, the prime character determines, roughly speaking, the complexity of minimal paths between minimal primes, and the filet character counts the maximum length of a chain of branching incidences. The first sections of this chapter are a review of two constructs essential to the discussion to follow. 2.1 Hahn Groups To begin, we recall a method of constructing examples of -groups having a specified root system as the skeleton of its prime spectrum. Let 6.. be a root system and define V(6.., JR) = { v: 6.. JR: supp(v) has ACC}, where supp(v) = { 8 E 6..: v(8)-:/= O}. V(6.., JR) is an -group under pointwise addition ordered by the relation: v > 0 if and only if v ( 8) > 0 for every maximal element 8 E supp(v). This -group is called a Hahn group. In the paper of Conrad, Harvey and Holland [CHH], it is demonstrated that any abelian -group can be embedded in a Hahn group of a more general description than we give here. Of interest to us is the -subgroup of maps with finite support denoted by E(6.., JR) and the subgroup of maps whose support is the join of finitely many inversely well-ordered sets, denoted by F(6.., JR). Clearly, E(6.., JR) F(6.., JR). 18

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19 The proof of the first statement of Proposition 2.1.1 is analogous to that of Theorem 6.1 in [CHH). This establishes that r(E(.6., R)), f(F(.6., R)) and .6. are isomorphic as partially-ordered sets. For the sake of completeness, we present an elementary proof of this fact for the case of F(.6., R), although the result is easily obtained from the theory of finite-valued -groups. The proof is identical in the case of E(.6., R). Recall that an -group is finite-valued if each element has only a finite number of values. A special value is a prime subgroup which is the unique value of an element. An -group G is finite-valued if and only if every value of G is special and if and only if every element of G is a finite sum of pairwise disjoint special elements; for details, see [AF, 10.10). HG has a set S of special values such that Sis a filter and ns = { 0}, then G is called special-valued. Proposition 2.1.1. Let .6. be a root system. For each 8 E .6. define = {f E F(.6., JR) : v(,) = 0 when 'Y 8}. Each is a special value. Further, every value of an element of F(.6., JR) is of the form for some 8. Thus, .6. is the skeleton ofSpec+(F(.6.,R)) and F(.6.,JR) is finite-valued Proof: Let 8 E .6. and let X6 E F( .6., JR) be the characteristic function on { 8}. Then X6 0. H 8 < 'Y, then O < X6 < v, a contradiction. H 8 = ,, then there exists a positive integer n such that O < X6 nv and hence x 6 E V by convexity, which is a contradiction. Thus V = Let v E F(.6., R)+ and let V be a value of v. Let D be the finite set of maximal elements of supp( v). Then the characteristic function xn is not in V; else,

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20 there exists an integer n such that O v n xv, a contradiction. Since V is prime and the set {x 6 : o E D} is pairwise disjoint, there exists a unique element o E D such that X6 V. By the above, we know that V Finally, since v we have that V = Vo, as desired. The final statement follows from [AF, 10.10] since we have shown that every value is special. 2.2 Lex Kernels and Ramification Throughout we will describe the location of a prime subgroup in the graph in reference to a designated convex -subgroup, called the lex kernel of an -group G and denoted by lex(G). It is the least convex -subgroup containing all the minimal prime subgroups of G. It is always the case that lex(G) is a prime subgroup [D, 27.2] which is normal in G [D, 27.13]. The following is a summary of a part of the discussion of lex kernels in [D, ] and gives a description of the -subgroup in terms of its generators. Proposition 2.2.1. Let G be an -group and let C be a convex -subgroup. The following are equivalent: 1. C = lex(G). 2. C is the least prime subgroup such that if O < g C then g > h for every hE C. S. C is the convex -subgroup of G generated by {g E G: g II O}. 4. C = {O} U {g E G : 391, 92, E G, g II 91 II 92 II II Un II O}. 5. C is the convex -subgroup of G generated by the nonunits of G. 6. C is prime and is the smallest among all convex -subgroups of G which are comparable with every convex -subgroup of G.

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21 7. C is the maximal convex -subgroup of G such that lex( C) = C. 8. C is the supremum of the proper polars of G in the lattice of convex -subgroups ofG. It is natural to now introduce a concept which we will discuss in more detail in Chapter 4. This is a generalization of a concept from [Le]. Let A be a commutative ring with identity and for each a EA, let Max(a) ={ME Max(A): a EM}. Recall that an ideal I of A is a z-ideal if a E J and Max(a) = Max(b) imply that b E J Definition 2.2.2. Let A be a commutative !-ring with identity. A prime l.-ideal P is ramified if it is the sum of the minimal prime ideals that it contains. A maximal l.-ideal M is totally ramified if every prime z-ideal contained in M is ramified. A completely ramified ring is one in which every prime z-ideal is ramified. Graphically, a prime -ideal P :5 A is ramified if and only if it is minimal or if the root system of prime -ideals of A branches at P. We begin with the -group characterization of ramification. It is the case that a ramified maximal -ideal M of A is the lex kernel of the local /-ring A/O(M). In order to discuss a proper lex kernel in an -ring, A, we must operate inside a localization. Henceforth, we will obtain results for local rings and tacitly extend to the general case by referring to localizations. The following characterization of ramified maximal -ideals is immediate from Proposition 2.2.1. Corollary 2.2.3. Let A be a commutative semiprime local !-ring with identity and bounded inversion and let M be the maximal ideal. The following are equivalent: 1. M is ramified. 2. M is the convex -subgroup of A generated by {f EA:/ II O}.

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22 3. M = {O} U {/EA: 3/1, /2, fn EA, f II /1 II /2 II 11 fn II 0}. ,4.. M is the convex -subgroup of A generated by the set {/EA: 3g EA, g > 0, g I\ f = 0}. 5. M is the convex l-subgro'Up generated by the elements of A which are not order units. 6. M is the smallest among all convex -subgroups of A which are comparable with every convex -subgroup of A. 7. Mis the supremum of the proper polars of A in the lattice of convexl-subgroups of A. It is well-known that the lex kernel of an -group is a prime subgroup (see [D, 27.2]). We now show that the lex kernel of a commutative local semiprime /-ring with identity is an ideal and then give conditions which guarantee that the lex kernel is a prime ideal. Let A be a commutative /-ring with identity. Recall that an ideal J 5 A is pseudoprime if ab = 0 implies a E J or b E J. An ideal J 5 A is semiprime if a E J whenever a 2 E J. A is square-root closed if for any 0 5 a E A, there exists 0 5 b E A such that a = b2. Let a, b E A, then A is n-convex if whenever 0 5 a 5 bn, there exists u E A such that a = bu. Proposition 2.2.4. Let A be a commutative semiprime local /-ring with identity. Then lex(A) is a prime s'Ubgroup which is a pseudoprime -ideal. If, in addition, A is square-root closed, then lex(A) is a semiprime -ideal. Proof: Let f E lex(A). Then there exists g > 0 such that f I\ g = 0. If af = 0 then af E lex(A); else, af I\ g = 0 and we conclude again that af E lex(A). Hence lex(A) is an ideal.

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23 Let N be the set of nonunits of A and recall that lex(A) = A(N). Let ab = 0. If a or b is 0, then there isThen by convexity, we see that a+, a E A(N). Hence a E A(N) and we have that the lex kernel is pseudoprime. Since any prime ideal is semiprime and the lex kernel is the sum of the minimal prime ideals, the lex kernel is semiprime, if A is also square-root closed; see [HLMW, 2.12(d)]. Corollary 2.2.5. Let A be a commutative semiprime local f-ring with identity and bounded inversion and let M be the maximal -ideal. M is ramified if and only if lex(A) is a z-ideal. Proof: Since the maximal -ideal is the only z-ideal of a local f-ring, this is immediate. Corollary 2.2.6. If A is a commutative local 2-convex semiprime / ring with iden tity which is square-root closed, then the lex kernel of A is a prime -ideal. Proof: By the remark after [La4, 4.2], under these hypotheses, we have that an -ideal is a prime ideal if and only if it is pseudoprime and semiprime. For the remainder of this section, let G be an abelian lattice-ordered group. Recall the following for Han -subgroup of G His rigid in G if for every h EH there is g E G such that h.1.1 = g.1.1_ It is shown in [CM, 2.3] that if His rigid in G then the contraction of minimal prime subgroups of G to minimal prime subgroups of H is a homeomorphism of minimal prime spaces. If H E ~( G), then H is very large in G if it is not contained in any minimal prime subgroup of G. It is shown in [CM] that if HE ~(G) then His very large in G if and only if His rigid in G. It turns out that ramification in a rigid subring indicates global ramification and vice versa. This is a direct consequence of the lex kernel correspondence demonstrated below.

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24 We will also need the following facts (see [BKW, 2.4.7, 2.5.8]): Proposition 2.2.7. Let HE t(G). 1. The contraction map from the set of prime -subgroups of G not containing H to the set of prime -subgroups of H is an order-preserving bijection. 2. IfV is a value of h EH in G, then V HV n H is a bijection between the set of values of h in G and the set of values of h in H. Proposition 2.2.8. Let H G be a convex -subgroup. Assume that u E H is a weak unit of H and a weak unit of G. Let V be a value of u in G. Then we have that V n H = lex(H) if and only if V = lex(G). Proof: Assume that H P E Min+ ( G) then u E P and we have that u.1 c/:. P by [AF, 1.2.11). This is a contradiction since u.1 = 0 E P. Thus His rigid in G since it is a convex -subgroup which is very large in G. Assume that V n H is the lex kernel of H. Then V n H is the least convex -subgroup of H containing all the minimal prime subgroups of H. Since His rigid in G, by the bijection given in the first part of Proposition 2.2.7, all the minimal prime subgroups of G are contained in V, and V is the least such convex -subgroup of G. That is, if W V also contains the minimal prime subgroups of G, then W n H is a convex -subgroup of H containing all the minimal prime subgroups of Hand hence W n H = V n H. But this says that W n HE r(H) is a value of u and hence, WE r(G) is a value of u Therefore V = W If Vis the lex kernel of G then VnH contains all the minimal prime subgroups of H. Thus the lex kernel of H is contained in V n H. Let P c V n H be a prime -subgroup of H containing all the minimal prime subgroups of H. Then by Proposition 2.2. 7, there exists a prime convex -subgroup Q G not containing H such that P = Q n H and since we have a rigid embedding, Q contains all the

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25 minimal prime -subgroups of G. Hence, Q = V, P = V n H and V n H is the lex kernel of H. Let A be a commutative semiprime f-ring with identity and bounded in version. If M E Max(A) then AM is semiprime with bounded inversion. This is a result of the well-known facts that the -homomorphic image of an !-ring with bounded inversion has bounded inversion and that AM A/O(M); see the proof of [La3, 2.7]. Since we must localize an f-ring in order to have a proper lex kernel, the following allows application of Proposition 2.2.8 to f-rings. Proposition 2.2.9. Let B be a commutative semiprime !-ring with identity and let A be a rigid convex f-subring of B. If M E Maxt(A) is such that M = N n A for some NE Maxt(B), then AM is a rigid convex f-subring of BN, Proof: Recall that AM~ A/O(M) and BN ~ B/O(N). Define a map: A BN by a~ a+O(N). This map is an !-ring homomorphism. We show that the image is convex in BN and that the kernel is O(M). For a EA, let O b+O(N) a+O(N). Then there exists n E O(N)+ such that O b a+n. If b-n 0 then O < b n and hence b E O ( N) since O ( N) is convex in B. Thus we may assume that O b-n a. Then bn EA and b+ O(N) = bn + O(N) Elm(). Therefore the image of is convex in B N. The kernel of is O(N) nA. It is easy to show that O(N) nA O(M) since M = N n A. For the reverse inclusion, assume that a E A and a O(N). Since O(N) is the intersection of the minimal prime ideals of B contained in N, there exists P EMin(B) such that P ~Nanda P. By the rigidity of A in B, PnA EMin(A), and therefore a O(M). We now have that Ker() = O(N) n A = O(M) and therefore AM is a convex f-subring of BN. Since AM contains the identity element of B N, AM is very large in B N. For rigidity, we need only recall that very large convex embeddings are rigid, [CM].

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26 Corollary 2.2.10. Let B be a commutative semiprime !-ring with identity. Let A be a rigid convex f-subring of B. Let M E Maxt(A) be such that M = N n A for some NE Maxt(B). Then M is ramified in A if and only if N is ramified in B. Let A* denote the f-subringofbounded elements of the commutativesemiprime f-ring A with identity. Note that A* is convex and rigid in A. In his dissertation [Wo], Woodward proves the following fact: Theorem 2.2.11. Let A be a semiprime !-ring with identity and bounded inver sion. Let M be a maximal ideal of A and let M be the unique value of A* containing Mn A*. The map M M gives a homeomorphism between Max(A) and Max(A*). That is, Max{A*) is the subspace consisting of values of l in A. In particular, if M is a real maximal ideal of A, then Mn A E Max(A*). Corollary 2.2.12. Let A be a commutative semiprime !-ring with identity satisfy ing the bounded inversion property. Let ME Max(A) be real. Then M is ramified if and only if M = M n A is ramified in A. 2.3 Rank The first character on Spec+ ( G) that we consider is simply one which counts the minimal prime subgroups contained in a convex -subgroup. Definition 2.3.1. The rank, rkG(H) of a convex -subgroup H G is the cardi nality of the set of minimal prime subgroups of G contained in H. If that cardinal is not finite, then we will say that H has infinite rank; we may choose to specify the cardinal when its value is of significance in a discussion. If H is a minimal prime subgroup of G, then we define rkG(H) = 0. This is a variation of the following definition given in [HLMW): Let A be a commutative !-ring with identity and M a maximal -ideal of A. The rank of M, denoted rkA(M), is the cardinality of the subspace of minimal prime ideals of A

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27 contained in M. By convention, if the rank of Mis infinite and we don't necessarily care about the exact cardinality, we write rkA(M) = oo. The rank of a point p EX, rkx(p), is the rank of Mp. The rank of the /-ring A is the supremum of the ranks of the maximal -ideals of A, when it exists; the rank of a space X is the rank of C(X). We begin with illustrations of the extremal values of ranks. An -group is semiprojectable if for any g, h E G+, (g I\ h)1= G(g1U b1-). In [BKW, 7.5 1], it is proved that G is semiprojectable if and only if each prime subgroup contains a unique minimal prime subgroup, which is equivalent to rka(P) 1 for every PE Spec+(G). Thus, it is evident that a space Xis an F-space if and only if C(X) is semiprojectable which is equivalent to rk(C(X)) 1. In particular, Xis a P-space if and only if C(X) is von Neumann regular, which is equivalent to rk(C(X)) = 0. The one-point compactification of the natural numbers, aN, is an example of a space for which C(X) has infinite rank, [GJ, 14G]. In fact, if a is the point at infinity, then the maximal ideal corresponding to a contains 2' minimal prime ideals one for each free ultrafilter on N. Moreover, by [HJ, 4.8], this subspace of minimal prime ideals is homeomorphic to the corona, {JN \ N. Proposition 2.3.3 describes a general situation in which we have infinite rank. We recall some definitions. From [LZ, 39.1]: let G be a vector lattice, v Ea+, and let {gn}~=l G be a sequence. We say that the sequence converges relatively uniformly to g E G along the Te!]'Ulator v, and write gn g, if for every > 0 there exists NE> 0 such that for all n NE, we have that jg gnl V. The sequence is relatively uniformly Cauchy with respect to v if for every > 0 there exists NE > 0 such that for all n, m NE, we have that jgm gnl V. G is called uniformly complete if for every v E a+, every sequence which is relatively uniformly Cauchy with respect to v relatively uniformly converges along the regulator v.

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28 Lemma 2.3.2. Let G be a uni/ ormly complete vector lattice with weak order unit u Ea + For any set {g;};ew, there exists g E G such that g = n;ewYT Proposition 2.3.3. Let G be a uniformly complete complemented vector lattice with weak order unit u E a+. If for some Q E Spec+(G) we have rkG(Q) ;?: w, then Q contains at least 2' minimal prime subgroups Proof: Note that, by [CM, 2.2], Min+(G) is compact since G is complemented. Let P = { Pn}neN be a countably infinite set of minimal prime subgroups which are contained in Q which is discrete in the hull-kernel topology on Min+(G). We first show that P is c -embedded in Min+ ( G) and conclude that the minimal prime space contains an homeomorphic copy of {IN. Then we describe the elements of Min+ ( G) that correspond to the points in this copy of /JN \ N. Let A, B P be completely separated in P and index them by I, J N as A = { Ai : i E J} and B = { B; : j E J}. Fix i E J and let Ai E A. For each B; E B, Let 'i ; E A+\ B; and b; E Bf\ Ai. Then E n;eJ U(b;J def Ki and B U;eJ U(D.i ; ) = U(E;eJ 2-i('i; I\ u)) def Li. Then Ki, Li are disjoint closed sets in Min+(G). For each i E J, generate the disjoint pair Ki, Li. Then A~ cl(UieJKi) def K and B nie1Li def L. By Urysohn's Lemma, the disjoint closed sets K, L are completely separated in Min+(G) since Min+(G) is normal. Consequently, A, Bare completely separated in Min+ ( G) and therefore P is c -embedded in Min+ ( G) by the Urysohn Extension Theorem. Finally, by [GJ, 6.5], the closed subset of Min+(G) of minimal prime subgroups in Q contains an homeomorphic copy of /JN. Let Ube a free ultrafilter on N. For g E G, let N(g) = {n: g E Pn}Define a new prime subgroup P by g E P if and only if N(g) EU. We show that Pis a

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29 minimal prime subgroup. The following proof is the same as that for [HJ, 4.8] and for [HLMW, 4.1). Let g, h E P. Pis a subgroup since N(g h) ::) N(g) n N(h) EU implies g h E P by filter properties. By convexity and since the Pn are prime subgroups, N(gV h)::) N(g) EU and N(gAh)::) N(g) EU, we have that Pis a sublattice of G. Thus Pis an -subgroup of G. Let O g h E P. Then N(g) ::) N(h) EU since each Pn is convex, and thus Pis convex. Let gAh E P, then N(g)UN(h)::) N(gAh) EU implies that N(g) or N(h) is in U since U. Thus Pis a prime subgroup of G. Let g E P. Since Pn is a minimal prime subgroup for each n, we have that for each n E N(g), there exists an kn E G \ Pn such that g A kn= 0. By Lemma 2.3.2, there exists h E G such that hJ. = nneN(g) h;. Then g A h = 0 and h rJ_ Pn for all n E N(g) since hJ. h; Pn for each n E N(g). Thus N(g) n H(g) rJ_ U and hence hr/. P and Pis a minimal prime subgroup of G. Recall that a space X is cozero-complemented if for any cozeroset U X there exists a cozeroset V X such that U n V = 0 and U U V is dense in X. A concrete example of a maximal ideal of infinite rank is found in C(X) where X is cozero-complemented and first countable. DeMarco shows in [Le, ] that rk(Mp) 2'.: 2, for any nonisolated point of a first countable space (the result actually says more than this, and we will discuss this in Chapter 4). By modifying DeMarco's proof, we show that Mp contains infinitely many minimal prime ideals and hence has rank at least 2'. Proposition 2.3.4. Let X be first countable and let p E X be nonisolated. For every m E N there exists a family of m distinct prime ideals which sum to Mp. Moreover, if X is also cozero-complemented, then there exist at least 2' minimal prime ideals contained in Mp.

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30 Proof: Since p has a countable base, there exists g E C(X)+ such that Z(g) = {p}. Let {}~ 1 be a neighborhood base at p and define a sequence of real numbers { 1 be given and let U 1 ,U 2 ... ,Um be distinct free ultrafilters on the sequence {xn}Define P; = {f E C(X): 3A E U,A Z(f)}, for each i. DeMarco shows that each of these sets is a prime z-ideal of C(X) and that MP is the sum of any two. Thus these prime ideals are noncomparable. We will demonstrate, as DeMarco has done for m = 2, that Mp is the sum of these m noncomparable prime ideals. Let {Ai}~ 1 be a collection of m pairwise disjoint subsets of {xn} such that A; EU, for each 1 i m. If Bi = g(Ai), then Bi U {0} is a closed subset of JR. Thus Bi U { 0} is a zero-set of JR. Choose 'Pi E C (R.) for each 1 i m such that Z(cpi) = Bi U {0} and E~ 1 'Pi= lJR. Let ui = 'PiY Then Ai C Z(g) = Z(ui), hence ui E Pi. Finally, we have g = u1 + u2 ++Um. If h E Mp, then Z(h+ g) = {p} and Z(h+ + g) = {p} By the above, h+ + g, h+ g E E~ 1 P; and hence h+, h-, h E E~ 1 P;. The final statement of the proposition follow from the previous proposition, since the cardinality of /JN is 2', by (GJ, 9.2]. Now that we have illustrated the extreme cases of 0,1 and infinite rank, we present a result of (HLMW, 3.1], which gives a test for finite rank of a point of a compact space.

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31 Proposition 2.3.5. Let X be a compact space. Then p E X has rank n < oo if and only if there exist n pairwise disjoint cozerosets {U;}j=l with p E n;=l cl(U;), and no larger family of pairwise disjoint cozerosets has this property. An /-ring A is called an SV-ring if A/Pis a valuation domain (i.e. principal ideals are totally-ordered) for every prime ideal P. A space X is an SV-space if C(X) is an SY-ring. We will discuss this class of rings in more detail in Chapter 3. However, using the above, it is shown in [HLMW, 4.1] that any compact SY-space has finite rank. The validity of the converse of this result is unknown. Presently, our objective is to show that the result in Proposition 2.3.5 does not hold for infinite rank. To demonstrate this, we define a cardinal function on compact spaces and compare its value with a known cardinal invariant. Definition 2.3.6. Let K be a cardinal and let X be a compact space with p EX. Let {Ua}a<~ be a family of pairwise disjoint cozerosets of X and call the set na<~ cl(Ua) a K-boundary. Define p(p, X) to be the infimu.m over all {infinite) cardinals K such that p is not contained in a K-bou.ndary and let p(X) be the supremum over all the points p E X of the cardinals p(p, X). Recall that the cellularity of a space X, denoted c(X) is the infimum over all (infinite) cardinals K such that every family of pairwise disjoint open sets of X contains at most K many sets. Proposition 2.3.7. Let X be compact Then p(X) c(X)+. Proof: If p(X) > c(X)+, then there exists a x;-boundary of cardinality greater than the cellularity of the space, which is nonsense. Example 2.3.8. Let > No. The product space, 2r, where 2 is the two-point discrete space is called the Cantor space of weight ,. We show in Example 4.3.4

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32 that every point of 2-r has infinite rank, thus rk(C(2-r)) 2' by Proposition 2.3.3. By [E, 3.12.12(a)], we have that c+(2-r) = N 1 Thus, Proposition 2.3.7 gives us that N 0 < p(2-r) N 1 Therefore, since N 1 =I= 2', we see that p(2-r) i= rk(C(2-r)). D 2.4 Rank via z#-Irreducible Surjections We must first recall a few definitions from (Ha] and (HVW2]. Let X, Y be Tychonoff Hausdorff spaces. Let f : Y X be a surjective continuous map. Then f is perfect if it is a closed map such that the inverse image of any point is compact. A perfect map is irreducible if proper closed sets of Y map to proper closed sets of X. The pair (Y, !) is a cover of X if/ is a perfect irreducible surjection from Y to X. Let (Yi, / 1 ) and (Y2, fi) be covers of X. We define (Yi, !1) ~ (1';, !2) if there exists a homeomorphism g : Y 1 Y 2 such that fig = / 1 Order the set of ~-equivalence classes of covers via: (Y 1 Ji) (Yi, fi) if and only if there exists a continuous map g : Y 1 Yi such that fig= J 1 A class of spaces C is a covering class if for any space X there exists a least cover (Y, !) of X such that Y E C. The minimal extremally disconnected, basically disconnected and quasi-F covers of compact spaces are described in (PW], (V], and, respectively, in (DHH], [HudP], and [HVWl]. Certain covering maps allow us to compute the rank of a space externally. A perfect irreducible surjection : Y X is z#-irreducible if for each cozeroset U Y, there is a cozeroset V X such that cly(U) = cly(4>1 (V)) This condition on maps is also known as sequential irreducibility and w 1 -irreducibility. It turns out that a map is z#-irreducible if and only if C ( ) is a rigid embedding of C ( X) inside C(Y), by (HaM, 2.2]. Hence we have a homeomorphism Min(C(Y)) ~ Min(C(Y)) via contraction, by (CM, 2.3]. It is therefore not surprising that these maps are useful for calculating rank.

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33 Example 2.4.1. Let X be a compact space. The quasi-F cover of X, (QFX, x), constructed in [HVWl] has the property that x is z#-irreducible. We summarize this construction. Let z#(X) = { clxintx(Z) : Z E Z(X)}. For A E z#(X) denote the set of ultrafilters on z#(X) containing A by A. The authors of [HVWl] show that T(Z#(X)), the compact space of ultrafilters on z#(X) whose topology has a closed base given by { A : A E z# ( X)}, is quasi-F and define a perfect irreducible surjection x: T(z#(X)) X by a E T(Z#(X)) maps to the unique point in n{A: A Ea}. Then QF X = T(Z#(X)) with the map x is the quasi-F cover of X. The map x is z#-irreducible: It is shown in [HVWl, 2.9] that if we have A E z#(QFX), then x(A) E z#(X), which is equivalent to the property of z#_ irreducibility. In fact, the quasi-F cover of X is characterized up to equivalence in [HVWl, 2.13] as the only cover (Y, J) of X for which Y is quasi-F and f is z#-irreducible. Before we continue, we discuss the question (now answered) which was our motivation for considering this line of investigation. Let X be a compact space of finite rank and W = {x EX: rkx(x) > 1}. In [La2], Suzanne Larson asks if Wis always closed in X. The answer is no. Her counterexample, presented at ORD98 (a conference on l-groups held in Gainesville, FL in 1998) follows: Example 2.4.2. Let Ube a free ultrafilter on N. Let E =NU {a} where points of N are isolated and neighborhoods of a are of the form U U {a}, where U E U. Let E; = E for j = 1, 2 and define Y = (E 1 II E 2 )/(a 1 ~ o2 ). Let Y,. = Y for each r E R, and let X = (llrelR Y,.) U { oo} where neighborhoods of oo contain all but countably many copies of the Y,.. Then oo is a P-point which is in the closure of the set W = {x EX: rkx(x) > l}.

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34 We provide a characterization of points of finite rank of a compact space X via z#-irreducible maps onto X. Let B be an f-subring of A and let O denote the natural surjection Max(A) Max(B). The following is proved in [HaM, 2.5]. Lemma 2.4.3. Let A and B be commutative !-rings with identity and bounded inversion. Let B be an f-subring of A. Then if B is rigid in A, we have that O(OM) = n{O(N) n B : ON= OM}, and if ONi = OM (for j=l,2} with Ni II N2, then (O(N1) n B) II (O(N2) n B). Let X and Y be compact spaces and : Y X a z#-irreducible map. Then 0: Max(C(Y)) Max(C(X)) is given by M .-+ {f E C(X) : f EM}. Lemma 2.4.4. Let X and Y be compact and : Y X a z#-irreducible map. Let p E X then O(Mq) = Mp if and only if (q) = p. Therefore, we have that Op= n{Oq n C(X) : q E <1>1 {p} }. Proof: Let q E <1>1 {p}. If f E O(Mq) then f E Mq and therefore we have that 0 = f(q) = f(p) and f E Mp. Let g E Mp; then (g)(q) = g(p) = 0, and hence g E O(Mq)Thus O(Mq) = Mp. Conversely, assume (q) = r -f. p. By complete regularity, there exists f E C(X) such that f(p) = 0 and f(r) = 1. We have / E Mp, but f(q) = J(r) = 1 -f. 0 and hence f O(Mq)The final statement then follows from the first and Lemma 2.4.3. Proposition 2.4.5. Let X and Y be compact spaces and : Y X z#-irreducible. 1. If p EX such that rkx(p) = n, then l1 {p}I = n. 2. If Y is an F-space and l<1>1 {p}j = n then rkx(p) = n. S. If Y has finite rank and l1 {p}j = n, then rkx(p) < oo. Explicitly, we have that rkx(p) = E~ 1 rky(qi) where <1>1 {p} = {qi}f=i

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35 4. Ifp EX is an F-point of X, then q E 4>1 {p} is an F-point of Y. Proof: (1) Let 4>1 {p} = {qi}iEI for some index set J. For each i E J, choose Qi E Min(C(Y)) such that Oqi Qi. Then by Lemma 2.4.4, it follows that we have Op= niEr(Oqi n C(X)) Qin C(X) Mp, for every i E J. By the bijection described in Proposition 2.2.7, the set of minimal prime ideals contained in Mp is given by { Qi n C(X)}iEI and III = n. ( 2) Let 4>1 {p} = {Qi} i=l. If Y is an F-space, then O qi is prime for each i and we have that Op= nf= 1 (Oqi n C(X)) Oqi n C(X) Mp for each i. Thus, by the bijection described in Proposition 2.2.7, the subspace of minimal prime ideals contained in Mp is {Oqi n C(X)}f= 1 and rkx(p) = n. (3) Let 4>1 {p} = {qi}f= 1 and let the subspace of minimal prime ideals in Mq be given by {Qi;: 1 j rky(qi)}i=l Then as above, the subspace of minimal prime ideals contained in Mp is given by {Qi; n C(X) : 1 j rky(qi)}i=: 1 and hence we see that rkx(p) = E:= 1 rky(qi) < oo, as desired. (4) Let p EX be an F-point and 4>1 {p} = {q}. Then Op= Oq n C(X) Mp is a minimal prime. There exists a unique Q E Min(C(Y)) such that OP= Q n C(X). Since Oq n C(X) = Q n C(X) and Q is unique, Oq = Q E Min(C(Y)). If Y = QF X in Proposition 2.4.5, then the third statement is a partial converse of [HLMW, 5.1] which states that if Xis compact and has finite rank then QF X has finite rank. The final statement is an extension of [HVWl, 3.12] in which it is shown that the preimage of a P-point of Xis a P-point of QFX. The second statement says that if QF X is an F-space, then the points of X of rank one are precisely the points with unique preimage under the covering map x. In this light, one should ask when a quasi-F cover of a space is an F-space. Recall that a space X is fraction dense if the classical ring of quotients of C(X) is rigid in the maximal (Utumi) ring of quotients of C(X). In [HVWl, 2.16],

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36 it is demonstrated that Q F X is basically disconnected if and only if X is cozero complemented. In fact, the basically disconnected cover is the quasi-F cover in this case, see [HaM, 2.6.2). The fact that QFX is realized by the extremally disconnected cover if and only if Xis fraction dense is proved in [HaM, 2.4). By [HudP, 6.2), QFX is an F-space if and only if for any two disjoint cozero-sets Ci, C2 X, there exist Z 1 Z 2 E Z(X) such that Ci Zi for i=l,2 and int(Z1 n Z2) = 0. We now provide an example of a space X such that QF Xis an F-space which is not basically disconnected. Recall that a space is a-compact if it is a countable union of compact spaces. Lemma 2.4.6. Let X be a noncompact a-compact locally compact F-space which is not basically disconnected. Let X 1 = X 2 = {3X and define Y be the quotient space of the topological sum of X1 and X2 where pairs of corresponding points of X; \ X (j=1,2} are collapsed to a single point. Then Y is not quasi-F and QFY is an F-space which is not basically disconnected. Proof: The disjoint union U = X II X is a dense cozero set of Y which is not C*-embedded in Y. Thus Y is not quasi-F. The quasi-F cover of Y is X1 II X2, which is an F-space but not basically disconnected. Example 2.4. 7. Let X be the disjoint union of a countable number of copies of the corona {3N \ N. Then Xis a a-compact noncompact F-space which is not basically disconnected. Construct Y as defined in Lemma 2.4.6. Then QFY is an F-space which is not basically disconnected. 2.5 Prime Character The second character we consider counts the minimum number of minimal prime subgroups that we must sum in order to obtain the lex kernel of an -group.

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37 Definition 2.5.1. Let S be a family of minimal prime subgroups of G. We call S ample if VS= lex(G). The prime character of G, denoted 1r(G), is the least cardinal so that any family of distinct minimal prime subgroups of that cardinality is ample. Note that iflex(G) = 0, i.e., if G is a totally-ordered group, then we say 1r(G) = l. A prime subgroup properly contained in lex( G) is called embedded. Proposition 2.5.2. Let G be an l-group and m a positive integer. The following are equivalent: 1. 1r(G) = m < oo. 2. m = 1 + sup{rk(P) : PE Spec+(G) is embedded}. S. m is minimal with respect to the property that lex(G) = G(LJ; 1 af) for any m pairwise disjoint positive elements, {a;}f= 1 lex(G)+. 4m is minimal such that for any embedded prime P, the chains of proper polars in P have length at most m 1. Proof: (1) =} (2) : Let P be embedded If P contains the m minimal prime subgroups {Q;}f= 1 Min+(G) then Vf= 1 Q; P lex(G). Hence 1r(G) > m. Thus we have shown that 1r(G) m implies rk(P) < m. Thus by (1), rk(P) m 1 and sup{rk(P) : P embedded} m 1. If sup{rk(P) : P embedded} < m 1, then for any family S of m 1 minimal prime subgroups of G, VS is not embedded since rk(VS) m-1. Thus VS= lex(G) and 1r(G) m-1. Thus 1r(G) = m implies that sup{rk(P) : P embedded} m 1. Therefore sup{rk(P) : P embedded} = m + 1. (2) =} (3): Let {a;}f= 1 lex(G)+ be pairwise disjoint. Let g E lex(G) \ G(LJ; 1 af) and let V be a value of g such that G(LJ; 1 af) C V C lex(G). Then by the polar characterization of O(V), [BKW, 3.4.12}, we have that a; (j_ O(V) for each j such that 1 j m. By (2), V contains at most m 1 minimal prime subgroups, {Qi}~ 1 1 Since O(V) = n~~ 1 Qi, we have by the pigeonhole principle that there

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38 must be a minimal prime subgroup, Q, contained in V which fails to contain two of the elements of {a;}j=i But since these elements are pairwise disjoint, this contra dicts the fact that Q is prime. Therefore, lex(G) = G(Uf: 1 a,f ). For the minimality of m, let P be an embedded prime of rank n < m. Then by the Finite Basis Theorem, P/O(P) contains n pairwise disjoint elements of corresponding to elements of P which are not in O(P), say {bAJ:= 1 s; P. Then bf C P for each k such that 1 :5 k \$ n and G(UZ=i bt) s; P lex(G). (3) :::} (1) : Assume that 1r(G) > m. Then there exists S = {Q;}j= 1 C Min+(G) which contains m elements and is not ample. Let Q = VS lex( G). For each j, let q,; E Qt\ Q;. Then q, = Vfa:= 1 q,; E Qt\ (UA:;ei QA:)Disjointify by defining q, = A,;e;Q.; Af= 1 Qk r/. Q,. Then Q; E Q, for every j =/: i and then we obtain that G(U;: 1 q/) s; VS~ lex(G). Hence, 1r(G) \$ m. By the minimality of min (3) and by (1):::} (3), we must have 1r(G) = m. (3) # ( 4) : Follows directly from the Finite Basis Theorem [D, 46.12] applied to P/O(P) for any embedded prime P. The following is immediate: Corollary 2.5.3. Let G be an l-group. The following are equivalent: 1. 1r(G) < oo. 2. sup{rk(P): P embedded}< oo. 3. There exists m E N such that lex(G) = G(lf= 1 af) for all families of m pairwise disjoint elements {a;}f=i We now consider some -group-theoretic properties of the prime character. Note that for -groups A and B, AEBB denotes the -group Ax B with componentwise operations and is called the cardinal sum.

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39 Proposition 2.5.4. Let G be an -group. 1. For any -homomorphic image, H, we have 1r(H) :5 1r(G). 2. For any CE
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40 character. I '--------(2.1) Let r be the root system above, where the subgraphs r A and r B each have infinitely many identical branches descending from its maximal vertex. Define G = E(r, JR), A= {v E G: supp(v) c r A} and B = {v E G: supp(v) c rB}Then we have that A~ E(r A,1R), B ~ E(rB,1R) and AV B ~ E(r Au fn,lR). Now, it is evident that 1r(G) = oo, 1r(A) = 3 = 1r(B), and 1r(A VB) = oo. D Recall that an -group G has a finite basis if it contains a finite maximal set of elements {b; }j= 1 such that the set {g E a+ : g b;} is totally ordered for each j. The following indicates when we can expect 1r(A VB) < oo: Proposition 2.5.6. Let G be an -group and let A, B be convex -subgroups such that 1r(A) = m < oo and 1r(B) = n < oo. If lex(A VB) = lex(A) V lex(B), then 1r(A V B) < oo. Otherwise, 1r(A V B) < oo if and only if each of A and B has a finite basis. Proof: Assume that lex(A VB) = lex(A) Vlex(B) and let P lex(AV B) be a prime subgroup of G. Then either lex(A) %. P or lex(B) %. P, or both. Say lex(A) %. P. Then P n lex(A) is an embedded prime subgroup of A, and since 1r(A) = m, we have that rkA(P n lex(A)) < m 1. Then by [BKW, 2.4.7], rkAvB(P) < m 1. Likewise, if lex(B) %. P then rkAvB(P) n 1. Thus, for every prime subgroup P lex(A VB), we have that rkAvB(P) max{m-1,n -1} < oo. Therefore, by Proposition 2.5.2, 1r(A VB) < oo.

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41 Note that we always have that lex(A) V lex(B) lex(A VB). We assume now that lex(A VB) =/= lex(A) V lex(B) and let PE Spec+(A VB) have the property that lex(A) V lex(B) P lex(A V B). If each of A and B has a finite basis, then rkAvn(P) = IMin+(A)I + IMin+(B)I < oo and for any embedded prime subgroup Q of AV B, we have that rkAvn(Q) ::; IMin+(A)l+IMin+(B)IThus, by Corollary 2.5.3, we have that 1r(Av B) < oo. Conversely, 1r(AV B) < oo implies that rkAvn(P) < oo. Hence, since IMin+(A)I, IMin+(B)I < rkAvn(P), we have that each of A and B has a finite basis by the Finite Basis Theorem. The proof of the following is evident: Proposition 2.5.7. Let G be an -group and let A, BE v(,) = O}. If v He,

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42 then there exists E C such that v(,) -:/= 0. Since supp(v) is finite, v is in all but finitely many minimal prime subgroups of E(r, JR). Thus by Proposition 2.5.8, E(r, R.) has Property F. Proposition 2.5.10. If G is a finite-valued -group of finite prime character, then lex(G) has Property F. Proof: Let g E lex(G). Then any minimal prime subgroup oflex(G) not containing g is contained in a value of g. Since each value of g contained in lex(G) contains a finite number of minimal prime subgroups and there are only finitely many values of g, there are only finitely many minimal prime subgroups of lex(G) not containing g. Thus lex( G) has Property F. Proposition 2.5.11. Let G be an -group and let m be a positive integer. If m is minimal such that every pairwise disjoint subset of G contains at most m 1 elements, then 1r( G) = m. Proof: Any proper prime subgroup of G is contained in a value of G, hence proper prime subgroups of G contain at most m 1 minimal prime subgroups. Thus 1r(G) :'.5 m. Let n < m and assume that every family of n minimal prime subgroups of G is ample. Then every proper prime subgroup (in particular, every value) of G contains at most n 1 minimal prime subgroups. This contradicts the minimality of m. Thus 1r( G) = m. Example 2.5.12. The following is an example of an -group of finite prime char acter such that lex( G) has Property F but G has pairwise disjoint sets of any size m. Thus the converse of Proposition 2.5.11 does not hold. Let r be the root system: I\ I\ I\ (2.2)

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43 and let G = E(r,R). Then 1r(G) = 3 and G is finite-valued. Thus, by Proposi tion 2.5.10, lex(G) has Property F. We demonstrate that there are bounded disjoint families of any given size. Index the maximal elements of r by;, where j is a positive integer, and let v; E G be such that v;(;) = 1 and supp(v;) c {'YE r: 'Y ~;}where v;(6) = 0 for all 6 < ;. Then { v;};~ 1 is an infinite pairwise disjoint family in G. Let a positive integer m be given. Choose any m elements from this set, { v; 1 vh, ... v;,..} and let s = uzi=l supp(v;.). Let V be the characteristic function on the finite set s, then v E E(r,R) and v v;. fork= 1,2, ... m. D Example 2.5.13. The converse of Proposition 2.5.10 does not hold. That is, we present an example of a finite-valued -group with Property F and infinite prime character. Consider the following root system r which is indexed by the positive integers: I /\ Ii\ (2.3) Then each prime subgroup of the -group G = E(r, R) contains a finite number of minimal prime subgroups, yet there is no bound on the number of minimal prime subgroups in each prime subgroup. Thus 1r( G) = oo. G has Property F by Corollary 2.5.9. D 2.6 Filet Character The third character that we define measures the length of a chain of incidents of branching. Definition 2.6.1. Let G be an -group. C = {11,Q; E Spec+(G): i O,j 1} is called a filet chain of prime subgroups if Po 2 P1 2 P2 ... for all i, Ps II Qi, and

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44 ~+i V Qi+l ~~for all i 2:: 0 {see below}. (2.4) The length of a filet chain is given by l(C) = max{j E N : 3Q; E C}. If the maximum does not exist, we write l(C) = oo. The filet character (G) is given by ( G) = sup{ l ( C) : C is a filet chain}. If Spec+ ( G) has no filet chains, i.e., if G is semiprojectable, then we say that ( G) = 0. Proposition 2.6.2. Let G be an -group. Then (G) \$ 1 if and only if 1r(G) \$ 2. Proof: Suppose that 1r( G) \$ 2. If ( G) > 1 then there exists a filet chain C of length 2 in which we may assume that P 0 = lex(G). Since rk(P 1 ) = 2, we have that 1r(G) > 2, by Proposition 2.5.2. Conversely, assume that 1r(G) > 2. Then there exist minimal prime subgroups P 2 Q 2 such that P 1 = P 2 V Q2 =f:. lex(G). Thus for any minimal prime subgroup Q 1 % P 1 the set C = {P 0 ,P 1 ,P 2 ,Qi,Q 2 }, where P 0 = lex(G), is a filet chain of prime subgroups in G oflength 2. Therefore (G) > 1. For larger filet character, the relationship between it and the prime charac teris more complicated, as the following example illustrates. Example 2.6.3. Let r be the following root system: (2.5)

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45 Then G = I;(r, R.) has (G) = 2 while 1r(G) = 4. The following relationships between the characters hold: Proposition 2.6.4. Let G be an -group. Then 1. (G) :5 rk(G) 1. 2. (G) :5 1r(G) 1 :5 rk(G). Proof: Let rk(G) = m. If G has a filet chain C = {Pi,Q; E Spec+(G): i 0,j 1}, then rk(Po) :5 m and hence l(C) :5 m-1. Therefore, (G) :5 m-1 = rk(G)-1. If 1r(G) = n and C = {Pi, Q 3 E Spec+(G) : i O,j 1} is a filet chain in G then l(C) < oo since rk(P 1 ) :5 n 1 by Proposition 2.5.2. Hence, in fact, l(C) :5 n 1 and therefore, (G) :5 n 1 = 1r(G) 1. Now, 1r(G) :5 rk(G) + 1 by Proposition 2.5.2. Thus, finally, (G) :5 1r(G) 1 :5 rk(G). At this time, any statement that we make about the filet character of an -group requires a restriction on the rank and prime character. Rather, we can not say much more than what we establish earlier in this chapter. We leave this investigation for a later date.

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CHAPTER 3 GENERALIZED SEMIGROUP RINGS Let A be a root system. Starting from the standard semigroup ring and Hahn group constructions and the investigation of the following section, we build an /-ring 'R having A as the skeleton of the graph of Spect('R). We follow up on ideas presented in two papers of Conrad [Cl],[C2], in the paper of Conrad and Dauns [CD], and in the paper of Conrad and McCarthy [CMc]. The first two papers look at the conditions on A which will yield a ring structure on V(A, JR) and on the subgroup F(A, R.) of elements v for which supp(v) is the join of finitely many inversely well ordered sets in ti. The paper [CD] focuses on the case when V(fi, JR) is a division ring, while in [CMc] the conditions are established for the ring to be an -ring its properties are studied when f1 is finite. Note that F(fi, :R) is denoted by W(fi, JR) in [Cl],[C2], and [CD]. 3.1 Specially Multiplicative {-Rings Let r be a partially ordered group which is a root system. Suppose also that r is torsion free, that the subgroup H of r generated by the positive cone is totally ordered and that r / H is finite. In the paper of Conrad and Dauns [CD, 2.2], it is shown that V(r, JR) = F(r, JR) and that V(r, JR) is a lattice-ordered division ring under the usual group-ring multiplication: for u, v E V(r, JR), and, Er u v(,) = I: u(a)v(,B). a+/J='Y It is easy to see that if an element of (V (r, JR), +, *) is special (i.e., has only one maximal component), then its multiplicative inverse is also special. Hence, the 46

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47 special elements of the ring V(r, R.) form a multiplicative group. Moreover, the following holds in general, [CD, Theorem I]: Theorem 3.1.1. Let R be a lattice-ordered division ring with identity. The follow ing are equivalent: 1. The special elements of R form a multiplicative group or the empty set. 2. If a E R+ is special, then a1 > 0. 3. If a E R is special, then a1 is special. 4. For all a ER+ special and x, y ER, a(x Vy)= ax Vay. Since the authors of the paper [CD] seek an embedding theorem for i-fields, their investigation in this realm is restricted to division rings. In this section, we consider a class of /-rings (which are not division rings) in which the special values form a partially ordered semigroup. The pursuit of a characterization similar to Theorem 3.1.1 of i-rings satisfying this condition is left for another time. For the/ rings that concern us here, some particular assumptions are needed on the associated semigroups. As a formality, we define: Definition 3.1.2. Call an f-ring A specially multiplicative if the special values of A, with an appropriately adjoined 0, form a partially ordered semigroup. Recall that an /-ring A is called an SV-ring if A/Pis a valuation domain (i.e., the set of principal ideals is totally ordered) for every prime ideal P. Let g be the class of commutative /-rings which are local, bounded (that is, A*= A), semiprime, finite-valued, finite rank and square root closed SV-rings with identity and bounded inversion. We demonstrate that the elements of g are specially multiplicative and then investigate the properties of the associated semigroups. We must first remind the reader of a couple of facts about special values and of the relatively deep theorem, recorded as [HLMW, 2.14]:

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48 Theorem 3.1.3. Let A be an !-ring of finite rank with identity and bounded inver sion which is local, semiprime and square root closed. Then A is an SV -ring if and only if whenever O a b and b is special, there exists x E A such that a = xb. The proof of the following lemma is well-known and routine, however, it is instructive, so we include it here. We remind the reader that we denote the root system of values of an -group G by r( G) = {V-r : 'Y E f}, where r is a partially ordered index set that is order-isomorphic to f(G). Lemma 3.1.4. Let G be an -group and let a, b E a+ be distinct special elements with values at V 0 V.8, respectively. 1. Vo II Vp if and only if a A b = 0. 2. V 0 < Vp if and only if a b. Proof: First, assume that Va II Vp. We show that a Ab is contained in each value of G and conclude that a and b are disjoint. Let V be a value in G and assume that b V. Then V Vp and a E V since, else, V Va, which contradicts the assumption of incomparability. Thus, by convexity we obtain that a A b E V. Again by convexity, if b E V then a Ab E V. Thus a Ab = 0, as desired. Conversely, assume that a and b are disjoint and, by way of contradiction, assume ( without loss of generality) that Va~ V.8 Since a V 0 and V 0 is a prime subgroup, b E V 0 Vp, a contradiction. Second, assume that Va< Vp. Then a E V.8 If there exists an integer n such that na > b, then since b V.8, we must have that na Vp, by convexity. But this contradicts that a E V.8, and so we conclude that a b. Conversely, assume that a~ b. We know that b V 0 and hence Va Vp. If Va = Vp, then by [D, 12.6], there exists an integer n such that na > b, which is nonsense. Thus V 0 < V.8.

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49 Combining Theorem 3.1.3 and Lemma 3.1.4 leads to the main result of this section: Theorem 3.1.5. Let A E <;;. 1. If a, b E A + are special and not disjoint, then ab is special 2. If a, a' are special with value Va, b, b' are special with value Vp and Va, V p are comparable then the special value of ab is the same as that of a'll. Proof: Let a, b E A+ be special with values at Va, Vp, respectively, and let M be the maximal ideal in A. Since A is bounded, we may assume without loss of generality that Va, Vp < M. This gives rise to the relations a, b 1 and ab a, by Lemma 3.1.4(2) Theorem 10.15 of [AF] states that an abelian -group is finite-valued if and only if each positive element is a finite disjoint sum of positive special elements. Thus we assume, by way of contradiction, that ab= c1 + C2 ++ck, where each c; E A+ is special and c; /\ Cj = 0 for all i =I= j. Then O < c; ab < a for each i and therefore Theorem 3.1.3 gives the existence of Xi E A + such that c; = ax i for each i = 1, ... k. Without loss of generality, we may assume that O < Xi b, by replacement with xi /\ b. Assume that xi = E~i Xi ; is a decomposition of X i into a sum of special elements. Then c; = E;~i axi; implies that axi ; = 0 for all but one j, by [AF, 10.15 ] Thus we may choose each xi to be special, without loss of generality. For each i, let Vo be the value of XiThen b Vo. and therefore for all i we have that Vo Vp. If Vo < Vp, then xi b. This gives a contradiction, since it implies that ax i ab. On the other hand, if Vo, = Vp, then there exist n, m E N such that x i ~ nb and b mx i Therefore, axi nab and ab~ maxi Thus by [D, 12.6] we have that the value of c; is the unique value of ab and ab= axi. Hence, ab is special.

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50 For the second statement, let be the value of the special element ab. We show that is the value of all and then a similar argument and transitivity gives that Vo is also the value of a'b', as desired. Towards this end, let V 7 be the value of ab'. H V 7 II Yo, then O = ab A all= a(b Ab') and hence O = a A (b All). On the other hand, if Va Vp, then since b All V 0 we have that a E Va, H Vp Va, then since a Vp, we have b Ab' and hence b or ll is in V13. These contradictions lead us to conclude that V 7 or V 7 V 0 If Vo < V-y then ab~ ab'. However, by [D, 12.6], there exists an integer n such that nb > ll and therefore, nab> nab'. Likewise, V 7 1'l,'o. Thus V 7 =,as desired. Let A E g, Then by [AF, 10.10], all the elements of r(A) are special since A is finite-valued. Abusing notation, we now identify values with their indices and define an operation on r. Let a, b E A be special elements with corresponding values at a, /3, respectively. Append a generic symbol, such that II a for all a E r and define a multiplication on r U {} such that a = = a and a. /3 = { if a II /3, the value of ab otherwise. By Theorem 3.1.5, this operation is well-defined. Some properties of this operation are recorded in the following proposition. Define a ~ /3 if and only if a and /3 are contained in the same maximal chains of r. Then~ is an equivalence relation on r. Let r O denote the ~--equivalence class of a. Proposition 3.1.6. Let A E g and let a, /3, ,, 8 E r correspond to the values of the special elements a, b, c, d E A+, respectively. Then 1. The operotion is associative, for every a, /3, 1 : ( a /3) 1 = a (/3 1 ) 2. If M is the maximal ideal of A, then M is a value of the identity. Let = M. Then a = a = a for every a

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S. If a {3 -::/=

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52 d = yb. Then O = ab I\ yb = ( a I\ y )b gives that O = a I\ y I\ b and hence O = ayb = ad and therefore al\d = 0. This says that a II 6, a contradiction. We now have that a /3 is comparable with any 6 < {3. It follows from this fact and from (3) that maximal chains are closed under the operation. (7) Let,, 6 Er 0 By (6), we have that, 6 is in the same maximal chains as, and 6. Hence 1 6 Er 0 (8) We know that, < a if and only if V-y < V 0 if and only if O < c a. This implies that there exists a positive element b such that c = ab, by Theorem 3.1.3. We have, by (AF, 10.15), that b is special. Thus, if V.B is the value of b, then = a {3, as desired. Corollary 3.1. 7. Let A E Q. Then A is specially multiplicative. Moreover the operation is a surjection onto r (A) U { }. 3.2 r-Systems and -Systems Let Ci be a given root system. The discussion of the previous section and of the papers (Cl], (C2], (CD], and (CMc] prepare us to construct an !-ring having Ci order-isomorphic to its root system of values. However, under the assumptions placed on Ci in these papers, these rings sometimes are -rings, and rarely are !-rings; in fact, as mentioned before, the paper (CD] focuses on the case when V(fi, JR) is a division ring. Also, contrary to our intentions, (CMc] concentrates on the properties of the ring when the root system is finite. We modify all these conditions on the root system in order to get a ring multiplication, *, yielding an f-ring structure on R = (F(fi,JR),+,*) such that the subring = (I;(~,JR),+,*), as defined in Chapter 2, is an f-subring. We start with (Cl,Theorem I] which establishes the ring structure. Proposition 3.2.1. Assume that Ci is endowed with a surjective partial binary op eration + : A Ci defined on A Ci x Ci. Let u, v E V ( Ci, JR), and define for

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53 8 E !l, u *v(8) = :Ea+.B=o u(a)v(.B). Then both~ and R are closed under* In fact, and 'R, are rings if and only if the operation on !l is (Baer-Conrad) associative: (a, /3), (a+ /3, 'Y) EA if and only if (/3, 'Y), (a, /3 +'Y) E A, and if (a, /3), (a+ /3, 'Y) E A then ( a + /3) + 'Y = a + (/3 + 'Y). Certain properties of these rings, and R, are completely determined by the operation + on /:l; commutativity is one such attribute. Definition 3.2.2. Let (!l, +) be a root system with a partial binary operation, +. We call (!l, +) an r-system if the operation is surjective and (Baer-Conrad) asso ciative. An r-system (!l, +) is commutative if (a, /3) EA if and only if (/3, a) E A and a + /3 = /3 + a for all such pairs. Proposition 3.2.3. Let (!l, +) be an r-system. (!l, +) is commutative if and only if R is a commutative ring. Proof: If (!l, +) is commutative, then for u, v E 7l we have u v(8) = L u(a)v(/3) = L v(f3)u(a) = L v(f3)u(a) = v u(8). a+,8=6 a+,8=6 .B+a=6 Thus 'R, is a commutative ring. Conversely, if a+ /3 =/= /3 + a, then Xa+.B = Xa X.B =I= X.B Xa = X.B+a In order to obtain an -ring, we must ask that the operation on /:l preserve the order of the root system and restrict the domain of the operation. AB we will indicate, the various strengths of order preservation and restrictions of domain yield varying richness of order structure on the rings. Definition 3.2.4. Let /:l be a root system. An r-system (!l, +) is an -system if it satisfies: a < /3 and (a, 'Y) E A implies that (/3, 'Y) E A and, in this case, a+'Y < /3+'Y; and if ( 'Y, a) E A implies that ( 'Y, /3) E A and, in this case, 'Y+a < 'Y+ f3. If every connected component of /:l has a maximal element, we call /:l bounded above.

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54 Note that the -system condition gives that no nonmaximal element of such a bounded above root system is idempotent. The following theorem is in (CMc, ). Theorem 3.2.5. Let (Ll, +) be an r-system. Then 'R, is an i-ring if and only if Ll is an -system. In the next section, we describe the conditions on an -system which induce an f-ring structure on 'R,. As mentioned, there are some restrictions that we need to place on the domain and range of the operation. Before we discuss this situation, we present an example of an -system that gives rise to an -ring which is not an f-ring. Probably, this is the simplest example of such an -system. Example 3.2.6. Let ,X > a > 2a > 3a. . and > /J > 2/3 > 3/3 .... Then we let Ll 0 = {.X} U {na}~= 1 let Llp = {} U {n/J}~=l and totally-order Ll 0 x Llp lexicographically such that (8 1 , 1 ) < (82, 2 ) if and only if 81 < 82 or we have 81 = 82 and 1 < 2 Identify pairs with sums and let Lla+/3 = {8+,: (8,,) E Ll 0 x Llp}. We define an associative and commutative addition on the root system given by disjoint union tl = tl 0 II tlp II Lla+/3 as shown in the following table, where v = A + and k, l, m, n are positive integers (note that the table is completed by reflection across the diagonal): + a /3 A V A+m/3 ka+ na +l/3 a 2a a+/3 a a+ a+ a+m/3 lk +lla+ ln+ lla +l/3 /3 2/3 A +/3 /3 A +/3 A+(m+l)/3 ka+/3 na + (l + 1)/3 A A V V A+m/3 ka+ na + l/3 V A+m/3 ka+ na + l/3 V V A+m/3 ka+ na + l/3 A+m/3 A+2m/3 ka+m/3 na+lm+fl/3 ka+ 2ka+ (k+n)a+l/3 na + l/3 2na + 2l/3 Let u, v E 'R,+. We need to show that u v E 'R,+ in order to conclude that 'R, is an -ring. Since u and v are positive, it is evident that U*v(.X), U*v(), U*v(v) 0. If 0 = u v(.X) = u(-X)v(.X), then assume that u(.X) = 0. For an integer n, u v(na) = u(-X)v(na) + u(na)v(.X) + L u(ia)v(ja). i+j=n i,#0

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55 H u(na) = 0 for all n, then u v(na) = 0. Conversely, if n # 0 is minimal such that na E supp(u), then u v(na) = u(na)v(A) 0. Thus, if v(A) > 0 then na is the maximal element of supp( u v) below A. ff v(A) = 0 and v(ma) = 0 for all m, then u v vanishes on ..6. 0 Otherwise, let m be minimal such that ma E supp ( v). Let k = n + m then ka is maximal in supp(u v) and u v(ka) = u(na)v(ma) > 0. A similar computation demonstrates that u v > 0 at the maximal element of its support in AfJ also. Assume that O = u v ( v) and, when they exist let Au= max(supp(u) n ..6. 0 ) Av= max(supp(v) n Aa) JJu = max(supp(u) n AtJ) ,; = max(supp(v) n AtJ) <>u = max(supp(u) n Aa+fJ) <>v = max(supp(v) n Aa+fJ) (3.1) Then the maximal element of supp( u v) lying below v is given by and so we are left to show that u v(,) > 0. We have three cases to check here, namely 'Y = A +m/3, ka+ or na+l/3. Upon consideration of the formulae below, it quickly becomes clear that the only nonzero summands of u v ( 'Y) are those of the form u(a)v(r), where a and r are one of the six maximal support elements listed above in (3.1), which implies that the summands are all positive. For the sake of a certain degree of completeness of exposition, we list the possible summands in the case that 'Y =A+ m/3. The analysis in the other cases is similar. u v(,\ + m/3) = u(,\)v(m/3) + u(mf3)v(,\) + u(,\)v(,\ + m/3) + u(,\ + mf3)v(,\) + u()v(A + m/3) + u(,\ + mf3)v() + u(v)v(,\ + m/3) + u(,\ + mf3)v(v) + L u(,\ + if3)v(jf3) + L u(i/3)v(,\ + j/3) + L u(,\ + if3)v(,\ + j/3) i+j=m i+j=m i+j=m i,j#) i,#0 i,#0 To compute u v(,\ + m/3), consider all the possible combinations of the following situations and then add.

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56 1. H u(-X) =/:0 then if v(m/3) =/:0, we have that m/3 v, If m/3 < v, then 'Y = ,\ + m/3
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u v(na + l/3) = u(na)v(lf3) + u(lf3)v(na) + u(A)v(na + lf3) + u(na + lf3)v(A) + u()v(na + l/3) + u(na + lf3)v() + u(v)v(na + l/3) + u(na + lf3)v(v) + u(na)v(A + l/3) + u(A + lf3)v(na) + u(na + )v(l/3) + u(lf3)v(na + + L u(ia + lf3)v(ja) + L u(ia)v(ja + l/3) + L u(na + if3)v(jf3) i+j=n i,#0 i+j=n iJ-f,O i+j=l iJ#O 57 + L u(if3)v(na + j/3) + L u(ia + )v(ja + l/3) + L u(ia + lf3)v(ja + i+j=l iJ#O i+j=n iJ#O i+j=n iJ#O + L u(A + if3)v(na + j/3) + L u(na + if3)v(A + j/3) i+j=l iJ-f,O i+i=l iJ#O + L u(ia + kf3)v(ja + m/3) i+j=n k+m=l iJ,k,m#O 'R, is not an f-ring: Let HllaH = {v E 'R,: 8 E /J.o.+fJ => v(8) = O} be the minimal prime subgroup. Then Xo., XfJ E Hlla+f:l' yet Xo. XfJ = Xo.+fJ Hlla+f:l Thus, Hlla+f:l is not an ideal and 'R, is not an !-ring by [BKW, 9.1.2]. The following two propositions record some consequences of the l-system condition. Note that none of the excluded conditions occur in Example 3.2.6. Proposition 3.2. 7. Let (JJ., +) be a bounded above l-system in which maximal ele ments act as an additive identity on elements below it. Then the Jollowing can not occur for nonmaximal elements a, f3 E JJ., 1. /3 +a< a, where (3, a are below the same maximal element, and f3 II f3 + a 2. /3 < a /3 + a 4If A, E /J. are maximal and (A,) EA, then A+ {A,.

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58 In particular, the second and third properties imply that if a, {3, {3+a are comparable, then {3 + a < a, {3. Proof: If {3 + a < a, {3 II /3 + a, and > a, {3 is maximal, then XfJ, X, Xa are positive. Yet, XfJ (X, Xa) = XfJ XfJ+a is negative, contradicting that R is an -ring. Assume that {3 < a {3 + a. Let {3 + a be maximal. Then as above, XfJ (X, Xa) < 0. The same contradiction is obtained in the case that we assume {3 {3 + a a < If)., are maximal and ). + < then X>. (X, 2X>.+,) = -x>.+, < 0. Proposition 3.2.8. Let(~,+) be an -system 1. There is no nonmaximal o E such that o is idempotent and o + a = o for all nonmaximal a > o. 2. If o E is nonmaximal and idempotent then there does not exist a maximal element > o. Proof: If o E such that o is idempotent and o + a = o for all nonmaximal a > o, Then we contradict the assumption that 'R, is an -ring since Xo (xa 2xo) = -xo If > o is maximal and o is idempotent, then Xo (X, 2xo) = -Xo 3.3 (-Systems We are much more interested in the /-ring situation. In [CMc, 2.1 ), the authors demonstrate a condition on~ which will give rise to an /-ring. Theorem 3.3.1. Let(~,+) be an -system. Then 'R, is an /-ring if and only i f the root system also satisfies: if a II /3 and (a,,) EA, then a+, II /3; and if ( 1 ,a) EA, then 'Y + a II {3.

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59 In this section we consider a condition on Li which is equivalent to the one stated above and proceed to investigate certain properties of the associated !-rings. Recall that r(F(Li, JR)) is order-isomorphic to Li by Proposition 2.1.1, where the values are of the following form, for 8 E Li : = {u E 'R:, 8 => u(,) = O}. Definition 3.3.2. Let (Li,+) be an -system such that 8 = a+ {3 implies a, {3 8. Then we say that (Li,+) is an J 0 -system. Proposition 3.3.3. If (Li,+) is an J 0 -system, then 'R is an -ring and for every 8 E Li, the subgroup Vo is an ideal. Hence, in particular, 'R is an !-ring and is an f-subring of 'R. Proof: Assume that (Li,+) is an J 0 -system. Let 8 E Li, v E u E 'R, and 8. Let,= a+/3, then by the J 0 -system assumption, {3 ~, 8 and hence v(/3) = 0, for all such {3 since v E Then U* v(,) = Eo+~=-r u(a)v(/3) = 0 and we conclude that U*V E Therefore, the values of'R are ideals. Moreover, since each minimal prime subgroup is an intersection of a chain of values by [D,10.8], each is an intersection of a chain of ideals; thus each minimal prime subgroup itself is an ideal. Finally, we have shown that 'R, and~ are !-rings by [BKW, 9.1.2]. Proposition 3.3.4. An -system (Li,+) is an J 0 -system if and only if 'R, is an !-ring. Proof: Sufficiency is shown in Proposition 3.3.3. Conversely, assume that there exist a, {3 E Li such that a+ {3 ;:. a or a+ {3 ;:. {3. Then by Proposition 3.2.7, we have that a, {3, a+ {3 are not all comparable. If a II {3, then we have that a II a+ {3 by Theorem 3.3.1. Assume that {3 II a+ {3 and let C be a maximal chain in Li containing a + {3. Let He = { v E 'R : 8 E C => v ( 8) = 0} be the associated minimal

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60 prime subgroup. Then X/3 E He, yet Xa X/3 = Xa+/3 ff. He. Thus He is not an ideal and hence n is not an /-ring by [BKW,9.1.2). Definition 3.3.5. An J 0 -system satisfying the following is calle,d an /-system: 1. If a and fJ are comparable, then (a, (J), ((J, a) EA. 2. If is maximal, then (8, ), (, 8), (, EA and 8 + = + 8 = 8 for every 8 In particular, + = We will shortly see that these additional assumptions on a bounded above / 0 system make maximal chains in t::,. into monoids. This is quite useful in our setting. For instance, it is not difficult to figure out when the /-rings have a multiplicative identity, under the /-system assumption. Proposition 3.3.6. Let (t::., +) be a bounded above /-system. and 'R, each have a two-sided multiplicative identity if and only if l::,. has a finite number of connected components Proof: Let (t::., +) contain only a finite number of connected components with maximal elements {, ... n}Let x.; be the characteristic function on the set {;} and let e = r:;;= 1 Xi Then for v E 'R, and 8 Et::., v e(8) = L v(a)e(fJ) = v(8)e(k) = v(8), a+/3=6 where k 8 is maximal. Likewise, e v(8) = L e(a)v(fJ) = e(k)v(8) = v(8). a+/3=6 Thus e is a two-sided multiplicative identity. Conversely, assume that e E 'R, is an identity and let E t::,. be maximal with characteristic function Xw Then 1 = = e = e(). Thus, supp(e) has

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61 as many maximal elements as there are maximal elements in A. Since e E 'R, we must conclude that there are only finitely many connected components in A. It is handy to have the following definition: Definition 3.3. 7. Call an r-system (A,+) unital if the ring 'R, has a multiplicative identity. The rings !Rand 'R, are semiprime, as we will now show. Thus the minimal prime subgroups are also prime ideals, by [BKW, 9.3.1]. Proposition 3.3.8. Let ( A, +) be an / 0 -system. Then 'R, is semiprime. Proof: Let u E 'R, where u = L; eI a; Xo; for an index set I of supp ( u) and for a; E JR\ 0 for all j E J. Then U*u(o) = Lo=oi+o; aia;. Let oi be maximal in supp(u). We show that oi + oi is maximal in supp(u u). Let 8 = 8; + Ok E supp(u u). If 8; II oi, then by the / 0 -system condition, 811 oi + OiOn the other hand, if 8;, Ok Oi, then 6 6i + 6i, as desired. Thus u u( Oi + oi) = a1 > 0 and we conclude that 'R, is semiprime. Corollary 3.3.9. Let (A,+) be an / 0 -system. Then maximal chains in A are closed under the operation +. Proof: Let C ~Abe a maximal chain and let He= {v E 'R: 8 EC=} v(o) = O} be the associated minimal prime subgroup. Let a, fJ E C and assume by way of contradiction that a+ fJ iC. Then Xa XfJ = Xa+fJ E He but Xa, XfJ iHe. Thus He is not a prime ideal. This contradicts Proposition 3.3.8, by [BKW, 9.3.1]. One should ask if the maximal -ideals of 'R are actually the maximal ideals; or equivalently, one should ask if 'R has the bounded inversion property. The answer is yes, if (A,+) is a unital /-system.

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62 Proposition 3.3.10. Let(~,+) be a bounded above unital /-system. An element u E 'R, is a multiplicative unit if and only if u (/. V for all maximal E ~Thus R satisfies the bounded inversion property. Proof: Let S be the set of maximal elements of and let e = LES X be the multiplicative identity. If u E 'R, is a multiplicative unit, then there exists v E 'R, such that u v = e. Hence 1 = e() = for every ES. Thus u() =/0 for all ES and hence u (/. V for all maximal E ~, as desired. Assume that u 8 E supp(u) and assume that v(,) is defined for all,> <5. Then u(a)v(T) is a summand of u v(8) only if <5 a, TE supp(u). Thus v(<5) is the only unknown in and is the unique solution of the equation O = u v ( <5). Hence, since if e is the multiplicative identity of 'R, and u e then u
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63 4> M K;z, = {g E G: all values of g are in 4>}. Proposition 3.3.12. Let (~, +) be a bounded above /-system. Any closed convex -subgroup of 'R is an ideal of 'R. Proof: Let K be a closed convex -subgroup of 'R; let u E K+ and v E 'R. If u is a unit, then 4>K = r(G) and by the preceding theorem, K = G. So assume that u is a nonunit. Let E be maximal and assume that u v(,) = 0 for all~,> 8 and u*v(8) #0 for some 8. By the preceding theorem, we need to show that 8 E 4>K Let 8 =a+ (3 such that u(a)v(/3) #0. Then there is a maximal a' a such that u( a') #0. Then a' = a' + a' + fJ a + fJ = 8 and a' E 4> K since u E K. Therefore, we conclude that 8 E 4> K since 4> K is an order ideal. Hence, u v E K by the preceding theorem. We now seek the prime -ideals and z-ideals among the prime subgroups V 0 and their associated value covers. The cover of V 0 is the set P 0 = { v E 'R : v(,) = 0 for all > 8} and is the smallest convex -subgroup properly containing and Xo First let us recall the most general definition of z-ideal. Let G be a vector lattice, v E a+, and let {gn}~=l G be a sequence. Recall that the sequence converges relatively uniformly to g E G along the regulator v and write gn g, if for every c > 0 there exists Ne > 0 such that for all n Ne, we have that lg gnl cv. Let H be a convex -subgroup (sub-vector lattice) of G. The pseudo-closure of H is H' = {g E G : 3{gn}~=l H, gn g for some v E n+}. Then His relatively uniformly closed if H = H'; let H denote the relative uniform closure of H. Then if G(g) denotes the convex -subgroup (sub-vector lattice) of G generated by g, we define a convex -subgroup (sub-vector lattice) H to be an abstract z-ideal if

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64 h E H,g E G and G(g) = G(h) imply that g EH. In fact, [HudPI, 3.4) says that this definition is equivalent to G(h) c H for all h E H. Proposition 3.3.13. Let(~,+) be an !-system. If E is maximal, then V is a prime ideal of 'R which is an abstract z-ideal. Proof: Let u v E V, and u r/. V, Assume without loss of generality that u+, v+ E V and ur/. V, We show that vE V, Since ur/. V, we have that u-() =I0. Thus since u+, v+ E V, 0 = u v() = L u(o:)v(,8) = L (u+ u-)(o:)(v+ v-)(,a), a+/J= a+/J= and since is maximal, we have that 0 = u-()v-() and therefore, v-() = 0. This gives that vE V and v E Vw Hence V is a prime ideal. Since 'R/V is isomorphic to lR via the evaluation map u u(), we have that V is uniformly closed by [HudPI, 2.1). Thus by [HudPI, 3.4), V is an abstract z-ideal. Corollary 3.3.14. If(~,+) is a bounded above unital !-system, then the maximal ideals are given by {V : E is maximal}. Proposition 3.3.15. Let(~,+) be an !-system and let 6 E be nonmaximal. Define Po= { v E 'R: v('y) = 0 for all > 8}. Then P 0 is an abstract z-ideal and it is a prime ideal if and only if o: + ,8 > 8 for all o:, ,8 > 8. Proof: Let v E Pf Then there exists { vn}~ 1 Po such that Vn v, for some w E 'R,+. Let,> 8, then for every c > 0 we have that jvj(,) cw('y). Thus v(,) = 0 and Po is relatively uniformly closed. Therefore, P 0 is a z-ideal by [HudPI, 3.4). Assume that Po is a prime ideal and that there exist o:, ,8 > 6 such that o: + ,8 8. Then Xa+/J E Po and Xa+/J = Xa X/J But Xa, X/J r/. P 0 which is a contradiction.

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65 Conversely, assume that o+/3 > 8 for all nonmaximal o, /3 > 8. Let U*V E Po and assume, by way of contradiction, that u, v P 0 Then u v ( 'Y) = 0 for all 'Y > 8 and there exist elements o E supp(u) and /3 E supp(v), each maximal in the support set and such that o, {3 > 8. Assume, without loss of generality that o {3. If o' o, {3' {3 and at least one of the inequalities is strict, then o' + {3' < o + {3 and we conclude that u*v(o+/3) = u(o)v({3)+u({3)v(o). If o = {3, then since o+/3 > 8, we have that O = u v(o + {3) = 2u(o)v(f3) -10, a contradiction. If o > {3, then v(o) = 0 and hence O = u v(o + {3) = u(o)v(/3) -10, another contradiction. Thus we conclude that either u or v is in P 0 and therefore P 0 is a prime ideal. Let S be a totally-ordered set. Recall that Sis an T/1-set if whenever A, B S are countable and A < B, then there exists c E S such that A < c < B. Since JR is not an 'f/ 1 -set, the ring 'R, is never an 'f/ 1 -set. To see this, let 8 E /:1 and consider the sets {x 0 } > {(1 0 : n EN}. But, 'R, can satisfy a related, slightly weaker condition. Definition 3.3.16. We call a totally-ordere,d set S an almost 'f/ 1 -set if A, B S are countable and if A < B, then there exists c E S such that A c B. Note that R. is such a set. Proposition 3.3.17. Let ( !:l, +) be a totally-ordered / 0 -system. 'R, is an almost 'f/1 -set if and only if !:l is an T/1 -set. Proof: Assume that 'R, is an almost 'f/ 1 -set. We first note that /:1 contains no suc cessor pair. Leto< {3 be a successor pair. Then the sets {nxa}nEN and {l/nx,0}nEN contradict the almost 'f/ 1 -set condition. Let A = { oi}iEN, B = {/3; };EN C /:1, where 01 < 02 < < /3,i < f3i. Then Xa 1 < Xa 2 < < X,8 2 < X.8 1 and there exists / E 'R, such that Xa. f X.8; for all i,j. Let 'Y be the maximal element of supp(!). Then / Xai 0 implies that 'Y = Oi and /('Y) 1 or "f > Oi and /('Y) > 0. If 'Y = oi, then / Xa.+i < 0, which

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66 is a contradiction. Thus 'Y > a; for all i. Similarly, 'Y < /3; for all j. Therefore, is an 111-set. Conversely, assume that is an 111-set and let Ii < h < < 92 < 91 E 'R. Let 1 2 'Y 2 'Yi E be the corresponding maximal elements of the support sets. Let 4> = {;};eN and r = {'YiheN We have a few cases to consider: 4> and r are the same constant sequence, one of the sequences is eventually constant, or neither sequence is eventually constant. If there exists n E N such that ; = a = 'Yi for i, j > n, then we get the following string of inequalities in R: /i(1) /2(2) .. Y2h2) Y1h1), Since JR is an almost 111-set, there is r E :R such that !;(;) r Yi('Yi) for all i,j. Then J; rx 0 < Yi for all i, j. If 4> is eventually constant and r is not, say ; = a for all i > n, for some n, then, by hypothesis, there exists /3 E such that a< /3 < r. Then /; < X/J Yi for all i, j. If neither sequence is eventually constant, then by the 111-set hypothesis, there exists /3 such that ; < /3 < 'Yi for all i and /; < X/J < 9i for all i,j. 3.4 Survaluation Ring and n th -Root Closed Conditions Recall that a commutative ring A is a survaluation ring ( or SV-ring) if A/ P is a valuation ring for every prime ideal P. In this section, we set down a character ization of those /-systems which give SY-rings. Let (~, <, +, be a totally ordered and cancellative abelian monoid with identity element We define the group of differences, q~, of as it is done in [Fu, X.4). Define an equivalence relation on~ x by (61, 6 2 ) ~ {'Yi, 'Y 2 ) if and only if 61 + 'Y2 = 'Y1 + 62. It is clear that the relation is reflexive and symmetric; the transitivity follows from the cancellation property of the monoid. We let q~ be the quotient x ~/ ~, denote the class of the element (6 1 6 2 ) by [6 1 6 2 ), and define an operation+ as is usual. That is, [61 -6 2 ) + ['Y 1 -'Y 2 ] = [(6 1 +'Yi)(6 2 + 'Y 2 )]. The

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67 cancellation in the monoid ensures that the operation is well-defined. The element [ ] is an identity and [6 2 6i] is an additive inverse of [61 62]. We define [61 62] ['Yi 12] if and only if 61 + '}'2 '}'1 + 62. By [Fu, X.4.4), this is the unique order on q~ which extends the order on ~Finally, o-embed in q~ via 6 [6 ]. Let H be a partially ordered groupoid. Then h E H is called negative if hx x or xh x, or both for all x E H. The groupoid H is called negatively ordered if every element is negative. Definition 3.4.1. Let be a partially ordered semigroup. is called inversely naturally ordered if it is negatively ordered and 6 < a implies that there exists /3 E such that 6 = a + {3. Example 3.4.2. Let~= {1 }~= 1 U [1,oo) CR. be inversely ordered with the usual addition in the reals. Then is an !-system which is not inversely naturally ordered. To see this, note that (1 + c = 1 if and only if c = n = 2 or c=l,n=l. Let (~, +,)be a totally-ordered abelian cancellative monoid with maximal element, such that(~,+) is an !-system. Let X = {x 6 : 6 E ~}Then (X,*,X) is a totally-ordered abelian monoid which is -isomorphic to (~, +, ). Since X is written multiplicatively, the elements of the group qX are quotients and we denote them as such in the following proof. Theorem 3.4.3. Let(~,+,) be a totally-ordered abelian cancellative monoid with maximal element Let X = {x 0 : 6 E ~} and 'R, = F(~, R.). The following are equivalent: 1. 'R, is a valuation ring. 2. 'R, is I-convex.

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68 S. 'R, is Bezout. 4'R, is convex in q'R,. 5. X is convex in qX. 6. D. is convex in qD... 7. D. is inversely naturally ordered. Proof: The equivalence of (1), (2), and (3) is [MW, Theorem 1]; the equivalence of (2) and (4) is [ChDi2, Lemma 2]. (2) :::} (5) : Let x 7 x 0 /x13 Xo Then O Xo XH/3 and by (2), there exists / E 'R, such that Xo = f Xo+P If f = E;EJ /;X; else Thus, for some j E J, we have that a= (3 + 6 +;,I;= 1 and fk = 0 for all k-/= j. Hence f = X
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69 Let w = LkEK bkx 7 ,.. Then w() = b 1 > 0 and hence b\ w is invertible in n. Let x = L;o a;x 61 Then by cancellation in .6., we have that 0 x w and 0 b\ x b\ w. If we let f = (b\ w)1 (b\ x), then b\ x = b\ w f and x = w f. Finally, u = X{Ji x = X{Ji w f = v f, as desired. The following lemma is well-known and routine to verify. Lemma 3.4.4. Let (.6., +) be an -system and C .6., a maximal chain. Denote the associated minimal prime subgroup by He= {u En: C n supp(u) = 0}. The map cp : F(.6., lR) -+ F(C, lR) given by restriction is a surjective l-ring homomorphism with kernel He. Thus, Rf He~ F(C,R) Corollary 3.4.5. Let ( .6., +) be a 1.mital !-system. Then n is an SV-ring if and only if each maximal chain in .6. is inversely naturally ordered. Example 3.4.6. 1. If .6.1 = (0, oo) c lR is inversely ordered with the usual addition of real numbers, then F(.6. 1 lR) is an SY-ring. 2. If .6.2 = {1 }~=l U [1, oo) c JR is inversely ordered with the usual addition in the reals, then F(.6. 2 JR) is not an SY-ring. 3. Let A= JR[[x, y]] be the ring of formal power series in the indeterminates x, y. Order the monomials lexicographically via 1 x, y and xiyi < xkyl if and only if k < i or k = i and l < j. The ring A is not an SY-ring since it is not 1-convex: note that O y x and the equation y = xf has no solution f EA. Let Zx = Zy = { n E Z : n O} be inversely ordered. In the lex-order described above, if .6. = Zx x Zy, then A~ F(.6., JR). We convexify A in qA by convexifying .6. in q.6.. That is, if .6.c = .6. U {(n, m) E Z x Z : n > 0, m < 0} then F(.6.c, JR) ~ A( { xiyi : i > 0 or i = 0 and j > 0}) U A is an SY-ring which is the convexification of A in qA.

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70 Recall that an f-ring A is n th -root closed if for every a E A+ there exists b E A such that a = bn. This property arises in 7l if there is a certain amount of divisibility in the arithmetic structure of ~Definition 3.4. 7. Let ( ~, +) be an r-system. is called n-divisible if for every 8 E there exists a E such that na = 8. We say that the system is divisible if it is n-divisible for all n E N. Proposition 3.4.8. Let ( ~, +) be a totally ordered !-system such that 1l is n th -root closed. Then is n-divisible. Proof: Let 8 E ~Then Xo = vn for some v = E;o a;Xa;If 8 = is maximal in ~, then 8 = n, so we assume that 8 -:/. If a 1 E supp(v) is maximal in the support set, then na 1 E supp(vn) is maximal. Therefore 8 = nai, as desired. Proposition 3.4.9. Let(~,+) be a totally ordered inversely naturally ordered! system with maximal element. If~ is n-divisible, then 1l is n th -root closed. Proof: We begin with square-roots Let u En,+ be given by u = E;EJ a;Xa ; for some index set J, a; E JR, and a; E for all j E J. If u() -:/0, then we define a square-root v recursively on ~To begin, let v() = v1u(ji') and assume that a 1 is maximal in supp(u) \ {}. Let v(8) = 0 for all a1 < 8 < Then we define v(ai) = u(ai)/2v(). Let S be the N-linear span of supp(u) and define v(8) = 0 for all 8 (j. S. If v(,) is defined for all > 8, then we define v(8) to be the unique solution of the equation u(8) = Eo+r= 6 v(a)v(r), where, necessarily, a, r E { a E S : a 8}. Then u = v v, by construction. Now, assume that u() = 0 and let a 1 be the maximal element in the support of u. Since~ is inversely naturally ordered, for every j E J, there exists 8; E such that a;= a1 +8;. Note that 81 =.Then u = Xai E;EJ a;Xo; Let w = E;EJ a;Xo;, then w() -:/0. Thus w = v 1 v 1 by the above construction. Since is 2-divisible,

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71 a 1 = 2 11 for some 11 ED.. Therefore x 01 = X,y 1 X,y 1 Letting v = X,y 1 vi, we then have that v v = X,y 1 v 1 X,y 1 v 1 = x 01 w = u. Thus n th -roots exist when n is a power of 2. Let n be odd and let v E 'R, be given by v = EkEK bkXp,. for some index set K, bk E R, and f3k E D. for all k E K. As with square roots we consider two cases. First assume that v() =/0 and define an n th -root recursively. Let w() be a real n th -root of v() and let v(c5) = 0 for all /3 1 < 8 < where /3i is the maximal element of supp( v) \ {}. Then the n-fold convolution product equation we must solve reduces to v(/3 1 ) = wn(/3 1 ) = n(w())n-lw(/3 1 ). In order to see this, we proceed by induction on n. If n = 2, then W*w(/31) = w()w(f31)+w(/31)w() = 2w()w(/31) Assume that the statement holds for n = m. Then wm+ 1 (/31) = (w wm)(/31) = w()wm(/3i) + w(/31)wm() = w()m(w())m1 w(/3 1 ) + w(f31)(w())m = (m + l)(w())mw(/3i) as desired. We may then define w(/3 1 ) = v(/3 1 )/(n(w())n1 ). In general, we let w(,) = 0 if 'Y is not in the N-linear span of the support of v Assume that w(,) is defined for all,> 8. We show that the equation v(c5) = wn(a) is linear in w(8); hence, we may define w(8) to be the unique solution of this equation. H n = 2, then w 2 (8) = Eu+T=o w(o-)w(-r) + 2w(8)w(). Assume that wm(a) is linear O',Tf.0 in w(c5). Then wm+l(a) = L w(a)wm(-r) + w()wm(8) + w(8)wm() O'+T=6 O',Tf.0 is linear in w(8), by induction, since w(8) will not appear in wm(-r), as 8 < 'T and D. is an !-system. Thus, in this case, v has an n th -root. Second, assume that v() = 0 and proceed as in the square-root case. Let /3 1 be the maximal element in the support of v. Since D. is inversely naturally ordered, for every k E K, there exists 8k E D. such that f3k = /3 1 + 8k. Note that 8 1 = Then

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72 v = XtJi LkEK bkX5,.. Let x = LkEK bkX5,., then x() -/:0. Thus x = wf, for some w 1 by the above construction. Since 6. is n-divisible, /31 = n-y 1 for some 'Y i E 6.. Therefore Xf3 1 = (x, 1 )'1'. Let w = x, 1 w1; then w" = (x, 1 w1)" = Xf3 1 x = v. Corollary 3.4.10. Let (6., +) be a totally ordered inversely naturally ordered! system with maximal element 6. is divisible if and only if 1l is n th -root closed for all n. Example 3.4.11. 1. Let 6. 1 = [0, oo) c JR be inversely ordered with the usual addition of the real numbers. Then F(6.1, R) is n th -root clo.5ed for all n. 2. Let 6.2 = {n E Z : n 2:: 0} be inversely ordered. Then 6.2 is not 2-divisible and Xi > 0 has no square-root. That is, if Xi = v 2 then v(O) = 0 and we then conclude that 1 = x1(l) = 2v(0)v(l) = 0, a contradiction. 3. Let 6. 3 = {l-}~=3U [ 1, oo) CR be inversely ordered with the usual addition in the reals, then 6. 3 is not inversely naturally ordered and similarly x 1 has no square-root since 1 has no nonzero summand. Proposition 3.4.12. Let (6., +) be a totally ordered /-system with maximal ele ment Then if has an immediate predecessor 1r, then X1r has no square-root. If there exists > 6 E 6. such that 6 has no nonmaximal summand other than itself, then X5 has no square-root. Based on the preceding examples and results on n th -roots, we formulate the following: Conjecture 3.4.13. Let (6. +) be a totally ordered /-system with maximal element F(A, R) is square-root closed if and only if 6. is 2-divisible and every nonmaximal element of 6. has a nonmaximal summand other than itself

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73 Recall that a field K is real-closed if every positive element is a square and every polynomial p E K[x] of odd degree has a root in K. An integral domain R is called real-closed if qR is a real-closed field. Let (~, +) be a totally ordered inversely naturally ordered !-system with maximal element Assume that is also 2-divisible. Then 'R, is a 1-convex and square-root closed !-domain. By [ChDil, Theorem 1], under these conditions, 'R, is real-closed if and only if every odd degree polynomial over 'R, has a root. What additional assumptions on are necessary to guarantee the real-closed property? Conjecture 3.4.14. Let (~, +) be a totally ordered commutative !-system with maximal element If~ is divisible and inversely naturally ordered, then 'R, is real closed We end this section by shedding a little light on this conjecture. Recall from [HLM] that a commutative f-ring A with 1 satisfies the Intermediate Value Theorem for polynomials (or is an !VT-ring, for short), if for every p(t) E A[t], and pair of distinct elements u, v E A such that p( u) > 0 and p( v) < 0, there exists w E A such that p(w) = 0 and u I\ v w u V v. We show that a totally ordered commutative semiprime valuation !-domain with identity is real-closed if and only if it is an IVT ring. It is necessary to record the following unpublished theorem of Suzanne Larson, which was communicated via electronic mail on April 17, 1997. Her proof follows. Theorem 3.4.15. Let A be a commutative semiprime !VT-ring with identity. If S is a multiplicatively closed subset of regular elements of A+, then the ring of quotients, s1 A is an !VT-ring. Proof: Let p(t) E s1 A[t] be given by p(t) = a 0 b 0 1 + a 1 b 1 1 t + + anb;/tn and assume that p( u1 v 1 1 ) 0 and p( u 2 v 2 1 ) 0. Let d = vfv 2 b 0 b 1 bn. Then d E S is regular Define a new polynomial q(t) E A[t] by

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74 Then q(u 1 v 2 ) = dp(u 1 v 1 1 ) 0 and q(u 2 v 1 ) = dp(u2v 2 1 ) ::; 0. Since A is an IVT-ring, there exists w E A such that u 1 v 2 I\ u 2 v 1 ::; w ::; u1 v2 V U2V1 and q( w) = 0. Then u 1 v 1 1 /\ u 2 v 2 1 ::; wv 1 1 v 2 1 ::; u 1 v 1 1 Vu2v 2 1, and dp(wv 1 1 v 2 1 ) = q(w) = 0. Since dis regular, p(wv 1 1 v 2 1 ) = 0 and we conclude that the quotient ring is an IVT-ring. Proposition 3.4.16. Let A be a totally ordered commutative semiprime valuation !-domain with identity. Then A is real-closed if and only if it is an /VT-ring Proof: If A is real-closed, then qA is a real-closed field and, by [ChDi2], qA is an !VT-field. Let p( t) E A[t] be such that p( u) > 0 and p( v) < 0, for some u, v E A. Then there exists w E qA such that p(w) = 0 and u I\ v ::; w ::; u V v. Since A is a valuation domain, A is convex in qA by [ChDi2, Lemma 2]. Hence, w E A and we have that A is an IVT-ring. Conversely, if A is an IVT-ring, then qA is an !VT-field, by the preceding theorem of Larson. Then, by [ChDi2], qA is real-closed and therefore A is real-closed.

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CHAPTER 4 RAMIFIED PRIME IDEALS In this chapter we expand on the notion of a ramified prime ideal, as defined in [Le], which we introduced in Chapter 2. We first examine the concept in general and then move to try to understand ramified maximal ideals which correspond to nonisolated G 6 -points. This result is then used to consider local versus global ramification conditions. 4.1 Ramified Points Definition 4.1.1. Let X be a completely regular space. A prime ideal of C(X) is ramified if it is the sum of the minimal prime ideals that it contains. We define p E X to be ramified if Mp is ramified. A point p E X is totally ramified if every prime z-ideal contained in Mp is ramified. The space X is (totally) ramified if every nonisolated point of X is {totally} ramified. A ramified -ideal of C(X) is a prime ideal, by Corollary 2.2.6. LeDonne proves that a ramified prime ideal of C(X) is necessarily a z-ideal. Let us consider two extreme conditions. Recall that we say a point p E X is an F-point if Op is prime. If pis an F-point, then since Op is the unique minimal prime ideal contained in Mp, Mp is not ramified Likewise, in this case, no prime z-ideal contained in MP is ramified. On the other hand, the condition of total ramification ensures branching at every prime z-ideal. Analytically, LeDonne shows [Le, 3]: Theorem 4.1.2. If Xis a metric space then every maximal ideal of C(X) is totally ramified. 75

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76 Note that this result says that every maximal ideal of C(X) (fixed or free) is ramified, if X is metric. We do not know of any weaker topological condition which guarantees total ramification of C ( X). Definition 4.1.3. Let A be a commutative /-ring with identity. For any integer n 2, call a prime l-ideal P n-limbed if P is the sum of n noncomparable prime l-ideals of A which are properly contained in P. A point p of X is n-limbed if M 11 is n-limbed. Note that any n-limbed l-ideal P is necessarily ramified and rk{P) n. Example 4.1.4. We now present an example of an /-ring 'R, in which a maximal ideal is ramified but not n-limbed for any n. Let Ao = [O, oo) R and define A~ = (1/n, oo) JR, for i = 1, 2, where each interval is inversely ordered. Let An = A~ II~ be the disjoint union and then let A = -6,o II (IInEwAn). We obtain a root system with the induced ordering which we describe in the following diagram: .L\o {4.1) We endow A with a partial commutative associative binary operation. Let ( )~ denote the sum in parentheses as the usual sum of real numbers residing in Ai n,

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77 the mark "-" signifies that the sum is undefined. Note, to conserve space, the table is completed by reflection across the diagonal. + r E Ao r EA:_ r EA:,. r E At r E Af s E Ao (r + s)o (r+s):, (r + s)!.. (r + s)t (r + s)f II E A:, (r + s); (r + s); (r + s)t ifk < n; else (r + s)t if l < n; else s E A:,. (r + s):,. (r + s)Z if k < m; else (r + s)j if l < m; else s EA! (r + s)t s E Aj (r + s)t Let 'R, = F(fl., R) and Vn = Llo II (IIm>nA~); let Co = Ao II (llnEwA~) and Cn = Vn II A!. Then the minimal prime ideals of 'R, correspond to these maximal chains and are given by Qn = { u E 'R, : u(Cn) = {O}}, where n = 0, 1, 2, .... Any supremum vjEJ Q; over a finite set J C w is the prime ideal Pm given by { u E 'R, : u(Vm) = {O}}, where m is the maximum element of J. Hence, for all n E w, the maximal ideal Yo is not n-limbed since it is not a finite supremum of minimal prime ideals. However, V 0 = V nEw Qn, and so it is ramified. We show in Proposition 2.3.4 that for any nonisolated point p of a first countable space, the maximal ideal MP is n-limbed for every n. H the space is also cozero-complemented, then rk(Mp) 2:: 2' and Mp is 2'-limbed. From this, we also obtain the following, which is weaker than Theorem 4.1.2: Corollary 4.1.5. Every metric space is ramified. The following theorem means that if X is a metric space that is not pseudocompact, then there exist points of f3X \ X such that MP is minimal. Hence, not every maximal ideal of C(X) branches nontrivially in the root system Spec(C(X)). Recall that we call a topological space X perfect if every closed set of Xis a G 6 -set. Note that any metric space is perfect. A point of /3X is remote if it is not in the /JX-closure of any nowhere dense subset of X. A point p E /3X \ X is a C-point if p intpx\x(clpxZ(/) n /3X \ X) for all / E C(X). A theorem similar to the following appears in [W, 4.40]. All the proofs there carry through here, verbatim, under our reduced hypotheses.

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78 Theorem 4.1.6. Let X be a completely regular space and consider the following conditions on a point p E {3X \ X: 1. pis a C-point. 2. Z[MP] contains no nowhere dense set. s. MP= QP. ,f.. p is a remote point Without additional assumptions, (3) => (1). Let X be perfect and assume the exis tence of a remote point, then (4) => (2) => (3). If X is perfect and the set of isolated points of X has compact closure in X, then (2) => (4). If X is realcompact and C-points exist, then (1) => (2). It is not known if a remote point p always has the property that MP = QP. We do know the following about the rank of a remote point: Proposition 4.1.7. Let X be a completely regular space. Let p E {3X be a remote point. Then rkc(x) (MP) = 1. Proof: In [vD, 5.2], it is demonstrated that no remote point is in the closure of two disjoint open sets of {3X. Thus, in particular, no remote point is in the closure of two disjoint cozero-sets of {3X. By [HLMW, 3.1], we have 1 = rktJx(p) = rkc(tJX)(M*P). Finally, since C(f3X) is rigid in C(X), we have rkc(x)(MP) = 1. Finally, we ask: does ramification of a point in X indicate ramification in {3X, or vice versa? Proposition 4.1.8. A point p EX is ramified in X if and only if it is ramified in {3X. Likewise, a point p E vX is ramified in vX if and only if it is ramified in {3X. Proof: This is a corollary of Proposition 2.2.12, since we know C(f3X) = C*(X) and MfX = M{ n C(f3X).

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79 4.2 Ramilied G 6 -points The main theorem, Theorem 4.2.5, of this section provides a good method for checking the ramification of G 6 -points. We will use it to characterize ramified G 6 -points in normal countably tight spaces and to find some ramified points in product spaces. We first discuss the following proposition, which we will obtain as a corollary to Theorem 4.2.5. Proposition 4.2.1. Let p E X be a G 6 -point. If X \ p is not c -embedded in X, then rk(p) 2. Since an F-point has rank 1, the preceding proposition, proved in [Le) and (in greater generality) by van Douwen in [vD), shows that a G 0 -point, p, is not an F-point if it has the property that X \pis not C*-embedded in X. We give a counterexample for the converse if the G 0 -condition is lifted. Example 4.2.2. Let X = B 2 (N, (3N \ N) be as defined in Example 5.2.5. There, we show that there exists a point p of the corona such that rk(p) = 2, X \ p is C*-embedded in X and pis not ramified. But no point of the corona is a G 6 The following two results are Theorems 2.1 and 2.2 of [K2); we will use these to prove our theorem on the ramification of G 0 -points. Theorem 4.2.3. Let p be a nonisolated G 0 -point of X. If Z E Z[C(X \ p)] then clx(Z) E Z[C(X)). Define : Z[C(X \ p)] -+ Z[C(X)] by ,(Z) = clx(Z). Let be the extension of the identity map X \p-+ X to the largest subspace X 1 f3(X \p) such that it is extendible as a continuous map into X.

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80 Theorem 4.2.4. Let p be a nonisolated G 0 -point of X then 1. The mapping 1 is one-to-one from the set of prime z-filters on X \p converging to points of 4>1 ({p}) onto the set of prime z-filters on X contained properly in Z[Mp]2. A prime z-filter W on X \p converging to a point of 4>1 ( {p}) is a z-ultrafilter if and only if ,(W) is maximal in the class of prime z-filters on X contained properly in Z[Mp]Theorem 4.2.5. Let p be a nonisolated G 0 -point of X. The point p is ramified if and only if X \ p is not C* -embedded in X. Proof: Let p be a nonisolated G 0 -point of X. If p is not ramified then the prime z-ideal P = EQEMin(Mp} Q is properly contained in Mp. We will show, in this case, that every point of Xis the limit of a unique z-ultrafilter on X \p. Then by [GJ, 6.4], X \pis C*-embedded in X. Let q E X \ p. Then Mq E Max(C(X \ p)) gives rise to the z-ultrafilter Uq = Z[Mq] on X \p. Clearly q E n{clx(U): U E Uq}The uniqueness of Uq is a standard result [GJ, 3.18]. By (GJ, 6.3(b)], there exists a z-ultrafilterU on X\p converging top. Assume that there exists another such z-ultrafilter, V. Let U = ,U and V = ,V. Then Qu = ZtU and Qv = ZtV are prime z-ideals of C(X) which are properly contained in Mp. If Qu P then U = Zx[Qu] Zx[P]. Hence U = ,1 u ,1 Zx[P], which contradicts that U is a z-ultrafilter on X \p. Likewise, Qv is not properly contained in P. Thus P Qu, P Qv and by [GJ, 14.8(a)], we must have either Qu c Qv ~ ~ or Qv C Qu. But Qu c Qv gives that U V and therefore U V. Since U is an ultra.filter, U = V, as desired. In a similar manner, if Qv C Qu, then V = U. Conversely, assume that p is ramified. Then Mp = E Min(Mp) and there exists more than one prime z-ideal in C(X) which is maximal in the class of prime

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81 z-ideals properly contained in Mp. These give distinct prime z-filters on X which are maximal in the class of prime z-filters on X properly contained in Z[Mp]Hence, via,, we have distinct ultrafilters on X \ p converging top. Again by (GJ, 6.4), X \ p is not C* -embedded in X. Corollary 4.2.6. If X is a metric space, then X is ramified. Corollary 4.2. 7. If X is first countable and p E X is nonisolated, then X \ p is not C* -embedded in X. Corollary 4.2.8. If p EX is a G 0 -point and X \pis not C*-embedded in X, then rk(p) 2. Let X and Y be completely regular spaces which are not P-spaces and let W =Xx Y. We conjecture that every nonisolated point of Wis ramified. We use Theorem 4.2.5 to deduce two partial answers to this question. Proposition 4.2.9. Let X and Y be completely regular spaces and let W =Xx Y. Let x E X and y E Y be nonisolated G 0 -points and let p = (x, y) E W. Then W \ {p} is not C* -embedded in W. Proof: Since X is completely regular and x is a G 0 -point of X, { x} is a zero-set of X. Say, {x} = Zx(f) for some f E C(X). Then we have for Ex= {x} x Y, Ex= {x} x Y = 1rt(Zx(/)) = Zw(/ o 1rx) where 1rx denotes the natural projection from W onto X. Likewise, we have that E 11 =Xx {y} = Zw(g o 1ry) where g E C(Y) such that Z 11 (g) = {y} and 1ry is the natural projection from W onto Y. Let f and g denote the restrictions off 01rx and go1ry to W\ {p}. Then we have that Ex\ {p} = Zw\{p}(f) and E 11 \ {p} = Zw\{p}(g) are disjoint zero-sets of W\ {p}. Thus Ex\ {p} and E 11 \ {p} are completely separated in W\ {p}. But p E clw(Ex \ {p} )nclw(E 11 \ {p}) and therefore Ex\ {p} and E 11 \ {p}

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82 are not completely separated in W. By the Urysohn Extension Theorem, W \ {p} is not c -embedded in W. Corollary 4.2.10. Let X and Y be completely regular spaces and let W =Xx Y. Let x EX and y E Y be nonisolated G 0 -points and let p = (x, y) E W. Then p is ramified in W Proof: Follows from Theorem 4.2.5. Proposition 4.2.11. Let X and Y be completely regular spaces and let W = Xx Y. Let p = ( x, y) E W be non isolated. If W \ {p} is normal, then W \ {p} is not c embedded in W. Proof: Let Ex = { x} x Y and E 11 = X x {y}. Then Ex\ {p} and E 11 \ {p} are disjoint closed sets in the normal space W \ {p}. Thus, they are completely separated in W\ {p}. But p E clw(Ex \ {p}) n clw(E 11 \ {p}), so Ex\ {p} and E 11 \ {p} are not completely separated in W. Therefore, by the Urysohn Extension Theorem, W\ {p} is not c -embedded in W. Corollary 4.2.12. Let X and Y be completely regular spaces and let W =Xx Y. Let p = (x, y) E W be a nonisolated G 0 -point of W. If W \ {p} is normal, then p is ramified in W Proof: Follows from Theorem 4.2.5. We now investigate the ramification of G 0 -points in a class of spaces more general than metric or first countable spaces. A topological space X is countably tight if for a subset U c X we have that any p E cl(U) is in the closure of a countable set S c U. A Frechet-Urysohn space is one in which every p E cl(U) is the limit of a sequence of distinct points {p,J c U. It is evident that any Frechet-Urysohn space is countably tight.

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83 Lemma 4.2.13. Let X be a normal topological space and let p EX be nonisolated. Then X \ {p} is c -embedded in X if and only if p cl x (A) n cl x ( B), whenever A and B are completely separated in X \ {p}. Proof: By the Urysohn Extension Theorem, X \ {p} is C*-embedded in X if and only if A and B are completely separated in X, whenever A and B are completely separated in X \ {p}. Assume that X \ {p} is c -embedded in X and let A and B be completely separated in X \ {p}. Then A and B are completely separated in X and hence are contained in disjoint closed sets of X. Thus p clx(A) n clx(B). Conversely, let A and B be completely separated in X \ {p}. We wish to show that A and Bare completely separated in X. By hypothesis, p
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84 Corollary 4.2.15. Let X be normal FrechetUrysohn space and let p E X be non isolated. Then X \ {p} is C* -embedded in X if and only if there do not exist two sequences in X which are completely separated in X \ {p} and converge to p. Finally, Proposition 4.2.14 and Theorem 4.2.5 imply the following. Corollary 4.2.16. If X is a normal countably tight topological space and p EX is a nonisolated G 0 -point, then pis ramified in X if and only if for every two countable sets S 1 and S 2 which are completely separated in X \ {p}, p clx(S1) n clx(S2)4.3 Ramification via C-Embedded Subspaces Let A and B be commutative rings with identity. Assume that we have a surjective ring homomorphism f: A B, with K = Ker(!). Recall that there is a one-to-one order-preserving correspondence between Spec(B) and the set of prime ideals PE Spec(A) such that K P. Let ME Max(B) be such that M = P1 + P2 for some nonmaximal proper primes P 1 P 2 E Spec(B). If NE Spec(A) corresponds to M, then we have B/M ~ (A/K)/(N/K) ~ A/N. So NE Max(A). Let Q 1 ,Q 2 be the prime ideals of A corresponding to Pi and P2. Then ( Qi + Q2) / K ~ P1 + P2 via the surjective map given by a + b f (a) + f (b) with kernel K. Thus, by the correspondence, we have that N/ K M = Pi +P2 ~ (Qi +Q2)/ Kand N = Qi +Q2. In fact, we have shown: Proposition 4.3.1. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A B with K = Ker(!). If P B is a prime ideal which is a sum of two distinct prime ideals then there exists Q E Spec(A) such that Q/ K ~ P and Q is a sum of two distinct prime ideals of A. Ramification in a C-embedded subspace implies global ramification. Corollary 4.3.2. Let X be a completely regular space and let Y be a C-embedded subspace. If a point p of Y is 2-limbed in Y, then p is 2-limbed in X.

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85 Proof: Since Y is C-embedded in X, we have a surjective ring homomorphism from C(X) onto C(Y), given by restriction with kernel {f E C(X) : Y Zx(f)}. Hence this result follows from Proposition 4.3.1. Proposition 4.3.1 also gives the following, since any compact subspace of a completely regular space is C-embedded. Corollary 4.3.3. 1. Let X be compact, Y X a closed subspace If a point of Y is 2-limbed in Y then it is 2-limbed in X. 2. Let X be a compact space consisting of more than one point. If every noniso lated point in a proper zeroset of X is 2-limbed, then every nonisolated point of X is 2-limbed. Example 4.3.4. The Cantor Set is metric, hence every point is 2-limbed. By [E, 3.12.12c], every point of 2r, the Cantor space of weight T, is contained in a closed set which is homeomorphic to the Cantor Set. Thus every nonisolated point of 2r is 2-limbed. In fact, induction on Proposition 4.3.1 gives that every point of the Cantor space is n-limbed for every n EN. Conversely, if a maximal ideal of A containing K is a sum of primes containing K, then by the correspondence given above, its image is a maximal ideal which is a sum of primes in B. That is, if NE Max(A), K N, and N =Qi+ Q2 such that K Qi and K Q2, then f(N) E Max(B) as B/f(N) Or! (A/K)/(N/K) A/N. And f(N) = f(Qi + Q2) = f(Qi) + f(Q2) ~ Qi/K + Q2/K. This gives a partial converse: Proposition 4.3.5. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A B with K = Ker(!). Let P be a prime ideal of A containing K. Then P is a sum of two distinct prime ideals

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86 containing K if and only if f(P)/ K is a prime ideal which is a sum of two distinct prime ideals of B. Corollary 4.3.6. The ring-homomorphic image of a commutative ramified ring with identity is ramified. Corollary 4.3.7. Let Y be a C-embedded subspace of X and let p E Y have finite rank in X. If p is ramified in Y, then p is ramified in X. If any function in C(X) vanishing on Y also vanishes on a neighborhood of p, then p is ramified in Y if and only if p is ramified in X. Proof: The first statement is an application of Proposition 4.3.1 by induction. The second statement follows from Proposition 4.3.5 by induction. Note that the hypothesis of the second statement of the above merely de mands that the kernel of the restriction map be contained in o;. This is satisfied if Y is open or if Y is a P-set. The preceding results indicate that ramification is a local property. Let A be a commutative ring with identity, let S be a multiplicative system in A such that 1 E S. Then there exists a one-to-one correspondence from Spec(s1 A) onto {PE Spec(A) : P n S = 0}. The proofs of the following are routine: Proposition 4.3.8. Let A be a commutative ring with identity, let S be a multi plicative system in A such that 1 E S. 1. If P E Spec(s1 A) is a sum of two proper primes, then the preimage of P, the set {x EA: x/1 EA}, is a prime ideal which is a sum of two proper primes in A. 2. If Q E Spec(A) is a sum of primes and Q n S = 0, then s1 Q is a sum of primes in s1 A.

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87 Corollary 4.3.9. Let Y be a subspace of X such that C(X) <-t C(Y) is a ring of fractions map. That is, there is a multiplicative system S C C(X) such that 1 E S and C(Y) = s1 (C(X)). Then: 1. For p E Y, rky(p) rkx(p). 2. Let p E Y. If M{ n S = 0, then there is PE Spec(C(X)) such that P M{ and rky(p) = rkx(P).

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CHAPTER 5 m-QUASINORMAL f-RJNGS In [Lal 3], Suzanne Larson defines the notion of a quasinormal !-ring; one in which the sum of any two distinct minimal prime ideals is a maximal -ideal or the entire f-ring. We generalize this definition and a few of her results. 5.1 Definitions Definition 5.1.1. Let A be a commutative f-nng with identity and let m be a positive integer. Call A m-quasinormal if the sum of any m distinct minimal prime ideals is a maximal -ideal or the entire f-ri,ng A. If X is a topological space such that C(X) is m quasinormal, then we call X an Fm-space. Note that the "2-quasinormal" is Larson's "quasinormal" condition, the "1quasinormal" condition is equivalent to von Neumann regularity, and if A is quasinormal then A is n-quasinormal for any n > m. Hence, the F 1 -spaces are exactly the P-spaces and any Fm-space is an Fn-space, when n > m. Theorem 5.1.5 generalizes [Lal, 3.3] and characterizes the m-quasinormal semiprime !-rings. Note that [Lal, 2.2], which we now state, gives necessary and sufficient conditions for a semiprime f-ring to have the property that the sum of any two distinct minimal prime ideals is a prime -ideal. This condition is stronger than the assumption we make in our theorem, but this result indicates when one can expect to be able to apply it. Theorem 5.1.2 Let A be a semipri,me f-ring. The following are equivalent: 1 The sum of any two semiprime -ideals is semiprime. 88

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89 2. The sum of any two minimal prime -ideals is prime. S. The sum of any two prime -ideals is prime. 4,. For any a, b E A+, the -ideal a.1..1. + b.1..1. is semiprime. 5. For any two disjoint elements a, b E A+, the -ideal a.l. + is semiprime. 6. For any a, b EA+, the -ideal a.1. + b.1. is semiprime. 7. When x,a,b,c,d E A+,a,b =/ 0 are such that x 2 = c+d and a/\c = b/\d = 0, there exist g, h E A such that x = g + h and g I\ a = h I\ b = 0. The theorem above holds for C ( X) since the sum of two prime ideals is prime by [GJ, 14B]. We will use the following lemmas. The first follows from the fact that a prime -ideal P of a semiprime /-ring is minimal if and only if for every p E P there exists q P such that pq = 0. The second lemma shows the existence of certain functions which we will take for granted in the proofs to follow. Lemma 5.1.3. Let A be a semiprime /-ring in which the sum of any m distinct minimal prime ideals is a prime -ideal. Let P be a prime -ideal. Then P is minimal with respect to containing EJ=i af if and only if for every p E P there exists q ff. P such that pq E Ei= 1 af. Lemma 5.1.4. Let A be a commutative /-ring. Let n 2:: 2 and let Q 1 .. Qn be distinct minimal prime ideals. Then there exists an element f E Qt\ LJ#i Q;, for each i = 1, ... n. Proof: Let fk E Qt \ Qk for k = 1, ... n. Then, by convexity, we have that f = v:=2/i E Qt\ LJ#i Q;, as desired. Theorem 5.1.5. Let m be a positive integer. Let A be a commutative semiprime /-ring with identity in which the sum of any m distinct minimal prime ideals is a prime -ideal. The following are equivalent:

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90 1. A is m-quasinormal. 2. For every nonmaximal prime -ideal P, rk(P) m 1. 9. Let { a; }~ 1 be a family of positive pairwise disjoint elements of A. Proper prime -ideals containing L.1=i af are maximal -ideals. 4. Let { a;}~ 1 be a family of positive pairwise disjoint elements of A, let M be a maximal -ideal containing the -ideal LJ=l af, and let p E M. Then there exists z rl. M such that zp E L.1= 1 af. Proof: (1) => (2): Let P be a nonmaximal prime -ideal of A such that rk(P) m and let Q 1 ... Qm be m distinct minimal prime ideals that are contained in P. Then LJ=l Q; P is not maximal, hence A is not m-quasinormal. (2) => (3) : Let P be a prime -ideal containing L.1=i af. Then af P for every j. Therefore, a; rl. O(P) for all j, and hence P contains at least m minimal prime ideals by the pigeonhole principle. Thus condition (2) gives us that Pis a maximal -ideal. (3) # (4): Follows from Lemma 5.1.3. (3) => (1): Let M be a maximal -ideal and let Q 1 ... Qm M be minimal prime ideals. Then by hypothesis, L.1=i Q; is a prime -ideal and we are left to show that it is a maximal -ideal. For each j = 1, ... m, let a; E Qt \ Uit;Qi and define b; = /\i#;ai-/\~ 1 ak E L.1=i Q;. Then {b;}.i=l is a pairwise disjoint set of m distinct elements of L.1=i Q; and by the choice of the a;'s, we have that /\;:1= 1 ak E n;:1= 1 Qk by convexity and /\i#iai rl. Q; for each j. Hence b; rl. Q; and bf Q;, for each j. Thus L.1=i Q; is a maximal -ideal by condition (3), since L.1=i bf~ L.1=i Q;. The quasinormal condition is a variation of the normal condition, which is that the sum of any two minimal prime ideals of a semiprime /-ring with identity is the entire !-ring. This is discussed in [Hu]. The expected generalized definition

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91 follows, along with a theorem recording two equivalent conditions. The result is a special case of Theorem 5.1.5 and the proof follows immediately from the one above and from [Hu, Theorem 8]. Definition 5.1.6. Let m 2 be a positive integer. An /-ring A is m-normal if for any painnise disjoint family {a;}j=:: 1 we have that A= Ei=l af. Theorem 5.1. 7. Let A be a commutative semiprime !-ring with identity and let m 2 be a positive integer. The following are equivalent: 1. A is m-normal. 2. For any maximal -ideal M, we have that rk(M) m 1. 3. The sum of any m distinct minimal prime ideals is A. Before we move to describe Fm-spaces, we first discuss a special class of m-quasinormal !-rings. Definition 5.1.8. Let A be a local /-ring. An embedded prime -ideal P is high if for every minimal prime ideal N E Min(A), either N P or NV P = Iex(A). Otherwise, P is low. Call an !-ring A a broom ring if for every maximal -ideal M every prime -ideal in AM is high. The following is immediate from Proposition 2.5.2 and Lemma 5.1.3. Proposition 5.1.9. Let A be a local commutative semiprime /-ring with identity and maximal -ideal M. The following are equivalent: 1. A is a broom ring. 2. 1r(A) 2. 3. If P Iex(A) is a prime -ideal, then rkA(P) 1.

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92 4. If a, b E A are disjoint and P is a proper prime -ideal containing a.1 + b.1, then lex(A) P M. 5. If a, b EA are disjoint and a.1 + b.1 lex(A), then for every p E lex(A), there exists z rt, lex(A) such that zp E a.1 + b.1. Proof: (1) # (2) : Since every prime -ideal of A is high, we have that for any two distinct minimal prime ideals Qi, Q2 that Qi V Q2 = lex(A). Hence 1r(A) :::; 2 by definition. Conversely, 1r(A) :::; 2 implies that every minimal prime ideal is high and therefore that every prime -ideal is high, as desired. (2) # (3) : Since 1r(A) :::; 2, we have that rk(P) :::; 1 for any embedded prime -ideal P, by Proposition 2.5.2; and vice versa. (2) (4) : By Proposition 2.5.2, we have that a.1 + b.1 = lex(A). (4) # (5): Follows from Lemma 5.1.3. (4) (3) : Assume that (4) holds. Let P be an embedded prime -ideal and assume, by way of contradiction, that rk(P) 2. Let Qi, Q2 P be minimal prime ideals. Let Q1 E Qf \ Q2 and let Q2 E Qt \ Q1. Disjointify by defining Qj = Qj Q1 /\ Q2 for j = 1, 2. Then Q1 .1 Q2 since Q1 rt, Q2 and Q2.1 Qi since Q2 rt, Qi. Then q 1 .1 + q 2 .l P and hence lex(A) P, which contradicts that P is embedded. Example 5.1.10. We now present an example of an /-ring which is 3-quasinormal but is not a broom ring. Let Ao = A1 = [0, oo) 1R, and A2 = Aa = (1, oo) 1R, where each is inversely ordered. Identifying the copies of 0 in the disjoint union A 1 = (Ao II A 1 )/(0 0 ~ 0 1 ) and letting A 2 be the disjoint union A 2 II A 3 we obtain a root system A= A 1 IIA 2 with the induced ordering which we now describe. That is, r < s if and only if either r, s E Aj for j = 0, 1, 2 or 3 and r
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93 We endow A with a partial binary operation. To begin, note that we define 0o + 0o = 0 1 + 0 1 = 0o + 0 1 = 0 0 ~ 0 1 Let r;, s; E A; be nonzero for j = 0, 1, 2, 3. Let ( ); denote the sum in parentheses as the usual sum of real numbers residing in A;; the mark "-" signifies that the sum is undefined. + ro so rt st r2 s2 rs 83 ro (2rolo (ro + solo so (ro + so)o (2so)o rt (2rth St +rt t (r2 + r1)2 (s2 + rt)2 (rs+ ri)s (ss + ri)s s1 (r1+sth (2sth (r2+st)2 (s2 + s1)2 (rs+s1)s (s3 + St)s r2 (rt + r2 2 s1 + r2 2 (2r2)2 (s2 + r2)2 s2 (rt + s2 2 s1 + s2 2 lr2 + s2)2 (2s2)2 rs (rt +r3J3 St + rs s (2rs)s (ss + rs}s 83 (rt+ s3 s St+ 83 3 (r3 + s3)3 (2s3)3 Let Co = Ao, C1 = A1, C2 = A1 II A2, and C 3 = A1 II A 3 Then the minimal prime ideals of F(A, JR) are of the form Q; = { u E F(A, JR) : supp(u) A\ C;} for j = 0, 2, 3. The similarly defined Q 1 is a prime ideal by Proposition 3.3.15. Now, it is evident that L = lex(F(A,JR)) = Q 0 V Q2 V Q 3 and Q2 V Qa = Q1 =IL so we know that 1r(F(A, JR)) = 3. Therefore, Proposition 5.1.9 shows that the ring is not a broom ring. Since L is the maximal ideal of the ring, we have shown that F(A, JR) is 3-quasinormal. We present a similar example of an broom ring that is not quasinormal. Example 5.1.11. Let Ao = (0, oo) lR, and A 1 = A 2 = (1, oo) lR, where each is inversely ordered. Let A 1 = A 0 and let A 2 be the disjoint union A 1 II A 2 in order to obtain a root system A= A 1 IIA 2 with the induced ordering which we now describe. That is, r < s if and only if either r, s E A; for j = 0, 1 or 2 and r < s in the inversely ordered real numbers; or if r E Ai, s E A 0 ; or if r E A 2 s E A 0 Explicitly, r II s if r E A1 and s E A2. We endow A with a partial binary operation. Let r;, s; EA; for j = 0, 1, 2. Let ( ); denote the sum in parentheses as the usual sum of real numbers residing in A;; the mark "-" signifies that the sum is undefined.

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94 + ro so r1 81 r2 s2 ro (2ro)o so +ro o (r1 + roh s1 + ro 1 (r2 +roh (s2 + roh so ro + so o (2so)o (r1 + soh 81 + so 1 1r2 +soh (s2 + soh r1 ro + r1 1 so+ r1 1 (2r1)1 s1 + r1 1 s1 ro + s1 1 so+ s1 1 (r1 + s1h (2s111 r2 ro +r2 2 so+ r2 2 (2r2)2 (s2 + r2)2 s2 ro + s2 2 so+ s2 2 (r2 + s2)2 {2s2)2 Let Co = .6.o, C 1 = .6. 0 II .6. 1 and C2 = .6.o II .6.2. Then the minimal prime ideals of F(.6.,R) are of the form Q; = {u E F(.6.,R.) : supp(u) .6. \ C;} for j = 1,2; the ideal Q 0 is a prime ideal by Proposition 3.3.15. Now, Qo = lex(F(.6., R)) = Qi V Q2 and so we know that 1r(F(.6., R)) = 2. Therefore, Proposition 5.1.9 shows that the ring is a broom ring. Since Q 0 is not the maximal ideal of the ring, we have shown that F(.6., :R) is not quasinormal. D 5.2 (B, m)-Boundary Conditions Definition 5.2.1. Let m be an integer greater than 1 and let {U;}~ 1 be a family of m pairwise disjoint cozerosets of a topologirol space X. The subspace nj=l clx(U;) is rolled an m-boundary in X. Let B be a topological property. We say that a space X satisfies the (B, m)-boundary condition if every m-boundary in X has property B. In [Lal, 3.5], Larson proves that if Xis completely regular, every point of X is a G5-point, and C(X) is quasinormal, then X satisfies the (discrete, 2)-boundary condition. This result is improved in [La3, 3.5) to say that if X is normal and for every p E /3X \ vX, the t'-ideal QP is prime then C(X) is quasinormal if and only if X has the (finite, 2)-boundary condition. Here we refine this theorem by removing the restriction on the points of the corona. First, a lemma, extending [La3, 3.1), which we henceforth refer to as "Larson's Lemma": Lemma 5.2.2. Let X be normal and let {g;}~ 1 C(X)+ be a family of pairwise disjoint functions. Define Ym = n;:1 cl X ( coz(g;)). Then E;:1 gf = r\eYm My.

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95 Proof: Each function in E1=1 9f must vanish on Ym, hence E1=1 9f nyEYm My. For the reverse inclusion, we use recursion. The proof of the base case of m = 2 is in [La.3, 3.1). Let f E nyEYm My and define fi(x) = {Jo I\ 1 if x E cl(coz(91)), if X E n;: 2 cl ( coz (9;)). Since X is normal and Ji is defined on a closed set, the function has a continuous extension, 1 1 E C(X). Then f /\1-1 1 E 9f and 1 1 E n{Mp: p E nf= 2 cl(coz(9;))}. Recursively define a function 1 k to be the continuous extension of fk(x) = {1k-l I\ 1 ~f X E cl~z(9k)) 0 If X E nj=k+l cl(coz(9;)). Then lk-1 I\ 1lk E 9f and 1k E n{Mp: p E nj=k+l cl(coz(9;))}. In particular, by the base case, we have that 1 m2 E n{Mp : p E ni=m-I cl(coz(9;))} = 9~-1 + 9~.i m .i But then f m-a /\ 1 f m2 E 9m2 implies that f m-a /\ 1 E E;=m2 9; and therefore m .i we have that f m-a = (f m-a /\ 1) (f m-a V 1) E E;=m2 9; Thus, by recursion, we m .1 m .i deduce that f 1 E E;= 2 9; Hence f I\ 1 = (f I\ 1f 1 ) + f 1 E E;= 1 9; and therefore, f = (f I\ l)(f V 1) E E1=i 9f, as desired. Let X be normal and let Ym, as above, be given. The set Ym is C-embedded in X by [GJ, 3D). Thus we have a surjective ring homomorphism cp: C(X) C(Ym) given by restriction of functions. The kernel of the homomorphism is m K= {f E C(X): Ym Zx(f)} = n My= L9f, yEYm j=l by Larson's Lemma. Thus by the First Isomorphism Theorem, it follows that m C(Ym) ~ C(X)/K = C(X)f L9f. j=l We utilize the one-to-one correspondence between the prime ideals of C(Ym) and prime ideals of C(X) which contain K, the kernel of cp.

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96 Theorem 5.2.3. Let X be normal and m 2 an integer. The ring C(X) ism quasinormal if and only if X satisfies the ( P, m )-boundary condition. Proof: Let Ym = ni=l cl(coz(g;)) be an mrboundary. Ym is a P-space if and only if the prime ideals of C(Ym) are both minimal and maximal by [GJ, 4J]. By the discussion above and Larson's Lemma, this is also equivalent to the condition that the prime ideals of C(X) containing EJ=l gf are maximal. In turn, this is equivalent, by Theorem 5.1.5, to the statement that C(X) is mrquasinormal. By [GJ, 4K.1], we have that countably compact P-spaces must be finite and discrete. In this light, the following is immediate. Corollary 5.2.4. Let X be normal. 1. If X is countably compact, then C ( X) is m-quasinormal if and only if X satisfies the (finite, m)-boundary condition. 2. If X is locally compact, then C(X) is m-quasinormal if and only if X satisfies the ( discrete, m )-boundary condition. S. If X is (1-compact, then C(X) is m-quasinormal if and only if X satisfies the ( countable discrete, m )-boundary condition. Example 5.2.5. (Butterfly Spaces) Let X be a noncompact completely regular space and let S {3X \ X be closed. Define m Bm(X, S) = (ll /3X;)/(S1 ~ S2 ~~Sm), j=l where we assume that X; = X, S; = S for all j = 1, ... m and Si ~ S; indicates that corresponding points of Sare identified, when i =I= j. Consider X = Bm(N, {3N \ N). Let g 1 (x) =when x EN; and let g 1 vanish elsewhere on X. Then {g 1 }J!, 1 is a pairwise disjoint set of functions such that we

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97 have Ym = ni= 1 clx(coz(g;)) = (3N \ N, which not a P-space. Thus Bm(N, (3N \ N) is not an Fm-space by Theorem 5.2.3. Every point of this space has rank less than or equal tom, thus it is an Fm+1-space, by Theorem 5.1.5. The minimal prime ideals contained in Mp for p E (3N \ N and j = 1, ... m, are given by: We show that C(Bm(N,/3N\N)) is a broom ring by demonstrating that Qf, f; Q~+Q; for all j 3. Let f E Qf, and represent the function by an m-tuple (/1, h, ... f m) where fk = f l.BN 4 for all k = 1, ... m. For some j 3, let 91 = (f;,h,h,.,.,fm) and 92 = (/1 -J;,0,0, ... ,0). Then Yi E Q! for i = 1, 2 and/= 91 + 92 E Q~ + Q;. Thus we have shown that 1r(C(Bm(N, (3N \ N))Mp) ::; 2 for all p E Bm(N, (3N \ N). Hence C(Bm(N, (3N \ N)) is a broom ring by Proposition 5.1.9. Let cp: C(Bm(N,/3N \ N)) C(Ym) be the canonical surjection with kernel K = Ei= 1 gf. We want to explicitly describe the root system structure at each point of the space Bm(N, (3N \ N). For each j, the points p E N; are isolated and hence the spectrum at each of these points consists only of the maximal ideal. If p E f3N\N is a P-point then Mp E Max(C(Ym)) is minimal. Hence Mp= cp+-(Mp) is a maximal ideal of C ( Bm (N, (3N \ N)) and is minimal with respect to containing K Thus, the maximal ideal Mp is ramified and the graph of Spec(C(Bm(N, f3N\ N).M) p has the form: (5.1)

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98 If p E ,BN \ N is an F-point which is not a P-point, then Mp E Max(C(Ym)) properly contains a unique minimal prime ideal, namely Op. Hence Mp = cp+--(Mp) is a maximal ideal of the ring C(Bm(N, ,BN \ N)) and the ideal Op= cp+--(Op) is the unique prime ideal of C ( Bm (N, ,BN \ N)) which is minimal with respect to containing K. Thus, Q;+Q; = E~ 1 Qf, =Op~ Mp and the graph ofSpec(C(Bm(N, ,BN\N)M,,) has the form: (5.2) The following is also immediate from Theorem 5.2.3: Corollary 5.2.6. Let X be normal. 1. Let X be a noncompact, locally compact, a-compact F-space and S a closed subspace of the corona. Then B 2 ( X, S) is an Fr space if and only if S is finite. 2. Let X be an F-space and S a nowhere dense zeroset of X. Let X; = X and let S = S; X; for j = 1, 2. Define Y = (X1 II X2)/(S1 ~ 82), where corresponding points of S are identified. Then Y is an Frspace if and only if S is a P-space in the subspace topology. Proof: (1) By [GJ, 6.9], the hypotheses on X imply that ,BX\ Xis compact and X is a cozeroset of ,BX. Thus S is compact and there exists f E C (,BX) such that coz(f) = X. Let fi(x) = {f 0 (x) ifx E ,BX 1 if XE X2 h(x) = {fo(x) if XE ,8X2 if XE X1

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99 Then cl(coz(f 1 )) n cl(coz(f 2 )) = S. If B 2 (X, S) is an F2-space, then Sis a compact P-space by Theorem 5.2.3. Thus, Sis finite by [GJ, 4K.l]. Conversely, assume that S is finite. Since X is an F-space, we know that any 2-boundary of B 2 (X, S) must be contained in Sand must therefore be finite. Thus, every 2-boundary is a P-space and we conclude that B2(X, S) is an F2-space by Theorem 5.2.3. (2) As above, any 2-boundary of Y is contained in S. In particular, S itself is a 2-boundary since it is a nowhere dense zeroset of X. Thus, if Y is an F2-space, then Sis a P-space, by Theorem 5.2.3. Conversely, since any subspace of a P-space is a P-space by [GJ, 4K.4], we know that every 2-boundary of Y is a P-space. Therefore, Y is an F2-space. Next, we show that it is the case that any normal P-space arises as a 2boundary in an F2-space. Proposition 5.2.7. Let Y be a normal P-space. Then there exists an F2-space X containing Y such that in the subspace topology Y = clx(coz(Je)) n clx(coz(J 0 )), where le, lo E C(x)+ are disjoint. Proof: Let Y be a normal P-space and for each y E Y, let a 11 N 11 be the one-point compactification of the natural numbers in which ay plays the role of the point at infinity. Define X = (YU (llyEY a 11 N 11 ))/(a 11 ~ y). Let the points of N remain isolated. A base for the X-neighborhoods of y E Y is given by the sets of the form Uy U (Uy'EU 11 Ny1 ), where Uy is a neighborhood of yin Y and Ny, is a neighborhood of ay' in ay, Ny, Define fe on X such that fe(n) = for all even n E Ny and fe(x) = 0 otherwise. Similarly, define a function f 0 on X such that Jo( n) = if n E Ny is odd, and Jo(x) = 0 otherwise. Then fe A J 0 = 0 and Y = clx(coz(Je)) n clx(coz(J 0 )).

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100 Since the only points of X having rank greater than 1 are contained in the subspace Y, we have that all the 2-boundaries are subspaces of a P-space. Thus by [GJ, 4K.4] and Theorem 5.2.3, we have that X is an F2-space. It is natural to consider the implications of boundary conditions other than for P-spaces. We now take a look at F-space boundary conditions. Theorem 5.2.8. Let X be normal. The following are equivalent: 1. X satisfies the (F-space, n)-boundary condition. 2. Let p E /JX. If { P 1 ... P 2 n} MP are pairwise incomparable prime ideals, then E;= 1 P; and E~n+i P; are comparable. 3. There does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. That is, the graph of Spec(C(X)) does not contain a copy of n n Proof: The equivalence of (2) and (3) is clear. (1) =} (3) : Let p E /JX be such that MP contains two incomparable prime ideals, P, Q, each of rank greater than or equal to n. Let P 1 ... Pn P and Q 1 ... Qn Q be distinct minimal prime ideals. For k = l, ... n, let Then by convexity and primeness, fk = f1k/\hk E (QtnP:)\((U#k Q;)U(LJ,# P,)) We disjointify by defining gk = /\ifkfi A'J= 1 f;Then gk (/. Qk U Pk and hence, gt Qk n Pk. Thus E:= 1 gt (E;= 1 Qk) n (E;= 1 Pk) P n Q.

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101 IT y = n~=l cl(coz(gk)), then by the prime correspondence discussed before Theorem 5.2.3, and by the construction of the functions {9kH1= 1 the incompa rable prime ideals P and Q of C(X) give rise to the incomparable prime ideals P/'E~= 1 gt,Q/"~= 1 gt MP/E~= 1 gt in C(Y). Therefore, Y is an n-boundary which is not an F-space. (3) ::::} (1) : Let Y = ni=l cl(coz(Jj) be an n-boundary in X and assume that Y is not an F-space. Then there exists p E (3Y such that Wy contains incomparable prime ideals P, Q. Since C(Y) ~ C(X)/ Ei=l ff, we have that M~ contains the two incomparable prime ideals P, Q such that P / Ei=l ff ~ P and Q/ Ej=l ff ~ Q. Thus, since Ei=l ff~ P, we know that rkc(x)(P) n, by the pigeonhole principle and [BKW, 3.4.12). Likewise, rkc(x)(Q) n. Example 5.2.9. Consider the ordinal X = w 2 + 1 in the interval topology. Let E be the subset { >. + 2n : n E w, >. limit ordinal} be the set of even ordinals less than w 2 + 1 and let D be the subset {>. + (2n + 1) : n E w, >. limit ordinal} be the odd ordinals less than w 2 + 1. Define fe(x) = if x = >. + 2n E E, and let fe vanish otherwise on X. Let f 0 ( x) = if x = >. + (2n + 1) E D and let f O vanish otherwise on X. Then Y = clx(coz(Je)) n clx(coz(/ 0 )) = {nw: n E w} U {w 2 } is homeomorphic to the one point compactification of w where w 2 acts as the point at infinity. This 2-boundary is not an F-space. Thus, via analysis similar to the proofs above, the canonical surjection and [GJ, 14G), we know that the spectrum at Mw2 in C(w 2 +1) contains the following subgraph: (5.3)

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102 In fact, if, form~ 2, we partition N into m countably infinite sets {A;}J!= 1 and let f-( x) = { if x = A + n for n E A; 3 0 else. Then Y = nJ=l cl(coz(f;)) = {nw : n E w} U {w 2 } is an m-boundary in w 2 + 1. Thus each branch in the graph above that emanates from the maximal ideal has at least 2' minimal vertices, by Proposition 2.3.3. The special F-spaces, extremally and basically disconnected spaces, have a different sort of boundary description. Let us review a couple of results. Recall that a completely regular space X is cozero-complemented if for any cozeroset U X there exists a cozeroset V X such that U n V = 0 and U U V is dense in X. Dually, an /-ring A is complemented if for every f E A+ there exists g E A+ such that f I\ g = 0 and / + g is not a zero-divisor. Lemma 5.2.10. Let X be completely regular. 1. The following are equivalent: (a) C(X) is complemented. (b) X is cozero-complemented. (c) Min(C(X)) is compact. 2. X is a cozero-complemented F -space if and only if X is basically disconnected. 3. X is extremally disconnected if and only if Min(C(X)) a,{ Max(C(X)) is ex tremally disconnected. Proof: (1) The equivalence of (a) and (c) is shown in [M, 1.5). The equivalence of (a) and (b) is from the definitions. (2) See [HVWl, 2.16]. (3) See [M, 2.6] and [M, 2.7].

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103 Let F C(X). Under the hull-kernel topology on Spec(C(X)), we define a subspace homeomorphic to Spec( C ( X) / E /EF f .1) by PF= {PE Spec(C(X)): Pis minimal with re.spect to containing Lf.1}. /EF Proposition 5.2.11. Let X be normal. The following are equivalent: 1. X satisfies the (basically disconnected, n)-boundary condition. 2. X satisfies the (cozero-complemented F, n)boundary condition. S. If F = {/;}7= 1 c C(X)+ is a pairwise disjoint family, then PF is compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. Proof: (1) <=> (2): Follows from (2) of Lemma 5.2.10. (2) <=> (3) : The compactne.ss of PF follows by (1) of Lemma 5.2.10 and the prime corre,spondence discussed before Theorem 5.2.3. The final statement of (3) is a result of Theorem 5.2.8. Proposition 5.2.12. Let X be normal. The following are equivalent: 1. X satisfies the (extremally disconnected, n)-boundary condition. 2. If F = {J;}J=l C C(X)+ is a pairwise disjoint family, then PF is extremally disconnected and compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. Proof: Follows directly from (3) of Lemma 5.2.10, the prime corre.spondence dis cussed before Theorem 5.2.3 and from Theorem 5.2.8. Next we describe the spectra of C(X) which guarantee the (Fm, n)-boundary condition on X.

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104 Definition 5.2.13. Let a EN and let n > 0 be an integer. Then we call then-tuple a = (a 1 "2, ... an) a good (ordered) partition of a if the following hold: for all j = 1, ... n, we have that O < a; E Z, a= a1 + a2 ++an, a1 > 1 and a1 "2 Construct an upward-directed graph ~a corresponding to a good partition a of a containing: nodes P1,P2, ... ,Pn such that P1 < P2 < < Pn, each P; lies above s; = a 1 + 02 terminal vertices and there is a node q such that q > Pi for every j = 1, 2, ... n. As follows: (5.4) Proposition 5.2.14. Let A be a commutative semiprime /-ring with identity in which the sum of any m distinct minimal prime ideals is a prime l-ideal. Then A is m-quasinormal if and only if Spect(A) does not contain ~a as a subgraph for any good ordered partition a of m. Proof: Assume that Spect(A) contains ~a as a subgraph for some good ordered partition a = ( a1, .. an.) of m. Let Pk be the prime -ideal at the node Pk for each k = 1, ... n. For a fixed k, let Qk = { Qk;} ;~ 1 be a family of distinct minimal prime ideals contained in Pk but not contained in P; for j < k. Then LJ~ 1 Qk is a family of m distinct minimal prime ideals whose sum is the nonmaximal prime -ideal Pn.. Thus A is not m--quasinonnal. Conversely, if A is not m-quasinonnal, then there exists a family of m distinct minimal prime ideals { Qk}r=l such that :E;;=l Qk is not a maximal -ideal. Let

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105 P 1 = E;:= 1 Q1r, and let M be the maximal -ideal containing it. Then the subgraph of Spec(A) bounded by the vertices corresponding to the prime -ideals in the set { Pi, M} U { Q1r,}~ 1 is of the form .6. 0 for some a. Example 5.2.15. The good ordered partitions of 4 are (4), (3, 1), (2, 2) and (2, 1, 1). If Xis normal, then it satisfies the (P, 4)-boundary condition if any only ifSpec(C(X)) does not contain any subgraphs of the following forms: (5.5) This follows from Theorem 5.1.5 and Proposition 5.2.14. Obtain the graph Ll~ by appending a graph having at least k 2 minimal vertices to each terminal vertex of the graph .6. 0 For instance: (5.6) The following is a consequence of Proposition 5.2.14 and the method of proof of Theorem 5.2.3. Corollary 5.2.16. Let X be nonnal. Then the following are e.quivalent: 1. X satisfies the (Fm, k)-boundary condition.

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106 2. Spec(C(X)) does not wntain a subgraph of the form~~ for any good ordered partition a of m. 9. Let a family P Spec(C(X)) of mk nonwmparable prime ideals and a good partition a= (ai, a 2 , an) of mk be given. Partition Pinto pairwise disjoint sets {P;}ff 1 where P; contains a; elements of P. Then at least two of the prime ideals given by E P; are wmparable. Proof: The equivalence of (2) and (3) is clear. (1) => (2) : Assume that X contains a k-boundary Y = f\~ 1 cl(coz(f,)) which is not an Fm-space. Then by Proposition 5.2.14, Spec(C(Y)) contains a subgraph of the form ~ 0 below some maximal ideal Mt for some good partition a= (a 1 ... an) of m and some p E f3Y. For each i = 1, ... n, let Pi be the prime ideal at the node Pi E ~ 0 and let {Qi;} i 1 be a family of distinct minimal prime ideals contained in but not contained in P, for l < i. Then, by the prime correspondence discussed before Theorem 5.2.3, there exist, for each i, prime ideals ~' Qi; in C(X), such that ~ I'i/ Ef= 1 f/and Qi; '.:::'. Qi;/ Ef= 1 J,-1for each j. Then by [BKW, 3.4.12), rk(Qi;) k for all i,j. Therefore, the spectrum below M~ contains a copy of~~ (2) => (1) : Assume that the spectrum below M~ contains a copy of~! for some good partition a= (ai, ... an) of m. For each i = 1, ... n, let~ be the prime ideal at the node Pi E ~! and let { Q;;} i 1 be a family of distinct nonmaximal as well as nonminimal incomparable prime ideals contained in~' but not contained in P,, for l < i. For each pair i,j such that 1 i n and 1 j ai, let {Qi;lf= 1 be a family of distinct minimal prime ideals contained in Qi; but not contained in Qr., if r < i or if r = i and s -/= j. Let !str E Q:; \ (U lfr Q~J Then, by convexity, we have that for all l
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107 E:=1 g; C Qi; for all pairs i,j. Then Y = n;=l cl(coz(gr)) is not an Fm-space since Mt contains b.. 0 as a subgraph, by our choice of the functions. 5.3 {jX, m-Quasinormal and SV Conditions First we give equivalent conditions under which one may expect that f3X is an Fm-space for some integer m. This improves [La2,4.3] and extends [La2, 3.3]. Lemma 5.3.1. Let X be normal and let Ym = {x EX: rkx(x) > m -1}. 1. Let y E f3X \ X. If rkpx(y) > m 1, then y E clpx(Ym) 2. If Ym is compact, then rkpx(Y) :Sm -1 for every y E f3X \ X. Proof: (1) Let U be a f3X-neighborhood of y. Then there exists a closed f3X neighborhood V of y such that V U and V n X is closed in X. Since V n X is C*-embedded in X, we have that y E f3(V n X) = clpx(V n X) V. Now, by [La2, 1.6], we have that V n Ym =J 0 since rkpx(Y) > m 1. (2) Let y E f3X \ X. If rkpx(Y) > m -1, then by (1), y E clpx(Ym) X, which is a contradiction. Hence, rkpx (Y) :S m 1. Using the lemma above and Theorem 5.2.3, we obtain: Theorem 5.3.2. Let X be normal and let m 2 be an integer. The following are equivalent: 1. C(X) is m-quasinormal and rkx(MP) :Sm 1 for every p E f3X \ X. 2. X satisfies the (finite, m)-boundary condition. 9. X contains only finitely many points of rank greater than m 1. 4f3X contains only finitely many points of rank greater than m 1. 5. C(f3X) is m-quasinormal.

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108 Proof: The implications (2) =} (3) =} (4) =} (5) have the exact same proofs as in [La2, 4.3], using Lemma 5.3.1. (5) =} (2) : Let { coz(fi) }~ 1 be a pairwise disjoint set of cozerosets of X, and for each i, let ff be an extension of Ji to (3X so that { coz(ff) }~ 1 is a pairwise disjoint family. Then n:, 1 clpx(coz(ff)) is finite by Theorem 5.2.3. Since n:, 1 clx(coz(fi)) is contained in this finite set, it too is finite. Thus, (2) holds by Theorem 5.2.3. (1) =} (5) : Follows from Theorem 5.1.5. (3) =} (1) : Since (3) implies (5), we have that C(X) is m-quasinormal. Condition (3) also implies that rkpx(p) = 1 for all p E (3X \ X, by Lemma 5.3.1. Recall that an /-ring A is called an SV-ring if A/ P is a valuation ring for every prime ideal P. The following arises when in pursuit of conditions which imply that C(X) is both m-quasinormal and SV. Theorem 5.3.3. LetX be normal andm 2. IJC(X) is m-quasinormal and every maximal -ideal has finite rank, then X satisfies the (finite, m)-boundary condition. Proof: Assune that C(X) is m-quasinormal, that every maximal -ideal has finite rank and that {U; : 1 :S j :S m} is a family of pairwise disjoint cozerosets such that the set W = nf= 1 clx(U;) is infinite. Then there exists a copy of Nin W. Denote this copy by Y = {x; : j EN}. Then Y is not closed in (3X, so there exists p E (3X such that p E clpx(Y) \ Y. Let {U_; : 1 :S j :S m} be cozerosets of (3X such that u; n X = U; for each j Then p rJ_ u; for all j = 1, 2, ... m and p E nf= 1 cl 13 xu;. For each i, Xi E W and hence rkx(xi) m. Let Pii, P 2 i, ... Pmi be distinct minimal prime ideals of C(X) such that we have P;i Mxi for all j = 1, 2, ... m and U; (/. ukhcoz(Pki)Let p E (3X and rkc(X)(MP) = n < 00. Then as shown in the proof of [L,4.2], there exists a minimal prime ideal Q MP such that for every Z E Z(Q) such that Zn Y is infinite.

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109 For every i E C(X) define Jj = { xi E Y : i E P;i} for j = 1, 2, ... m. Let :F be the z-ultrafilter on Y containing the z-filter {Zn Y : Z E Z(Q)}. Define for j = 1, 2, ... m, R; = {i E C(X) : J1 E :F}. As shown in [L,4.2], each R; is a prime ideal of C(X). First we show that the R; are noncomparable ideals. For each Xi E Y, let Ui be a neighborhood of Xi such that the collection of these neighborhoods is pairwise disjoint. For each i E N and for each 1 5 l, k 5 m, let iu:i E C(X) such that i1ki E Pii \ Pki, i(X \ Ui) = 0, and O 5 ilki 5 ~. Then let i1k = E: 1 i1ki Finally, i1k E R1 \ Rk and ikl E Rk \ R1 for all 1 5 l, k 5 m, l =I= k. Let R'; R; be a minimal prime ideal for each j = 1, 2, ... m. Larson demon strates that R; MP for all j. We will show that E;, 1 R'; is neither the maximal -ideal MP nor all of C(X). Define h E C(clx Y) by h(xi) = for i = 1, 2, .. and let h(x) = 0 otherwise. Then since X is normal and clx Y is closed, h extends to a function h E C(X) such that h E MP. We show that h 'IE;, 1 R;. For each j = 1, 2, ... m, let!; ER;. Then Jji E :F and hence ni== 1 J1i E :F and (E;, 1 !;)(xi) = 0 for every Xi E ni== 1 Jt If h = E;, 1 i; then since h(xi) =/= 0 for all Xi E Y, 0 = ni== 1 Jji E :F, a contradiction. Thus h E MP\ E; 1 R; and therefore, E; 1 If; E1=i R; MP and C(X) is not m-quasinormal, a contradiction. Thus for any m pairwise disjoint cozerosets {U;: 1 5 j 5 m} the set ni= 1 clxU; is finite.

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[AF] [AM] [BKW] [BP] [Ce] REFERENCES M Anderson and T. Feil, Lattice-Ordered Groups, Reidel (Kluwer), Dor drecht, 1988. M. Atiyah and I. Macdonald, Introduction to Commutative Alge bra, Addison-Wesley Publishing Company, Reading-London-Amsterdam Sydney, 1969. A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et Anneaux Reticules, Springer Lecture Notes in Mathematics (608), Springer Verlag, Berlin Heidelberg-New York, 1977. G. Birkhoff and R. S. Pierce, Lattice-ordered rings, Anais. Acad. Brasil. Cien. (29), 1956, pp. 41 69. E. Cech, On bicompact spaces, Annals of Math. (38), 1937, pp. 823 844. [ChDil] G. Cherlin and M. Dickmann, Real closed rings, II. Model theory, Ann. Pure Appl. Logic (25), 1983, pp. 213 231. [ChDi2] G. Cherlin and M. Dickmann, Real closed rings, I. Residue rings of rings of continuous functions, Fund. Math. (126), 1986, pp. 147 183. [Cl] [C2] [C] [CD] [CHH] [CM] [CMc] [D] P. Conrad, Generalized semigroup rings, J. Indian Math. Soc. (21), 1957, pp. 73-95. P. Conrad, Generalized semigroup rings, II, Portugaliae Math. (18), 1958, pp. 33 53. P. Conrad, Lattice-ordered Groups, Tulane Lecture Notes, Tulane Univer sity, 1970. P. Conrad and J. Dauns, An embedding theorem for lattice-ordered fields, Pacific J. Math. (30), 1969, pp. 385 398. P. Conrad, J. Harvey, and W. Holland, The Hahn embedding theorem for lattice-ordered groups, Trans. Amer. Math. Soc. (108), 1963, pp. 143 169. P. Conrad and J. Martinez, Complemented lattice-ordered groups, Indag. Mathern., N. S. (1), 1990, pp. 281 298. P. Conrad and P. McCarthy, The Structure of /-Algebras, Math. Nachr., (58), 1973, pp. 169-191. M. Darnel, Theory of Lattice-Ordered Groups, Marcel Dekker, New York Basel, 1995. 110

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[DHH] [vD] [E] [Fu] [GK] [GJ] [G] [Ha] [HaM] [HIJo] [HJ] [HJo] [HLM] 111 F. Dashiell, A. Hager, and M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math (32), 1980, pp. 657--685. E. van Douwen, Remote Points, Dissertationes Math. (Rozprawy Mat.) (188) 1981, pp. 1-45. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. L. Fuchs, Partially Ordered Algebraic Systems, Addison-Wesley, Reading Palo Alto-London, 1963. I. Gelfand and A. Kolmogoroff, On rings of countinuou.s functions on topological spaces, Dokl. Akad. Nauk SSSR (22), 1939, pp. 11 15. L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics, Springer Verlag (43), Berlin-Heidelberg-New York, 1976. R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics (12), Marcel Dekker, New York-Basel 1972. A. Hager, Minimal covers of topological spaces, Ann. NY Acad. Sci, Papers on Gen. Topol. and Rel. Cat. Th. and Top. Alg. (552), 1989 pp. 44 59. A. Hager and J. Martinez, Fraction-dense algebras and spaces, Canad. J. Math (45), 1993, pp. 977 996. M. Henriksen, J. R. Isbell, and D. G. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. (50), 1961, pp. 107 117. M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. (82), 1965, pp. 110 130. M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice ordered algebras, Fund. Math. (50), 1961, pp. 73 93. M Henriksen, S. Larson, and J Martinez, The Intermediate Value The orem for polynomials over lattice-ordered rings of functions, Papers on General ToEology and its Applications (Amsterdam, 1994), Ann. NY Acad. Sci. (788), 1996, pp 108-123. [HLMW] M. Henriksen, S. Larson, J. Martinez, and R. G. Woods, Lattice-ordered algebras that are subdirect products of valuation domains, Trans. Amer. Math. Soc. (345) 1994, pp. 195 221. [HVWl] M. Henriksen, J. Vermeer, and R. Woods, Quasi F-covers of Tychonoff spaces, Trans Amer. Math. Soc. (303), 1987, pp. 779-803. [HVW2] M. Henriksen, J. Vermeer, and R. Woods, Wallman covers of compact spaces Dissertationes Mat. (Rozprawy Mat.) (280), 1989, pp. 1 31. [Hu] C. Huijsmans, Some analogies between commutative rings, Riesz spaces, and distributive lattices with smallest element, Indag. Math. (36), 1974, pp. 132 147.

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(HudP] 112 C. Huijsmans and B. de Pagter, Maximal d-ideals in a Riesz space, Canad. J. Math. (35), 1983, pp. 1010 1029. (HudPI] C. Huijsmans and B. de Pagter, On z-ideals and d-ideals in Riesz spaces, I, Nederl. Akad. Wetensch. Indag. Math. (42), 1980, pp. 183 195. (Kl] (K2] [K3] (Lal] [La4] [La2] (La3] (Le] (LZ] (M] (MW] (PW] (Sh] [St] [VJ (Wa] (Wo] C. Kohls, Ideals in rings of continuous functions, Fund. Math (45), 1957, pp. 28 50. C. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. (2), 1958, pp. 505-536. C. Kohls, Prime ideals in rings of continuous functions, II, Duke Math. J. (25), 1958, pp. 447 458. S. Larson, A characterization of !-rings in which the sum of semiprime -ideals is semiprime and its consequences, Comm. Alg. (23), 1995, pp. 5461 5481. S. Larson, Convexity Conditions on f-rings Can. J. Math. (38), 1986, pp. 48-64. S. Larson, !-Rings in which every maximal ideal contains finitely many minimal prime ideals, Comm. Alg. (25), 1997, pp. 3859 3888. S. Larson, Quasi-normal !-rings, Ordered Algebraic Structures, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997, pp. 261 276. A. LeDonne, On prime ideals of C{X), Rend. Sem. Mat. Univ. Padova (58), 1977, pp. 207 214. W. A. J. Luxemburg and A. Zaanen, Riesz Spaces, I, North Holland, Amsterdam-London 1971. J. Martinez, On commutative rings which are strongly Prof er, Comm. Alg. (22), 1994, pp. 347~3488. J. Martinez and S. Woodward, Bezout and Prii.fer !-rings, Comm. Alg. (20), 1992, pp. 2975 2989. J. Porter and R. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer Verlag, New York-Berlin 1988. E. Shchepin, Real functions and near-normal spaces, Soviet Mathematics (13), 1976, pp. 820-830. M. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. (41), 1937, pp. 375-481. J. Vermeer, The smallest basically disconnected preimage of a space, Topol. and its Appl. (17), 1984, pp. 217 232. R. Walker, The Stone-Cech Compactification, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Verlag, Berlin-Heidelberg-New York, 1974. S. Woodward, On !-rings which are rich in idempotents, Doctoral Dis sertation, University of Florida, 1992.

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BIOGRAPIDCAL SKETCH Chawne Monique Kimber was born in Frankfort, Kentucky, on January 12, 1971. She was raised in Tallahassee, Florida, where she graduated from Leon High School in 1988. Chawne received the Bachelor of Science in Mathematics from the University of Florida in 1992, and the Master of Science in Mathematics from the University of North Carolina at Chapel Hill in 1995. 113

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