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Title:
Partially ordered ideal preserving groups
Creator:
Andrus, Jan Frederick, 1932-
Publication Date:
1958
Language:
English
Physical Description:
iii, 63 1 leaves. : illus. ; 28 cm.

## Subjects

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Algebra ( jstor )
Conceptual lattices ( jstor )
Connectivity ( jstor )
Integers ( jstor )
Mathematical theorems ( jstor )
Mathematical transitivity ( jstor )
Mathematics ( jstor )
Partially ordered sets ( jstor )
Symbolism ( jstor )
Dissertations, Academic -- Mathematics -- UF
Group theory ( lcsh )
Mathematics thesis Ph. D
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bibliography ( marcgt )
non-fiction ( marcgt )

## Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaf 62.
Also available on World Wide Web
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Manuscript copy.
General Note:
Vita.

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Full Text

PARTIALLY ORDERED IDEAL

PRESERVING GROUPS

By
JAN FREDERICK ANDRUS

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

UNIVERSITY OF FLORIDA
June, 1958

ACKNOWLE DGMENTS

The author wishes to thank Dr. Alton T. Butson, Assistant

Professor of Mathematics, for suggesting the problem treated in

this dissertation and for his encouragement and guidance in working

towards its solution.

He is also grateful to his wife and to George B. Morgan for

their invaluable work in preparation and reproduction of the

manuscript.

ACKNOWLEDGMENTS .. . . . ..........

I. INTRODUCTION . . . . . . . .

II. PRELIMINARY CONCEPTS . . . . . .

III. NON-DUALISTIC POIP-GROUPS . . . . .

IV. CONNECTIVITY . . . . . . .

V. A DECOiMPOSITION THEOREM . . . . .

VI. FINITE DIMENSIONAL POIP-GBUPS . . .

VII. THE ALGEBRAIC CHARACTER OF POIP-GROUPS. .

VIII. LOIP-GROUPS . . . . . . . .

IX. SUMMARY . . . . . . . . .

LITERATURE CITED . . . . . . . .

BIOGRAPHICAL SKETCH ...... . ........

iii

Page
ii

* .

* .

" .

I. INTRODUCTION

In recent years there has been considerable study of alge-

braic systems in which there is a relation of simple or partial

ordering which is closely related to the algebraic operations of the

system. Examples of such systems are partially ordered groups and

semigroups, ordered fields and rings, vector lattices, and partially

ordered linear spaces. It has been usual in studying such systems to

assume a very strict connection between the operations and the or-

dering, namely that one or all of the operations should preserve the

order relation. In a recent paper Frink (1) proposed a definition

of an ideal in a partially ordered set and suggested the possibility

of generalizing an ordered algebraic system by requiring the alge-

braic operations to preserve ideals (determined by the order relation),

rather than the order relation itself. Butson (2) obtained a struc-

ture theorem for simply ordered ideal preserving groups--the suggested

generalization of simply ordered groups. This dissertation is con-

cerned with partially ordered ideal preserving groups. The principal

results show how a general partially ordered ideal preserving group

may be decomposed into a partially ordered group and a trivial par-

tially ordered ideal preserving group. Consequently, many of the re-

sults concerning partially ordered groups are easily extended to

partially ordered ideal preserving groups.

The multiplicative group of real numbers ordered according

to magnitude is a significant example of a system which is a partially

ordered ideal preserving group (actually a simply ordered ideal pre-

serving group) but is not a partially ordered group. This system

contains a maximal partially ordered subgroup--namely the positive

real numbers ordered as above. Orderwise, the set of negative real

numbers can be considered as the dual of the set of positive real

numbers so that the structure of this particular partially ordered

ideal preserving group as an ordered system is completely determined

by the maximal partially ordered subgroup. Actually any lattice

ordered ideal preserving group is the ordinal sum of a maximal lattice

ordered subgroup and its dual. However, the structure of a general

partially ordered ideal preserving group is more complicated.

II. PRELIMINARY CONCEPTS

The basic concepts of group theory may be found in (3) and

those of partially ordered systems in (4). Some of the less widely-

known concepts are presented in this section.

Let P be a po-set (partially ordered set). If F is a non-

void subset of P, F* will denote the set fx P / x> f for every

fF F and F+ the set x (P / x< f for every f F) The sets

(F )+ and (F+ ) will be denoted by F*+ and F+ respectively. It

can be shown that FC F FC: F, (F* ) = F*, and (F+) = F+.

Definition 2.1: P is up-directed if and only if F # 0 for

every finite subset F of P. It is down-directed if and only if
F + ~f for every finite subset F of P.

Definition 2.2: A subset J of P is an order ideal if and

only if F* I ( and F+CJ for every finite subset F of J. A sub-

set J of P is a dual order ideal if and only if F+ # and F+CJ

for every finite subset F of J.

The above definition generalizes the concept of a lattice

ideal. It differs from the definition suggested by Frink (1) for

an ideal in a po-set in the requirement that F be non-void.

In the remainder of this thesis order ideals and dual order

ideals will be referred to more briefly as ideals and dual ideals,

respectively. No ambiguity will result since algebraic ideals will

not be considered.

The following properties of P are immediate consequences

of Definition 2.2.

2.3: If J is an order ideal of P, x C J for every x4J, and

dually.

2.4: F+ is an ideal for every non-void subset F of P, and dually.

If a is a minimal element of P, a = (a) so that the set

{a) satisfies the conditions of Definition 2.2 for an ideal. Con-

versely, if (a) is an ideal, by (2.3) it is a minimal element of P.

The dual statements are also true.

2.5: A set (a} of P is an ideal if and only if a is minimal,

and dually.

Definition 2.6: An element of P which is not comparable to

any other element of P is called an isolated element.

Definition 2.7: An ideal J of P is called a principal ideal

if J = x+ for some x P. A dual ideal J of P is called a dual prin-

cipal ideal if J = x* for some xfP.

Definition 2.8: A po-group P (partially ordered group) is

(i) a po-set, (ii) a group, in which (iii) a> b implies that

x + a + y > x + b + y for all x,yEP. If P is a lattice satis-

fying (ii) and (iii), it is called an 1-group (lattice ordered group).

If P is a simply ordered set satisfying (ii) and (iii), it is called

an o-group (simply ordered group).

The following definition generalizes the above concepts in

the manner suggested by Frink. This dissertation will be primarily

concerned with these generalizations.

Definition 2.9: A poip-group P (partially ordered ideal

preserving group) is (i) a po-set, (ii) a group, in which (iii) if

J is an ideal, x + J + y is either an ideal or a dual ideal for all

x,y fP. If P is a lattice which satisfies (ii) and (iii), it is

called a loip-group (lattice ordered ideal preserving group). If

P is a simply ordered set satisfying (ii) and (iii), it is called a

soip-group (simply ordered ideal preserving group).

Let P be a poip-group. Assume that there exists a minimal

element a(P and that P has a chain

xl < x2 < x3

of length two. There exists an element t P such that t + a = x2.

Since fa) is an ideal, fx2) is either an ideal or a dual ideal.
+ *
However, this is impossible because neither x2 nor x2 is contained

in the set {x2

2.10: P has a minimal element if and only if every chain of P is

of length less than two.

Suppose now that P has a chain

xl < x2

of length one and an isolated element x0. By (2.4) x2 is an ideal

containing xl and x2. There exists an element t P such that

t + X2 = X0. Since x0 is isolated, t + x2 = x0, t + x = and

It x2 = x0, t + x1 = q. By Definition 2.2 t + x2 is neither

an ideal nor a dual ideal--contradicting Definition 2.9. Thus if P

is of finite length, it is totally unordered--that is, it is a

trivial poip-group, or it is of length one and contains no isolated

elements. These statements are summarized in the following theorem:

Theorem 2.11: A non-trivial poip-group has either infinite

length and no minimal elements or length one and no isolated elements.

Definition 2.12: A poip-group P is dualistic if x + J + y

is either an ideal or a dual ideal for every dual ideal J of P and

all elements x,y (P. A poip-group which is not dualistic is said to

be non-dualistic.

The above classifications of poip-groups will serve as a

framework for the remainder of this dissertation.

All graphical representations of poip-groups in this work are

similar to the "Hasse diagrams" (4, p. 6).

Example 2.13: Any po-group P is a poip-group since, for any
*+ *+ *
elements x,y(P, x + F* + y = (x + F + y) whenever F and F are

non-void subsets of P.

Example 2.14: The multiplicative groups of rational and real

numbers ordered according to magnitude are soip-groups.

The additive group of rational integers and the group of

integers modulo p will be represented by I and Ip, respectively.

Example 2.15: I and 12k are poip-groups of length one when

ordered so that m> n if and only if m is even and n = m + 1. The

diagram of I ordered in this manner is shown in Figure 2.16.

-4 -2 0 2 4

-5 -3 -1 1 3 5

Fig. 2.16

It should be noted that there are no non-trivial po-groups

of finite length.

Example 2.17: The direct sum I2k ; P of I2k ordered as in

Example 2.15 and any po-group P forms a poip-group when ordered as

follows: (i) if m 4 n, (m, x) > (n, y) if and only if m> n, (ii) if

m = n and m is even, (m, x) > (n, y) if and only if x> y, and (iii)

if m = n and m is odd, (m, x) > (n, y) if and only if x < y. When

k = 1 and P is an 1-group, 12k + P is a loip-group under the above

ordering. Figure 2.18 is the diagram for the case in which k = 2

and P is the additive group of rational integers in their natural

order.

(0, 2) (2, 2)

(0, 1) (2, 1)

(0, 0) (2, 0)

(0,-1) (2,-1)

(0,-2) (2,-2)

(1,-2) (3,-2)

(1,-1)) (3.-1)

(1, 0) (3, 0)

(1, 1) (3, 1)
i a
(1, 2) (3, 2)

Fig. 2.18

The following is an example of a poip-group which is not

the direct sum of a poip-group of finite length and a po-group.

Example 2.19: The additive group of rational integers

ordered as follows is a soip-group.

. .> 2> 0> -2> . .> -3> -1> 1> 3> . .

III. NON-DUALISTIC POIP-GROUPS

In this section we completely characterize non-dualistic

poip-groups and in so doing obtain several important theorems con-

cerning dualistic poip-groups.

Definition 3.1: Elements a and b of a po-set P are said to

be weakly connected if and only if there exists a finite sequence

a = Xl, x2. . xn = b

of elements of P such that xi is comparable to xi+l (i = 1, 2,

. . n-1).

Definition 3.2: A set C is said to be weakly connected if

a is weakly connected to b for every pair of elements a,b C. A

maximal weakly connected set is called a weak component, and Ca

will denote the component containing a.

Weak connectivity is obviously an equivalence relation. It

will be discussed in greater detail in Section IV but is applied in

several theorems of this section.

Unless otherwise stated, all discussion will be concerned

with the elements of a poip-group P. The results concerning addition

of an element on the left of elements of P are true when restated in

terms of addition on the riaht.

Lemma 3.3: If a is weakly connected to b, x + a is weakly

connected to x + b.

Suppose that a is weakly connected to b. Then there exists

a sequence

a = xl, x2, .. xn = b

such that xi is comparable to xi+1. The elements xi and xil are
+ +
contained in either xi or xi+l. By Definition 2.9 either Cx + xi,

x + xi+1 q or x + xi, x + xi+I1 Hence x + xi is weakly

connected to x + xi+1. It follows from the transitivity of weak

connectivity that x + a is weakly connected to x + b.

Theorem 3.4: The weak component CO is a normal subgroup of

P whose costs are the weak components of P.

Let c and c' be elements of CO. Then c and c' are weakly con-

nected to 0. By Lemma 3.3 c + c' is weakly connected to c + 0 = c.

By transitivity c + c' is weakly connected to 0, implying that

c + c'(CO. Again by Lemma 3.3 -c + c = 0 is weakly connected to

-c + 0 = -c. Hence -c C0. Thus CO is a subgroup of P.

By Lemma 3.3 -a + Ca is a weakly connected set, all the ele-

ments of which are weakly connected to 0. Hence -a + CaCCO and

Ca Ca + CO. Likewise, a + CO is a weakly connected set, all the

elements of which are weakly connected to a. Thus Ca:a + CO so that

Ca = a + CO. Similarly, CO + a = Ca.

Lemma 3.5: Let P be a system which is (i) a po-set, (ii) a

group, in which (iii) for each weak component C of P and each pair of

elements x,yCP, either x + a + y> x + b + y for all pairs a,bEC

such that a > b or x + a + y< x + b + y for all pairs a,b 6C such

that a > b. Then P is a dualistic poip-qroup.

Let J be an ideal of P. Since (a, b # h for all a,bEJ,

J is a subset of C for some weak component C of P.

Let x be any element of P and F any finite subset of J. Con-

sider the case in which x preserves order when added to the left of

elements of C (that is, x + a > x + b for all a,b C such that a> b).

It can easily be shown that -x preserves order when added to the left
*+
of elements of x + C. Let x + d be any element of x + F and x + e

any element of (x + F) Since x + e is over every element of x + F,

e is over every element of F. Hence e EF*. Therefore, d < e so that

x + d< x + e--that is, x + d (x + F)* Hence x + F*+ (x + F)*.

Now let x + g be any member of (x + F) It is under every element

of (x + F) If we let h be any element of F it is over all members

of F so that x + h is over all members of x + F. Thus x + h (x + F)

and x + F C (x + F) ; in fact, it can be shown that the sets are

equal. Therefore, x + g is under all elements of x + F --that is, g

is under every element of F and gEF This means x + gfx + F

so that x + F )(x + F)*+. Hence x + F = (x + F) .

It can be shown in a similar manner that if x reverses

order when added to the left of elements of C, then x + F =

(x + F) Thus x + F+ is equal to either (x + F)+ or (x + F) .

*+ *+ *+
Suppose x + F = (x + F) Since x + F C-x + J,

(x + F)*+ C x + J. Now any finite subset of x + J can be expressed

in the form x + F where F is a finite subset of J. Therefore x + J

is an ideal.
*+
On the other hand, it can be shown that if x + F =

(x + F)+*, x + J is a dual ideal.

Hence for any ideal J of P and x P, x + J is either an ideal

or a dual ideal. Dually, x + J is either an ideal or a dual ideal if

J is a dual ideal of P. Clearly x + J + y is an ideal or a dual

ideal for any ideal or dual ideal J of P and any pair x,yc P. Thus

P is a dualistic poip-group.

The following four lemmas on poip-groups of finite dimension

will be used in the proof of Theorem 3.10, which completely char-

acterizes non-dualistic poip-groups.

Lemma 3.6: In a poip-group P of length one the figures

a c x+a x+b

M I I
b d x+d x+c

cannot exist simultaneously.

Assume that the figures do exist in P. By (2.4) (a, ci +

is an ideal containing b and d. The elements x + b and x + d are

in x + a, c Therefore x + (a, c is not an ideal since

x + c E(x + b)+ but c~ {a, c +, and it is not a dual ideal since

x + a E(x + d) but a4 fa, c This is a contradiction since

x + a+ must be either an ideal or a dual ideal.

Lemma 3.7: In a poip-group P of length one the figures

a c x+d

X I
b d x+c

cannot exist simultaneously if x + a and x + b are maximal.

Assume that the figures do exist simultaneously and that

x + a and x + b are maximal. The ideal (x + d)+ contains x + c and

x + d but contains neither x + a nor x + b. Hence -x + (x + d)+
contains c and d but does not contain a and b. By (2.3) -x + (x + d)+

is neither an ideal nor a dual ideal. Since it must be one or the

Lemma 3.8: In a poip-group P of length one x + a and x + b

are not both maximal if a > b.

Assume that a > b and that x + a and x + b are maximal.

Since x + a is either an ideal or a dual ideal and contains x + a

and x + b, there exists an element x + ce6 x + a, x + bj +. Ob-

viously (x + a)+ and (x + b)h are ideals containing x + c.

Since -x + (x + a)+ is either an ideal or a dual ideal, either

Ia, c3 + I or fa, c) f 1. Also -x + (x + b) is either an ideal
or a dual ideal, so that either Ib, cj + 4 4 or [b, c} *

Clearly this is impossible unless either c> b or c < a.

Case i: Suppose c > b. The set -x + (x + a)+ contains

a and c but not b since x + b is maximal. By (2.3) and the fact that

b a +, -x + (x + a)+ is a dual ideal. If (bj = ta, cj +, then

bE [a, c} *. By the definition of a dual ideal [b3 = [a, c}

implies be -x + (x + a) Hence {b}) # a, cj so that there must
} +
be an element dF a, cJ distinct from b. The set x + a contains

x + a and x + d, so that either tx + a, x + dj + 9 or
+
fx + a, x + d} q. Since x + c contains x + c and x + d, either

{x + c, x + d} I or {x + c, x + d* 4 q. The above conditions

can occur only if x + d< x + a or x + d> x + c. However, these re-

lations contradict Lemmas 3.6 and 3.7, respectively.

Case ii: Suppose c < a. The set -x + (x + b) contains b

and c but not a since x + a is maximal. By an argument similar to

that in Case i, there exists an element d such that d> b, d> c,
+
and d I a. Since x + d contains x + b, x + c, and x + d, either

(x + b, x + c, x + d + I q or (x + b, x + c, x + dj q. Hence

either x + d< x + b or x + d> x + c. As before, these conditions

contradict Lemmas 3.6 and 3.7, respectively. This establishes the

lemma.

Lemma 3.9: If P is a poip-group of length one in which

a > b implies x + a + y and x + b + y are comparable for all x,y P,

then P is dualistic.

Let c and c' be maximal elements in a weak component C of P.

Then there exists a sequence

c = x1, x2, . xn = c'

of distinct elements of C such that xi is comparable to xi+1. We

note that xi is maximal if and only if i is odd and in particular

that n is odd. The sequence

x + c = X + X1, X + x2, . x + xn = x + c'

is also a sequence of distinct elements such that x + xi is comparable

to x + xi+1. If x + c is maximal (minimal), then x + xi is maximal

if and only if i is odd (even). Since n is odd, x + c' = x + x is

maximal if and only if x + c is maximal.

Thus if c' is some maximal element of C and if x + c' is

maximal (minimal), then x + c is maximal (minimal) for all maximal

elements c CC. Since every element of P is either maximal or minimal,

x preserves (reverses) order when added on the left of the elements

of C. It follows that P satisfies conditions i-iii of Lemma 3.5 and

that it is dualistic.

Theorem 3.10: Any non-dualistic poip-group is (i) a po-set

of length one, in which (ii) the weak components are the costs of CO,

(iii) every weak component is a principal ideal, and (iv) CO contains

more than two distinct elements. Conversely, any group P satisfying

conditions i-iv for some normal subgroup CO is a non-dualistic

poip-group.

The latter statement will be proved first. Let P be a group

satisfying conditions i-iv for some normal subgroup CO of P, and let

J be an ideal of P. By Definition 2.2 J is contained in some weak

component C, which by condition iii is a principal ideal, say, c .

If J is an ideal (j} consisting of a single element, then x + J + y =

fx + j + y} is, according to (i), either an ideal or a dual ideal.

If J contains two distinct elements j and j', by Definition 2.9 it

S*+ +
contains j, j' Thus c (J so that J =c = C. Hence J = C.

Since x + J + y = x + C + y = x + Cc + y = Cx+c+y, x + J + y is a

principal ideal. Therefore, P is a poip-group.

In order to show that P is non-dualistic, consider a weak
+
component c and distinct elements cl and c2, which are properly

under c. That such elements exist follows from condition iv. Now

there is an x such that x + c2 = c. Since c, cl, and c2 are distinct,

x + c 4 c and x + cl I c. Thus x + c and x + cl are minimal elements
*
such that [x + c, x + C1 = {x + c2J = fc Hence [c}
*+ *
tx + c, + c but (c x + c, x + cl = x + cl This

implies that x + cl is not an ideal. It also is not a dual ideal

since {x + c, x + cll = . Thus P is a non-dualistic poip-group.

Now let P be a non-dualistic poip-group of finite dimension.

By the contrapositive of Lemma 3.9 there exist elements a,b,x P such

that a > b but x + a and x + b are not comparable. By Lemma 3.8 and

the fact that Ix + a, x + b C_ x + a there exists an element

x + c properly over x + a and x + b.

The elements a, b, and c are members of -x + (x + c) so

that either (a, b, cj*+ q or [a b c) 4 This implies either

c > b or c < a. If c > b, then a and c are maximal and x + c > x + a,

which contradicts Lemma 3.8. So we must have c < a.

The set x + a is not a dual ideal since it contains two mini-

mal elements, x + a and x + b. Thus it is an ideal. It cannot con-

tain a maximal element distinct from x + c or a minimal element

not under x + c. Hence x + a O(x + c)+. Clearly x + a D(x + c) ,
+ +
so that x + a = (x + c) .

Suppose there exists an element e such that e > c. By

Lemma 3.8, x + e is minimal. Since x + c, x + e j x + e ,

x + c > x + e. Thus x + e E(x + c)+ x + a, so that e a. Since

e is maximal, e = a. This shows that e = a, e .

Now assume that there is an element f such that f > b. There

exists y such that y + b = c. So y + a, y + b = c, y + c C y + a

which implies that either Iy + a, y + b = c, y + c) or

y + a, y + b = c, y + c ~ Now c is minimal and c = a, c .

This means that Iy + a, y + b = c, y + c, = = so that a =

fy + a, y + b = c, y + c3 Hence a 6 y + a, y + b = c, y + c ,

meaning a6y + a It follows from Definition 2.9 and the equality
Sn e + + +
y + a, y + b = c, y + c = that y + a is an ideal and aC y + a .

Since y + f and y + b are in y + f, +y + f, y + b = cj

or y + f, y + b = c Now c = a, c which means that
+ + +
y + f Ea Therefore y + f y + a and f a Because f = a, f is

maximal, which shows that b = a, b

Replacing b by d and repeating the above argument, it can be

shown that d = a, di for any d such that d< a. Therefore the

particular component Ca is the set a .

We will next show that every weak component of P is a prin-

cipal ideal.

Consider a weak component C and assume that it is not a

principal ideal. Then there are distinct elements c0, cl, c2 6C

such that cl > co and c2 > co. There exists an element z such that

z + c = a. By Definition 2.9 either z + co, + ci or

z + c0, z + c i (i = 1, 2). So z + Co, z + cl, z + c2 a .

Consequently, we have z + c < z + c = a and z + c2 < z c = a.

Since fz + CO, z + c2jC z + c2 and z + cl z + c2 it follows

that z + c2 is not a dual ideal. Then z + cl =

aE z + CO, z + c23 (- z + c2 so that cl < c2. This is a contra-

diction--implying that C is indeed a principal ideal.

So P satisfies condition iii. By Theorem 3.4 it satisfies

condition ii, and it obviously fulfills conditions i and iv.

The proof of this theorem would be complete if it were known

that every non-dualistic poip-group has finite dimension. This re-

sult, along with a more useful definition of a dualistic poip-group,

will be obtained in the remainder of this section. However, before

proceeding we give an example of a non-dualistic poip-group.

Example 3.11: Order the additive group of integers by

placing 0 over 2n and 1 over 2n + 1 for all integers n. The re-

sulting system is a non-dualistic poip-group with two weak components

one composed of the even integers and the other composed of the odd

integers.

Lemma 3.12: If a > b in a poip-group P, then x + a I_ x + b

when x + b+ is an ideal, and x + a x + b when x + b is a dual ideal.

Assume x + b+ is an ideal and x + a < x + b. Then

x + a x + b which implies a< b. Hence x + a x + b.

Suppose x + b is a dual ideal and x + a_> x + b. Then

x + aE x + b+ which means a < b. Hence x + aj x + b.

Lemma 3.13: If x + b+ is an ideal of a poip-group P and

b is not maximal, then x + b = (x + b) .
+ +
Since x + bEx + b obviously (x + b)+ (:x + b So

-x + (x + b)+'lb If -x + (x + b)+ were a dual ideal, we would
+ + +
have b C--x + (x + b) +Cb But b : b is possible only if b is

a maximal element. Thus -x + (x + b) is an ideal. Since
+ + + +
bE -x + (x + b)+, b 1 -x + (x + b)+ and x + b c (x + b). There-

fore, x + b = (x + b)+.

Lemma 3.14: Let a be a non-maximal element of a poip-group P.

If a> b, -x + (x + a) is a dual ideal, and x + b is an ideal, then

_x + a>x + b.

Since a is not maximal, there is an element c > a in P. The

dual ideal -x + (x + a)+ contains a and hence c. So x + c E(x + a)+

which gives x + c < x + a. By the contrapositive of Lemma 3.12,

x + a is not an ideal. Therefore it must be a dual ideal. Now

fx + a, x + b) x + a so that x + a and x + b have a lower bound,

say, x + g. By Lemma 3.13 x + b = (x + b) implying that x + ge
+ + +
x + b and b Eg Also x + g E(x + a) which means gE -x + (x + a)+
+ +
and g *CZ -x + (x + a) Hence bE -x + (x + a) implying that

x + b (x + a) which gives x + b < x + a.

Lemma 3.15: If a > b in an infinite dimensional poip-group P,

and a is not maximal, then x + a and x + b are comparable.

Obviously a c-x + (x + a)+. If -x + (x + a) is an ideal,

it contains a and hence b. Thus if -x + (x + a)+ is an ideal,

x + b E(x + a) and x + b < x + a. So assume -x + (x + a)+ is a

dual ideal.

Clearly b C-x + (x + b)+. If -x + (x + b)+ is a dual ideal,

it contains b and thus a. Hence if -x + (x + b)+ is a dual ideal,

x + a E (x + b)+ and x + a < x + b. So assume that -x + (x + b)+ is

an ideal.

Case i: Suppose x + b is not maximal. By Lemma 3.13

-x + (x + b)+ = b+ (replace x by -x and b by x + b). Hence (x + b)

x + b so that x + b is an ideal. It follows from Lemma 3.14 that

x + a> x + b.

Case ii: Suppose x + b is maximal. If x + b+ is an ideal,

then by Lemma 3.14 x + a and x + b are comparable. Therefore, assume

x + b+ is a dual ideal. Since P is of infinite length, it follows

from (2.10) that it has no minimal element. So there must exist an

element d< b. Now -x + (x + b)+ is an ideal containing b and

hence d. Thus x + d E(x + b)+ and x + d< x + b. Now a> d where

a and x + d are not maximal. Replacing b by d in the proof of

Case i, it follows that x + a is comparable to x + d. If x + a <

x + d, then x + a < x + b. So assume that x + a > x + d. Since

deb+, x + dex + b+. Then (x + d)*cx + b)4 (x + b)+. This

gives x + a E(x + b)+ and x + a < x + b, completing the proof.

In Section II it was shown that an infinite dimensional poip-

group P contains no minimal elements. If we can show that such a

poip-group contains no maximal elements either, we shall be able to

conclude from the above lemma that comparability is preserved under

addition in an infinite dimensional poip-group.

Suppose a is a maximal element of a poip-group P of infinite

length. Since P has no minimal elements, there exist a chain

a>e > f>g

and elements t,b,c eP such that t + a = f, t + b = e, and t + c = g.

The above chain now becomes

a>t + b> t + a> t + c.

Since t + b> t + a, it follows from Lemma 3.15 that a and b are

comparable. Moreover, a > b because a is maximal.

By Lemma 3.12, -t + (t + a)+ is a dual ideal (replace x by

-t, a by t + b, and b by t + a). Now t + c E (t + a)+, so that
+ *
c E-t + (t + a) Thus c C -t + (t + a)+ and t + c c:(t + a)+.

If b Ec then t + b E(t + a)+, which is impossible since t + b>

t + a. Hence b c. However, by Lemma 3.15 b and c are comparable, so

b < c. The elements a and c are also comparable because t + a >

t + c and t + a is not maximal. Since a is maximal, a > c. From

Lemma 3.12 and the relations t + a > t + c and a > c, it follows

that -t + (t + c)+ is an ideal. By Lemma 3.13 -t + (t + c)+ = c

Since b cc t + b E (t + c)+ and t + b < t + c, a contradiction.

Hence P cannot contain a maximal element, and we can now state the

following theorems.

Theorem 3.16: An infinite dimensional poip-group contains

no maximal or minimal elements.

Theorem 3.17: Comparability is preserved under addition in

an infinite dimensional poip-group--that is, if a and b are comparable,

then x + a + y and x + b + y are comparable for all x,y eP.

Lemma 3.18: If a> b in an infinite dimensional poip-group,

then (i) x + a is an ideal if and only if x + a > x + b, (ii) x + a

is a dual ideal if and only if x + a < x + b, (iii) x + b+ is an ideal

if and only if x + a > x + b, and (iv) x + b+ is a dual ideal if and

only if x + a < x + b.

Conditions iii and iv follow from Theorem 3.17 and Lemma 3.12.

Suppose x + a is an ideal. By Lemma 3.13 x + a = (x + a) .

Since b Ea x + b ((x + a)+ and x + b< x + a.

Now suppose that x + a+ is a dual ideal and x + a> x + b.

By Lemma 3.12 x + b+ is an ideal. It follows from Theorem 3.16 that

there is an element c > a. Again by Lemma 3.12 x + c< x + a and

x + c > x + b. Hence x + a > x + c > x + b. The inequalities

x + a> x + c, c > a, and Lemma 3.12 imply that -x + (x + c) is a

dual ideal. Now x + b 6 (x + c)+, so that b& -x + (x + c)+. Then

b* Z-x + (x + c)+, a e-x + (x + c)*, and x + a (x + c)W. This

implies x + a < x + c, a contradiction.

Thus if x + a+ is an ideal, x + a > x + b, and if x + a is

a dual ideal, x + a < x + b. These statements and their contra-

positives give conditions i and ii.

An immediate consequence of this lemma is the following

result.

Lemma 3.19: Let a and b be comparable elements of an

infinite dimensional poip-group. Then x + a+ is an ideal if and only

if x + b+ is an ideal, and x + a+ is a dual ideal if and only if

x + b+ is a dual ideal.

Lemma 3.20: An infinite dimensional poip-group P is (i) a

po-set, (ii) a group, in which (iii) for each weak component C of P

and each pair of elements x,y eP either x + a + y_> x + b + y for

all pairs a,bEC such that a> b or x + a + y < x + b + y for all

pairs a,bE C such that a> b. Conversely, any system P satisfying

conditions i-iii is a dualistic poip-group.

Let x be any element and C any weak component of P. First
+
suppose that x + c0 is an ideal for some c0 gC. For any other

c EC, there exists a sequence

cO = xl, x2, . xn = c
+
such that xi and xi+l are comparable. If x + xi is an ideal, by

+ +
Lemma 3.19 x + xi+1 must be an ideal. Hence x + c is an ideal.
+ +
Similarly, if x + c0 is a dual ideal for some c0 GC, x + c is a

dual ideal for all c eC. That P satisfies condition iii now follows

from Lemma 3.18 and the extension of the above argument to the case

in which an arbitrary element y is added on the right of elements

of C.

This lemma and Lemma 3.5 enable us to conclude the following

result which we noted before was necessary to complete the proof of

Theorem 3.10.

Theorem 3.21: Every infinite dimensional poip-group is

dualistic.

Theorem 3.22: A dualistic poip-group P is (i) a po-set,

(ii) a group, in which (iii) for each weak component C of P and each

pair of elements x,y EP either x + a + y_> x + b + y for all pairs

a,b C such that a> b or x + a + y x + b + y for all pairs a,b C

such that a> b. Conversely, any system P satisfying conditions i-iii

is a dualistic poip-group.

The latter statement of the theorem is merely Lemma 3.5.

A dualistic poip-group P obviously satisfies conditions i and

ii. By Lemma 3.20, if it is of infinite dimension, it also satis-

fies condition iii. Hence it remains to be shown that any dualistic

poip-group of dimension one satisfies condition iii.

So assume that P has length one. Consider elements a,be P

such that a> b and suppose x + a and x + b are not comparable for

some x EP. Now [x + a, x + b3 C- x + a so that either

(x + a, x + b] i Q or (x + a, x + b} + i 0. Since x + a and x + b

are not comparable, there exists an element which is either properly

over them or is properly under them. Hence x + a and x + b are

either both maximal or both minimal. By Lemma 3.8 and its dual, this

is impossible. Therefore x + a and x + b are comparable.

In the proof of Lemma 3.9 it was shown that if P is a poip-

group of length one in which a > b implies x + a + y and x + b + y

are comparable for all x,y EP, then P satisfies condition iii. This

completes the proof.

Since non-dualistic poip-groups were described completely in

Theorem 3.10, only dualistic ones will be considered in the sequel.

Also, the above theorem provides us with a more useful definition of

a dualistic poip-group, which will be employed in the following

sections.

IV. CONNECTIVITY

The immediately preceding theorem suggests the following

definitions, which will lead to an even finer partitioning of a

dualistic poip-group than that determined by the weak components.

Definition 4.1: An element x is said to be order preserving

on the left relative to the weakly connected set C if x + a > x + b

for all pairs a > b in C. It is said to be order reversing on the

left relative to C if x + a < x + b for all pairs a > b in C.

By Theorem 3.22 any element of P is either order preserving

or order reversing on the left relative to any given weak component.

Henceforth there will be little occasion to consider ideals.

However, it should be noted that if J is an ideal contained in the

weak component C and if x is order preserving (reversing) on the left

relative to C, then x + J is an ideal (dual ideal). Also, if F is

a non-void subset of the weak component C, then x + F = (x + F)
+ *
or x + F = (x + F) depending on whether x is order preserving or

order reversing on the left relative to C. These results are con-

tained in the proof of Lemma 3.5.

Definition 4.2: An element x is of order preserving type 1

relative to the weakly connected set C if it is order preserving on

the left and on the right relative to C. It is of order preserving

type 2 relative to C if it is order preserving on the left and order

reversing on the right relative to C. It is of order preserving

type 3 relative to C if it is order reversing on the left and order

preserving on the right relative to C. It is of order preserving

type 4 relative to C if it is order reversing on the left and on the

right relative to C. An element of order preserving type 1 (type 2)

relative to Ca and an element of order preserving type 4 (type 3)

relative to Ca are said to be of opposite order preserving types

relative to Ca. The set of all elements of Cx which are of order

preserving type i relative to Ca will be denoted by Cx(Ca) (i =

1, 2, 3, 4).

We note that Cx(Ca) and C (Ca), i 4 j, are disjoint.

Any weakly connected set partitions P into from one to four

classes of element types. If it is commutative, all elements of P

are either of order preserving type 1 or order preserving type 4

relative to any given weakly connected set.

Example 4.3: Let P be generated as a group by a and x under

the conditions x + x = 0 and x + a = -a + x. When it is ordered as

shown in Figure 4.4, P is a non-commutative dualistic poip-group.

It can .be verified that a is in CO (CO) and C1 (C) and that x is a
2 2
member of Cx (CO) and Cx (Cx).

2a x + 2a = -2a + x

a x + a =-a + x
I i
0 x

-a x- a= a + x

-2a x 2a = 2a + x
I i

Fig. 4.4

Lemma 4.5: If x is order preserving (reversing) on the left

relative to Ca, -x is order preserving (reversing) on the left rel-

ative to Cx+a'

Let al and a2 be elements of Ca such that al> a2 and suppose

that x is order preserving on the left relative to Ca, so that

x + al > x + a2. The element -x is order preserving on the left

relative to Cx+a because -x + (x + al) = al a2 = -x + (x + a2).

The remaining case can be proved in a similar manner.

Theorem 4.6: C = C1 (C)UC (C
C 0 0 0 0
For any element y EC0 there exists a sequence

0 = X1, x2, . ., xn = y

of distinct elements of Co such that xi is comparable to xi+l.
1
Clearly 0 ECO(CO), so to complete the proof it is necessary to show
1 4 1 4
only that xi ECO(CO)UCO(CO) implies xi+1 EC0(Co)UC (Co).
1 2 1
Assume xi ECo(CO) and xi+1 CO(Co). By Lemma 4.5 -xiECO(C0)

and -xi+1 ECO(CO)

Case i: Suppose xi > xi+1. It follows that xi xi+1 < 0.

However, it is also true that 0 > xi+1 xi whence -xi+1 > -xi and

xi xi+l > 0, a contradiction.

Case ii: Suppose xi < xi+I. Obviously xi xi+l > 0. But

0 < xi+l xi, so that -xi+1 < -xi and xi xi+1 < 0, which is

impossible.
1 2
This proves that xi 6CO(CO) implies xi+l CO0(C0). Continuing

with this procedure it can be shown that xi 6C(CO)U CO(CO) implies

that xi+1 C0(CO)LC (CO) and, therefore, that xi+1 C0(C0O CO(C).

The theorem follows.

Theorem 4.7: For all e EP, C (C) = C(C ) and
0 0 e
4 4
CO(CO) = CO(Ce).

Let el, e2 ( Ce be such that el > e2 and consider an element

s ECC0(C0).

Case i: Suppose el is order preserving on the right relative

to CO. By Lemma 4.5 -el is order preserving on the right relative to

C so that 0 > e2 el. Therefore, s > s + (e2 el) = (s + e2) -e1

and s + el> s + e2.

Case ii: Suppose eI is order reversing on the right relative

to C0. By Lemma 4.5 0 < e2 el. Thus s < s + (e2 el) = (s + e2) el

and s + el > x + e2.

These arguments show that s is order preserving on the left
1
relative to Ce By left-right symmetry, s ECO(Ce).
4 4
Similarly, s ECO(CO) implies s eCO(C ), enabling us to con-

clude that CO(CO)CIC(C ) and C4(C0)CCO(Ce). However, by

Theorem 4.6 CO = CO(Co)UC4(Co). Since CO(C ) O (C)=

C (C) = C0(Ce) and C0(C0) = C (C).

As a result of this theorem, no ambiguity will arise if

C 0(C eand C (C ) are denoted by C0 and C ,respectively.
1 4
Theorem 4.8: The set C is a normal subgroup of P, and C0

either is empty or is one of its costs. For any e (P,

Ce = (e + C4) U (e + Cg). Furthermore, for any f 6P, e + CO = Ce(CF)
4 j
and e + CO Ce(Cf) where i is the order preserving type of e relative

to Cf and j is the opposite order preserving type relative to Cf.

Obviously Ce = e + CO = e + (C UC4) = (e + C) U (e + C)

and Ce = C0 + e = (C1 UC4) + e = (Cl + e)U (C + e). Now let f and

f2 be elements of Cf such that fl > f2 and let e + s be any member of

e + CO. Since sEC1 = C0(C ), s + fl > s + f2. Clearly s + fl E

CO + f1 = Cf, so that (e + s) + fl > (e + s) + f2 if e is order pre-

serving on the left relative to Cf, and (e + s) + fl < (e + s) + f2

if e is order reversing on the left relative to Cf. In other words,

e + s is order preserving (reversing) on the left relative to Cf

when e is order preserving (reversing) on the left relative to Cf.

Similarly, it is order preserving (reversing) on the right relative

to Cf when e is order preserving (reversing) on the right relative to

Cf. This shows that if i is the order preserving type of e relative

to Cf, then e + C CCi(Cf). By left-right symmetry, C1 + eCC (Cf).

It may be proved in much the same manner that if i is the

order preserving type of e relative to Cf, then e + Cg CCJ(Cf) and

C4 + eCCJ(C ) where i and j are of opposite order preserving types

relative to Ce. Since (e + C01)U (e + C4) = C and Ci(C ) c C(C) = ,

it is true that e + C = C(C) and e + C= (C).
it is true that e + C1 = Ci(Cf) and e + 0 = i(C).

Likewise C + e = Ci(C) and C + e = C (C). It follows that
0C + e + e e f
1 1 4 4
e + C1 = C1 + e and e + C4 = C4 + e.

1 1
If s and t are any members of CO, then s + t Es + Co =

C(CO) = CO, By Lemma 4.5 s C1 implies -s C1. Hence C1 is a

normal subgroup of P.

If C4 is not empty, there exists an reC4. Then r + C1

r + C(CO) = C4( C), so that C4 is a coset of C. This completes

the proof.

Corollary 4.9: For any weak component C and any f E P, either

C = C1(C )UC4(Cf) or C = C2(Cf)UC3( Cf).

The decomposition of P into the costs of Co yields little

information concerning the structure of P, but it motivates the

investigation of another decomposition determined by the maximal
1
weakly connected set in C0.

Definition 4.10: The element a is strongly connected to b

if and only if there exists a sequence

a = xl, x2, . xn = b

such that xi E C(CO) for some j and i = 1, 2, . n and such that

Xi is comparable to xi+l for i = 1, 2, . n 1.

Obviously if two elements are strongly connected, they are

weakly connected. Note that strong connectivity is an equivalence

relation on a poip-group but does not necessarily have meaning for

a po-set per se.

Definition 4.11: A set S is strongly connected if and only

if every two elements of S are strongly connected. A maximal

strongly connected set is called a strong component of P, and Sa

will denote the strong component containing a.

Any strongly connected set consists of elements of only one

order preserving type relative to any weakly connected set. Ob-

viously S0CCI CO.

Lemma 4.12: If a is strongly connected to b, then x + a

is strongly connected to x + b.

There exists a sequence

a = x1' x2, xn = b

of elements of Sa such that xi is comparable to xi+l. Now

S aCj(CO) for some j so that x + Sa x + CJ(CO) = Ck (C ) for
a a 0 a a 0 x~a 0

some k. In particular, the members of the sequence

x + a = x + xl, x + x2, . x + x = x + b

are contained in k +a(C0). Furthermore, by Theorem 3.22 x + xi is

comparable to x + xi+l. Hence x + a is strongly connected to x + b.

Let a and b be elements of S0. Then a and b are strongly

connected to 0. By Lemma 4.12 a + b is strongly connected to

a + 0 = a, so that by transitivity it is strongly connected to 0

and is therefore in 80. Again by Lemma 4.12 -a + a = 0 is strongly

connected to -a + 0 = -a whence -a ES0. Thus SO is a subgroup of P.

The set a + S0 is contained in Sa because a ESa and all ele-

ments of a + SO are strongly connected to a according to Lemma 4.12.

Likewise, -a + Sa is contained in 50--that is, Sa is contained in

a + S and, therefore, a + SO = Sa. Similarly, S + a = S so that

SO is a normal subgroup of P, proving the following theorem.

Theorem 4.13: SO is a normal subgroup of P, and the strong

components of P are the costs of SO.

In the next section the strong components are used to char-

acterize the order structure of dualistic poip-groups. This section

is concluded with the following example illustrating the relationships

among the costs of SO, CO(CO), and CO.

Example 4.14: Let P be the poip-group composed of the

additive group of integers ordered as shown in Figure 4.15. Any

block of integers in the diagram is a class of integers modulo 12.

In this poip-group

soUs Us8 = C(Co),

S2US6US10= CO(CO),

S3US7US11= Ci(CO),

s1Uss s = C4(Co),

CO(CO)UC4(Co) = Co,

C(CO)UC4(CO) = C.

34

27 31 35

15 19 23

Sg 3 S3 7 S7 11 S11
I I S
-9 -5 -1

-21 -17 -13

-33 -29 -25
I I I

-31 -35 -27

-19 -23 -15

-7 -11 3

S10 S5 1 SI 9 S
I I I
17 13 21
29 25 I33
29 25 33
I i 1

Fig. 4.15

V. A DECOMPOSITION THEOREM

In this section it will be shown that a dualistic poip-group

is order isomorphic to the ordinal product of a certain finite di-

mensional poip-group and a po-group.

Lemma 5.1: If SxCC (CO) and S(-CJx(CO), where i 4 j, and

if xl > Yl for some xl in Sx and some yl in Sy, then x' > y' for

every x' in Sx and every y' in S .

Consider the case in which i = 1 or 2 and j = 3 or 4. Let

x2 be a member of S which is comparable to x If x 2> xl, then
x 2-

X2 > Y'. So assume that x2 < xl.

Case i: Suppose xlECx(CO). Then -xl is order preserving

on the right relative to Cx, and xI > yl implies that 0 > yl xl.

Now x2 (C(C0), so that x2> (y xl) + x2. On the other hand,

xl x2 and the fact that -xl is order preserving on the left rel-

ative to Cx imply that 0> -xl + x2. Since y1 is order reversing

on the left relative to CO, we have yli< (yl xl) + x2 and, there-

fore, yl < x2.
2
Case ii: Suppose xl EC2(CO). Then -xl is order reversing

on the right relative to Cx, and xl > yl implies that 0 < yl xl.
2
Now x2 EtC(CO), so that x2 > (yl xl) + x2. However, xl) x2 and

-X1i C (C ) give 0 > -xl + x2. Now since y is order reversing on

the left relative to Cg, it is true that yl < (yl x2) + x2 and

hence that yl < x2.

The above argument proves that every element of Sx which is

comparable to xl is greater than yl. It follows that any x' in S ,

being strongly connected to xl, is also greater than yl. Using a

similar procedure it can be shown that any y' in S is less than xl.

So each y' S S is under xl and hence under every x' E Sx. This

establishes the cases in which i = 1 or 2 and j = 3 or 4. The dual

argument establishes the remaining cases.

Definition 5.2: Order the elements of P/S0 as follows:

Sx> S if and only if S = S or every x' in S is greater than

every y' in Sy. This ordering will be called the natural ordering

of P/S0.

The natural ordering of P/SO is obviously a partial ordering,

and henceforth the symbol P/SO will represent the group P/S0 ordered

in this manner.

Theorem 5.3: The mapping h given by h(x) = Sx is a group

homomorphism of P onto P/S0 such that x_> y implies h(x) > h(y) and

h(x) > h(y) implies x > y. A set C is a weak component of P if and

only if h(C) is a weak component of P/S0. Furthermore, when some

weak component of P/SO contains at least two distinct elements, every

weak component of P/S0 contains at least two elements; for any a,xE P

x is of order preserving type i relative to Ca if and only if h(x)

is of order preserving type i relative to h(Ca) (i = 1, 2, 3, 4).

Clearly h is a group homomorphism. Consider elements x > y

inCx 0= C(CO)U C (CO). If x and y are both elements of C (C0) or

both elements of Cx(CO), then they are members of the same strong

component Sx, so that Sx = Sy. However, if x (Cx(Co) and yE Cx(C0),

where i f j, then according to the preceding lemma every element of

Sx is over every element of S so that Sx > Sy. This shows that

x > y implies h(x) > h(y).

Suppose now that h(x) = Sx > S = h(y). Then Sx S Sy. By

Definition 5.2 every x' in Sx is greater than every y' in Sy' so that

in particular x > y.

It follows from the isotone property of h (that is, the

property that x > y implies h(x) > h(y)) that if C is a weak com-

ponent of P, then h(C) is a weak component of P/SO. Conversely,

since h(x) = S x> S = h(y) implies that x is connected to y, it is

clear that if h(C) is a weak component of P/S0, then C must be a

weak component of P.

By Theorem 4.8 if some weak component of P/S0 contains at

least two distinct elements, then all of them contain at least two

elements. Suppose h(a) > h(b). When h(x) + h(a) > h(x) + h(b), it

is true that h(x + a) > h (x + b) and x + a > x + b. On the other

hand, h(x) + h(a) < h(x) + h(b) implies h(x + a) < h(x + b) and

x + a < x + b. Thus when h(x) is of order preserving type i rel-

ative to a weak component h(Ca) having at least two elements, x EC(C0).

Similarly, if x EC(Ca), then h(x) is of order preserving type i

relative to h(Ca).

Corollary 5.4: The system P/SO is a dualistic poip-group of

finite dimension.

If each weak component of P/SO consists of merely a single

element, P/SO is totally unordered and is obviously a dualistic poip-

group. Otherwise, the result follows from Theorems 3.22 and 5.3.

Suppose that Sx > y > Sz in P/S0. Then by Theorem 5.3,

x> y> z. However, x, y, and z are all in Cx = C (CO) U Cx(CO),

so that at least two of them are members of either Cx(C0) or C (CO).

These two elements are thus strongly connected--meaning that they

are in the same strorg component. This is impossible since

Sx 4 Sy i Sz j Sx. Hence there is no chain of length greater than

one in P/S0.

Corollary 5.5: Let x and y be weakly connected elements

of P. If P/SO is of length one, x and y are of the same order pre-

serving type relative to any given weak component of P if and only

if S, and S are both maximal or both minimal elements of P/S0.

This corollary follows from the observation that when

x f Sy then Sx is comparable to Sy only if x and y are of dif-

ferent order preserving types relative to any given weak component

of P and the fact that any weak component of P contains elements of

only two order preserving types relative to any other weak

component.

Corollary 5.6: If P/S0 is of length one, the elements of Ex

have the same order preserving properties in P as Sx has in P/"0--

that is, x is of order preserving type i relative to C if and only

if S is of order preserving type i relative to h(C).
x

Corollary 5.7: If P/S0 is commutative and of length one,

then Cx = Cx(Ca) U Cx(Ca) for all a,x P.

Theorem 5.8: The strong component S0 is a self-dual po-

group under the relativized ordering. Furthermore, every coset of

SO is order isomorphic to SO and if a is any element of the coset S

of SO, such an isomorphism may be described as follows: (i) if a is

order preserving on the left relative to CO, then bES corresponds to

-a + bES0, and (ii) if a is order reversing on the left relative to

C0, then bES corresponds to -b + aESO.

Obviously SO is a po-set under the relativized ordering.

Moreover, SOCCO(CO), so that every element of SO is order preserving

on the left and right relative to So. Hence SO is a po-group.

To show that So is self-dual, let x correspond to -x for all

x ES0. If y > z in SO, then 0 > z y, so that -z > -y--that is,

the correspondence is an anti-isomorphism (in the order sense) and

S0 is self-dual.

Now let S be some coset of SO and a some element in S.

Suppose that the members of S = a + So are order preserving

on the left relative to Co and let bES correspond to x = -a + bE S

(denote this by b <--> x). The correspondence is clearly one to

one. If y and z are members of SO such that y > z where c <--> y

and d<---> z, then -a + c = y_ z = -a + d and, since a is order

preserving on the left relative to CO, c d. Now if c > d in bg,

where c <-> y and d <--> z, then y = -a + c > -a + d = z since

-a is order preserving on the left relative to Ca = Cc. Therefore,

the correspondence is an order isomorphism.

If the elements of S are order reversing on the left relative

to So, a procedure similar to that above shows that SO is anti-

isomorphic to S under the correspondence which carries bES into

-a + b ES. Since So is anti-isomorphic to itself under the cor-

respondence which carries every element of So into its inverse, this

indicates that S is order isomorphic to SO under the correspondence

which carries bES into -(-a + b) = -b + aES0.

Theorem 5.9: A dualistic poip-group P is order isomorphic

to the ordinal product (P/SO)So of the finite dimensional dualistic

poip-group P/SO and the po-group S3. If A is a set of coset repre-

sentatives, then the isomorphism may be described as follows: (i) if

bEP is order preserving on the left relative to 50, then b cor-

responds to (Sb, -a + b) where a is in Sbh A, and (ii) if bEP is

order reversing on the left relative to SO, then b corresponds to

(Cb, -b + a) where a is in Sbf A.

When b P and (Sb, y) correspond in the manner described in

(i) and (ii), denote this correspondence by b <-> (C., x).

Suppose b <-> (Sb, y) and c <-> (Sb, z). Then b and c

are both members of SI-. It follows from Theorem 5.8 that b> c if

and only if y > z. Hence b> c if and only if (Sb, y) > (Sb, z) since

(according to the definition of the ordinal product of two po-sets)

when u = v, (u, y) > (v, z) if and only if y > z.

Suppose now that b<-> (5b, y) and c <--> (S, z), where

Sb c .c Then b # c. If b > c, it is implied by Definition 5.2

that Sb > Sc. Hence if b > c, (b, y) > (5c, z). When (Sb, y) >

(Sc, z), it follows that Sb > Sc. Thus by Definition 5.2,

(Sb, y) > (Sc, z) implies that b > c. Therefore, b > c if and only

if (Sb, y) > (Sc, z), completing the proof.

The above theorem is illustrated by Example 4.14 in which

P/SO is composed of the integers modulo 12. When it is ordered in

the natural manner, P/SO may be represented by the diagram below.

The elements of the figure are, of course, added modulo 12.

4 0 8 3 7 11

2 6 10 5 1 9

Fig. 5.10

It is easily verified that when P is of finite length,

SO = 0o This enables us to give the following definition.

Definition 5.11: A poip-group P is said to be upright if

P/So is of length one and SO is maximal in P/S0. It is said to be

inverted if P/SO is of length one and SO is minimal in P/S0.

The dual of any theorem concerning upright poip-groups

holds for inverted poip-groups.

Theorem 5.12: Let P be a dualistic poip-group, P/S0 be of

length one, and Sa be a maximal (minimal) element of ?/S0. Then x is

order preserving on the left relative to Ca if and only if x + Sa

is maximal (minimal), and x is order reversing on the left relative

to Ca if and only if x + Sa is minimal (maximal).

As before, let h be the mapping defined by h(x) = Sx. Con-

sider P/SO = h(P) where Sa = h(a) is maximal (minimal). Then

Sx = h(x)6 P/S0 is order preserving (reversing) on the left relative

to h(Ca) if and only if Sx+a = h(x + a) is maximal (minimal), and it

is order reversing on the left relative to h(Ca) if and only if x + a

is minimal (maximal).

The theorem now follows from Corollary 5.6.

Corollary 5.13: Let P be an upright dualistic poip-group

such that P/SO is of length one. An xE P is of order preserving

type 1 (order preserving type 4) relative to CO if and only if 'x is

a maximal (minimal) element of P/S0.

Many of the results of this section have been concerned only

with the case in which P/SO is of length one. When it is of length

zero, P has a simpler form since then Sx = Cx for all xCP. This

means that the elements of any weak component of P are all of the

same order preserving type relative to any other weak component of P.

Furthermore, every weak component is order isomorphic to the po-

group CO.

43

It was shown in this section that when P/SO is of length one,

many of its properties are carried over to P. Also, the order

structure of P can be determined if the order structure of So and

P/SO are known. Since Sg is a po-group we need characterize only

finite dimensional poip-groups.

VI. FINITE DIMENSIONAL POIP-GROUPS

The only finite dimensional poip-groups which need to be

given further consideration are the dualistic poip-groups of length

one. These systems are characterized below.

6.1: When a is a maximal element of a poip-group of length one,

Ta will represent the set x EP / x < a) and when a is

minimal, it will denote the set Ix-P / x > a .

Theorem 6.2: An upright dualistic poip-group P satisfies

the following conditions: (i) if t ETO, then -tET0; (ii) for any

element s of s' + CO, every element of s' + CO can be written in

the form s + t1 + t + . + tn, tiET0 (i = 1, 2, . n),

where the elements t. are not necessarily distinct; (iii) if
1

t + t2 + + tn = 0, ti T0 (i = 1, 2, ..,, n), then n is

an even integer; (iv) a + T0 = T0 + a for all aEP; (v) a + Tg = Ta

for all a P; and (vi) if s is maximal, then s + t + . + tn,

tiETO (i = 1, 2, . n), is maximal (minimal) if and only if

n is even (odd). Conversely, let G be a group and CO a normal sub-

group of G which contains a non-void subset To satisfying condi-

tions i-iv. Define an ordering on G using conditions v and vi,

first assigning some representative element sCs' + C (let 0
0

be the representative element of CO) the role of a maximal element.

Under this ordering G is an upright dualistic poip-group in which C0

is the set of all elements weakly connected to 0 and TO is the set of

elements of G which are less than 0.

Let P be an upright dualistic poip-group.

If t < 0, then -t + t = 0 is comparable to -t, so that,

since 0 is maximal, -t < 0. This establishes condition i. If

b ETa, by (6.1) b is comparable to a. .Hence -a + b < 0 and

-a + b = t for some tfT0. Consequently, b = a + t, so that b is

in a + TO, proving that T C a + T0. Now let t be a member of TO'

whence t < 0. Therefore a + t is comparable to a but a + t f a--

that is, a + t is in Ta. This implies that Ta)a + To and thus

Ta = a + T0. Similarly, Ta = TO + a, proving conditions iv and v.

Next suppose that a is an element of Co not of the form

t1 + t + + tn, ti (T0. There exists a sequence

0 = xl, x2, . xn = a

such that xi is comparable to xi+1. Assume that k is the minimum

integer such that xk is not of the form tI + t2 + + tn,

t iCTO. Then xk-l = t + t2 + . + t for t.i T0 and Xk1 is

comparable to xk, By condition v xk = xk,1 + t for some t( TO,

so that xk = t + t2 + . + tn + t, a contradiction. Hence

every element of s + CO can be expressed as in condition ii.

Suppose a = s + tI + t2 +

ti TO. Now tl< 0 means s + t1

maximal,

S. tn where s is maximal and

comparable to s. Since s is

s + t1 < s,

s + t + t2 > s + tl*

s + tI + t2 + t < s + tl + t2

s + tI + t2 + t + t > s + t1 + 2 + t3

and so on. Thus s + +l + t2+ . + t > s + tI + t2 + .. + tn-

if n is even and s + t + t + . + tn < s + t + t2 + . + tn-

if n is odd. Hence a is maximal (minimal) if and only if n is even

(odd), establishing condition vi from which condition iii follows

immediately.

Conversely, let G be a group containing a normal subgroup CO

which has a subset TO satisfying conditions i-iv. Suppose that

s + tI + t2 + + tn = s + t1+ t2' + . + t where

ti, tj' ET0 (i= 1, 2, . m; j = 1, 2, . n). Then

t + t2 + .. t tn tn . t' = 0. This equation,

along with conditions i and iii, implies that m + n is even--that is,

that either m and n are both even or both odd. Thus conditions v

and vi establish a well-defined partial ordering for G. Under this

ordering it is clear that G is a po-set of length one with no

isolated elements.

Let t1 + t2 " + tn, ti. TO, be an element of CO.

By (i) and (v), t + t + . + t is comparable to

c + t1 + t2 + . + tn1; t + t2 + + tn1 is comparable to

c + t + t. + t n21 tI + t2 + + tn2 is comparable to

c + tl + t2 + . + tn3. Continuing in this manner, it follows

that tI + t2 + + tn is weakly connected to 0. Similarly, any

element weakly connected to 0 is of the form ti + t2 + + tn,

t E T0. Hence C0 is a weak component of G.

Let x be some element of G. If a > b, then x + a is com-

parable to x + b since a = b + t for some t 4T0 implies x + a =

(x + b) + t. The remainder of the proof is that of Theorem 3.9.

Example 6.3: Let G be a group which contains a commutative

normal subgroup CO with basis elements gl, g2, . qk each of

even order. Every element of CO can be expressed uniquely (apart

from order) as a summation nlg1 + n292 + . + nkgk with

0 < ni < qi where qi is the order of gi. Let T 0=

[1. . gk, g.1 . g) Suppose that

mlgl + .. + mkgk + n1(-g1) + . + nk(-gk) = 0. Then

(ml- n1g1 + . + (m nk)gk = 0, and (m ni)i= 0 (i =

1, 2, . k). Hence mi ni = piqi for some integer p., and

mi ni is even since qi is even. Thus mi + ni is even, implying

that mi + . + mk + n1 + . + nk = (ml + nl) + . + (mk + nk)

is even. Therefore, TO satisfies conditions i-iv of Theorem 6.2.

As a particular example, consider a group G in which CO is

generated by a and b under the conditions a + b = b + a, 2a = 0, and

4b = 0. Let TO = (a, b, 3bi Then when CO is ordered according

to conditions v and vi of Theorem 6.2, CO becomes the poip-group

represented by the following diagram.

0 a + b a + 3b 2b

a b 3b a + 2b

Fig. 6.4

Assign s + aEs + CO s C0, the role of a maximal element

of G and order s + C0 according to Theorem 6.2. Then s + C0 is as

indicated below.

s + a s + b s + 3b s + (a + 2b)

s s + (a + b) s + (a + 3b) s + 2b

Fig. 6.5

VII. THE ALGEBRAIC CHARACTER OF POIP-GROUPS

It has been shown that many of the properties of P, in

particular its structure as a po-set, can be determined when So and

P/SO are known. In this section more is said about the relation-

ships between the latter systems and the algebraic nature of P.

Theorem 7.1: Let G be a group satisfying (i) G contains a

normal subgroup S ordered so as to form a connected po-group, such

that (ii) G/S is ordered so as to form an upright dualistic poip-
*
group of length one, and (iii) a + 0 = 0 + a for all a(G. Then

G can be ordered in one and only one way to form a dualistic poip-

group such that the po-groups So and S coincide and the poip-groups

P/S0 and G/S coincide. This ordering is described as follows:

(1) if a + S # b + S, then a> b if and only if a + S> b + S in

G/S, (2) if a + S = b + S is maximal in G/S, a > b if and only if

a b (0 and (3) if a + S = b + S is minimal in G/S, a > b if

and only if b a 0 .

The ordering defined on G is obviously a reflexive and anti-

symmetric relation. In order to show that it is transitive, con-

sider a > b and b > c in G.

Case i: Suppose a + S = b + S = c + S is maximal in G/S.

Then a b > 0 and b c > 0. Since S is a po-group, we have a c =

(a b) + (b c) > (a b) + 0 = a b> 0, implying that

a cE0 and a > c.

Case ii: Suppose a + S = b + S = c + S is minimal in G/S.

Then b a > 0 and c b > 0. Therefore, c a =

(c b) + (b a) > (c b) + 0 = c b > 0, implying c aE0*

and a > c.

Case iii: Suppose a + S = b + S is maximal and c + S is

minimal in G/S. Then a + S = b + S> c + S, proving that a > c.

Case iv: Suppose a + S is maximal and b + S = c + S is

minimal in G/S. Then a + S> b + S = c + S, giving a > c.

Therefore, G is a po-set, and we want to show that under

this ordering G is a dualistic poip-group.

First observe that if x yEO then -yE-x + 0 = 0 -x,

implying that -y + x0 .

Now consider any element xEG and a > b in a weak component C.

Obviously a + S> b + S.

Suppose x + S is of order preserving type 1 relative to the

weak component of G/S which contains a + S. It will be shown that

x itself preserves order when added to the left and to the right of

elements of C.

Case i: Suppose a and b are ordered according to (1). Then

a + S> b + S and (x + a) + S = (x + S) + (a + S) > (x + S) + (b + S) =

(x + b) + S. Hence x + a > x + b. Similarly, a + x > b + x.

Case ii: Suppose a and b are ordered according to (2).

This means a bE0 and a + S = b + S is maximal in G/S. Then

-(x + b) + (x + a) = (-b x) + (x + a) = -b + a f0 Therefore,

(x + a) (x + b)0E Since x + S is order preserving on the left

relative to the weak component of G/S containing a + S, and a + S

is maximal, then (x + a) + S = (x + S) + (a + S) must be maximal.

Thus x + a > x + b. On the other hand, (a + x) (b + x) =

(a + x) + (-x b) = a bF0 However, (a + x) + S is maximal in

G/S, so that a + x > b + x.

Case iii: Suppose a and b are ordered according to (3).

The proof that x + a > x + b and a + x > b + x is similar to that

of the preceding case.

Continuing in this manner, it can be shown that the order

preserving properties of xEG are the same as those of x + S in G/S.

It follows from Theorem 3.22 that G under the ordering described by

(1)-(3) is a dualistic poip-group P. It is easily shown that SO

and P/SO coincide with S and G/S, respectively.

Now consider an upright dualistic poip-group P. Obviously

it satisfies (1). If a > b, where a + S = b + S is maximal in P/S0,

then bCb( CO) and -bECb(Cb). Hence a b > 0 and a b O*.

Moreover, a bE 0 implies a b> 0. Thus if b + S is maximal,

a bC0 if and only if a> b. We have proved that P satisfies (2).

Similarly, it can be shown that P satisfies (3). This establishes

the uniqueness of the ordering of G described by (1)-(3).

Theorem 7.2: Let G be a group such that (i) G contains a

normal subgroup S ordered so as to form a connected po-group, and
(ii) for each cost S' of S either a + 0 + a for every aS'
(ii) for each coset S' of S either a + 0 0 0 + a for every a ES'

0+
or a + 0 = 0 + a for every a S'. Then G can be ordered to form a

dualistic poip-group P such that the po-groups So and S coincide and

P/S0 is totally unordered. All such orderings are effected by

leaving elements not in the same coset of S unordered and ordering

each coset S' f S in either of the following ways: (1) for all

a,bES', a> b if and only if a b0 *, or (2) for all a,bES',

a> b if and only if b aE0 .

The proof that G is a po-set under any one of these orderings

is similar to that in the preceding theorem. We now show that G is

a dualistic poip-group under any one of them.
+ +
Suppose a + 0 = 0 + a and consider a + s where s0 .

There exists t (G such that a + s = t + a. Then a s = -t + a
+ + *
and, since -sE0 -tC0 It follows that tEO and a + + C0 + a.
+ + *
Likewise, a + 0 0 + a, so that a + 0 = 0 + a.
*
A similar argument will show that a + 0 = 0 + a implies

a + 0 = 0+ + a.

Suppose a> b according to (1). Then a bfO Clearly
*
a*0 + b and a + xq0 + (b + x) for any x. This means

(a + x) (b + x)E0 which implies that x is either order pre-

serving or order reversing on the right relative to any coset of S.

Case i: Suppose b + 0 = 0 + b. Their aEO + b = b + 0

and x + a (x + b) + 0*. If (x + b) + 0 = 0 + (x + b), then

(x a) (x + b)60 If (x + b) + 0 = 0 + (x + b), then

(x + a) (x + b)0+ and (x + b) (x + a)(0 .

0+ 0+
Case ii: Suppose b + 0 =0 + b. Then a 0 + b = b + 0.
+ 0+ *
Clearly x + a (x + b) + 0+. If (x + b) + 0 = 0 + (x + b), then

(x + a) (x + b)E0 If (x + b) + 0 = 0+ + (x + b), then

(x + a) (x + b) + whence (x + b) (x + a)E0 .

The proofs of the above cases imply that x is either order

preserving or order reversing on the left relative to any coset of S.

The case in which a > b according to (2) can be proved in a similar

manner. By Theorem 3.22, G is a dualistic poip-group under any one

of the orderings described in (1) and (2).

We now state an important corollary of the above theorems.

Corollary 7.3: Necessary and sufficient conditions that a

group G can be ordered to form an infinite dimensional poip-group

are: (1) G contains a normal subgroup S #f 0) which can be ordered

so that it forms a weakly connected po-group, such that (ii) for
*
each coset S' # S, either a + 0 = 0 + a for every a S' or
+
a + 0 = 0 + a for every aES'.

The sufficiency of the conditions has been shown. Let P be

an infinite dimensional poip-group. To establish the necessity

we need show only that P satisfies condition ii, since it obviously

satisfies (i).
2 *
Suppose that aCCa(C0). If bfa then b> a, so that

+
-a + b > 0 and b a < 0. Hence b(a + 0 and b 0 + a, implying
+ +
that a C a + 0 and a C0 + a. If cE 0 and dO0 then c> 0

and d<_0, so that a + c> a and d + a> a. Thus a + c(a and
S + +
d + a(a giving a = a + 0 and a = 0 + a.

Similarly, afCl(Co), a CC3(Co), and aCC4(CO) imply
+ + + *
a + 0 = a = 0 + a, a + 0 = a = 0 + a, and a + 0 = a = 0 + a,

respectively. Since Sa CC(C0) for some i, the result follows.

The following theorem occurs in (4, p. 214).

Theorem 7.4: Any po-group S is determined to within iso-

morphism by the set R = 0 since a > b, a bIR, and -b + aER

are equivalent conditions. Moreover, (i) OCR, (ii) if a,bR, then

a + b R, (iii) if a,bER and a + b = 0, then a = b = 0, (iv) for

all aCS, a + R = R + a. Conversely, if S is any group, and R is a

subset of S satisfying (i)-(iv), then S can be ordered to form a

po-group by defining a > b in S to mean a bC R.

It is obvious that a po-group S is weakly connected if and

only if, for every pair a,b CS, there exists a sequence

a = x1, x2, . xn = b

of elements of S such that either xi xi+lFR or xi+1 xiE R

(1 = 1, 2, . n 1).

The previous corollary can now be stated in purely algebraic

terms.

Theorem 7.5: Necessary and sufficient conditions that a

group G can be ordered to form an infinite dimensional dualistic

polp-group are: (i) G has a normal subgroup S 0 { 0 which contains

a subset R satisfying (ii) OFR; (iii) if a,bE R, then a + bCR;

(iv) if a,bR and a + b = 0, then a = b = 0; (v) for all a S,

a + R = R + a; (vi) for every pair a,b(S, there exists a sequence

a = x, x2, . xn = b

of elements of S such that either xi xi+,1 R or xi+1 xi R

(i = 1, 2, . n 1); and (vii) for each coset S' i S, either

a + R = R + a for all a S' or a + R = -R + a for all a fS'.

Theorems 7.1 and 7.2 also lead to methods for constructing

infinite dimensional poip-groups from a given dualistic poip-group

of finite dimension and a given weakly connected po-group. These

are described below.

In the discussion of the direct sum of two groups A and B,

the symbol (a, B'), where a EA and B' is a subset of B, represents

the set of all pairs (a, b') such that b'E B'. The symbol (A', b),

where A' is a subset of A and b B, is defined accordingly.

Theorem 7.6: Let P be the direct sum G + S of an upright

dualistic poip-group G and a weakly connected po-group S. Then P

is a dualistic poip-group when it is ordered as follows: (i) if

gl g 2 (g' sl) > (g2, s2) if and only if gl > g2 in G; (ii) if

gl= g2 is maximal, (1, s1) > (g2' s2) if and only if s1 > s2 in S;

and (iii) if gl = g2 is minimal, (g1, s1) > (92, s2) if and only if

s1 < s2 in S. Furthermore, SO coincides with (0, S), and P/S0

coincides with (G, 0).

The theorem follows directly from Theorem 7.1 once we have
shown that (, s) + (0, 0) (0, 0) + (g, s) for every element
shown that (g, s) + (0, 0) = (0, 0) + (g, s) for every element

(g, s) EG i S. Here it is understood that (0, S) is ordered such

that (0, sl) (0, s2) if and only if sl s2, and that (G, 0) is

ordered such that (gl, 0) > (g2, 0) if and only if gj_ g2. But
*
(g, s) + (0, 0) = (g, s) + (0, 0 ) = (g, s + 0 ) = (g, 0 + s) =

(0, 0 ) + (g, s) = (0, 0) + (g, s).

Theorem 7.7: Let P be the direct sum G + S of a group G

and a weakly connected po-group S. Then P is a dualistic poip-

group when it is ordered in any one of the ways indicated below:

(i) 91 9 g2 implies (gl, sl) and (g2, s2) are not comparable;

(ii) (0, sl) > (0, s2) if and only if sl > s2; g 4 0 implies either

(iii) for all sl, s2 S, (g, s) > (g, 2) if and only if sl > s2

or (iv) for all sl, s2 (S, (g, sl) > (g, s2) if and only if

si < s2. Moreover, for any one of these orderings, So coincides

with (0, S) and P/SO = (G, 0) is totally unordered.

The proof of the theorem is essentially that of Theorem 7.6.

Example 7.8: Let G be the poip-group consisting of the

group 12 ordered by setting 0 > 1, and let S be the po-group con-

sisting of the even integers in their natural order. When G + S

is ordered according to Theorem 7.6, we have

S. > (0, 2) > (0, 0) > (0, -2) > . >

(1, -2) > (1, 0) > (1, 2) > . .

Comparing this example to the poip-group P of Example 2.19

which is not the direct sum of P/SO and S0, even though P/SO and SO

are order isomorphic and group isomorphic (under the same correspond-

ence) to G and S, respectively, shows that the algebraic structure of

a poip-group P is not completely determined by the systems S0 and P/S0.

In this section we have given necessary and sufficient con-

ditions that a group can be ordered to form an infinite dimensional

poip-group. In addition, we have exhibited the relationships be-

tween the algebraic properties of a dualistic poip-group P and the

structures of Sg and P/SO, and have shown that the latter two systems

do not always completely determine P as a group.

VIII. LOIP-GROUPS

The preceding results are much simpler to state if it is

assumed that the poip-group itself is weakly connected.

A weakly connected dualistic poip-group P is a po-group if

and only if P/S0 is of length zero--that is, if and only if P = SO.

Otherwise, P is the union of CO and its coset CO. Moreover, it can

easily be shown that, under the relativized ordering, C1 is a po-

4 1
group and CO is order isomorphic to CO under the correspondence

carrying afC1 into -a + rfC4, where r is the coset representative

of C4 As a consequence of Corollary 5.5, no member of C1 is less
0 0

than an element of C0 when P is upright. These results are sum-

marized in the following theorem.

Theorem 8.1: If P is an upright dualistic poip-group which

is weakly connected, then (i) P = C1 U C4 where (ii) C1 is a normal

subgroup of P with costs C1 and C4, (iii) under the relativized

ordering Cg is order isomorphic to the po-group C1, and (iv) no

element of C1 is under an element of C4

When every pair of elements of P is bounded in some manner,

conditions i-iv are strengthened as shown in the corollaries below.

Corollary 8.2: If P is an upright dualistic poip-group

in which every pair of elements has either an upper or a lower bound,

it satisfies conditions i-iii, and (iv)' every element of C1 is
0
greater than every element of C4.

Clearly P satisfies conditions i-iv. Now let a and b be
1 4
members of C0 and C0, respectively, and suppose that they have an
C4sohtcC and cES How-
upper bound c. By condition iv, c C4, so that cC-1 and c(Sa How-

ever, c > b and Sa = Sc > b whence a > b. Similarly, if a and b

have a lower bound, a > b.

Corollary 8.3: If P is an upright dualistic poip-group

which is either up-directed or down-directed, it satisfies (i),

(ii), (iii), (iv)', and (v) C1 and C4 are strong components of P.
0 0

Corollary 8.4: If P is an upright loip-group, it satisfies

(i), (ii), (iv)', (v), and (iii)' under the relativized ordering C1

is an 1-group and C4 is order isomorphic to it.

By condition iii, CO is a po-group under the relativized

ordering. If a and b are elements of Cg, by (iv) a U b(C. Now
4 4
assume a (f bECO, so that every member of C6 must be less than

a () b, in which case a C) b is a maximal element of C4. Condition iii

thus implies that CO has a maximal element. However, the only

1-group with a maximal element is the group of order one. Thus

a bCCO, and C1 is a loip-group.

As a result of this corollary many of the results concerning

1-groups are easily extended to loip-groups. We note several

examples. In an 1-group L, x + (a U b) = (x + a) U (x + b) for all

a,b,x EL. In a loip-group P, this becomes x + (a U b) =

(x + a) U (x + b) if x C1 and x + (a U b) = (x + a) n (x + b) if
0

xEC4. Every 1-group and every loip-group is a distributive lattice.

In (4, p. 222) it is shown that the congruence relations on an

1-group L are the partitions of L into the costs of its different 1-

ideals. On the other hand, the congruence relations on a loip-group P

are the partitions of P into the costs of its k-ideals, where the

1k
k-ideals are P and its normal subgroups which are 1-ideals of CO'

These results are all easily established.

IX. SIThfr4ARY

The order structure of partially ordered ideal preserving

groups was determined in terms of partially ordered groups; many

results concerning these latter systems can easily be extended to

the more general partially ordered ideal preserving groups. Nec-

essary and sufficient conditions that a group can be ordered to

form a non-trivial partially ordered ideal preserving group were

obtained. The principal concepts employed in this dissertation are

the equivalence relations called weak and strong connectivity.

The investigation of partially ordered ideal preserving

rings and fields is suggested as a problem for further research.

LITERATURE CITED

1. O. Frink, Ideals in partially ordered sets, Amer. Math. Monthly.
vol. 61 (1954) pp. 223-34.

2. A. T. Butson, "Simply ordered ideal preserving groups." Paper
read before the meeting of the American Mathematical Society,
Rochester, N. Y., December, 1956.

3. C. C. MacDuffee, An introduction to abstract algebra. New
York, J. Wiley and Sons, 1940.

4. G. Birkhoff, Lattice theory. American Mathematical Society
Colloquium Publications, vol. XXV. New York, American Mathe-
matical Society, 1948.

BIOGRAPHICAL SKETCH

Jan Frederick Andrus was born September 17, 1932, in

Washington, D. C. He was graduated from the College of Charleston

in Charleston, South Carolina, in May, 1954, with the degree

Bachelor of Science. Work for the degree Master of Arts was under-

taken at Emory University, Emory University, Georgia, and the degree

was granted in August, 1955. The author's undergraduate major was

in chemistry with a minor in mathematics; his graduate major has

been mathematics, and his minor work has been in the field of physics.

During his studies the author has held assistantships in

chemistry at Emory University and in mathematics at the University

of Florida. He was employed in the summer of 1957 as an aeronautical

research engineer with the National Advisory Committee for Aero-

nautics at Langley Field, Virginia.

While pursuing undergraduate study, the author was elected to

membership in Sigma Alpha Phi, College of Charleston honorary soci-

ety. He was awarded the S. Keith Johnson medal for outstanding

achievement in science and a similar award for the highest achieve-

ment in mathematics. He was elected to membership in Alpha Sia

Phi, national physics honorary, while a graduate student at Emory

University.

This dissertation was prepared under the direction of the

chairman of the candidate's supervisory committee and has been

approved by all members of that committee. It was submitted to the

Dean of the College of Arts and Sciences and to the Graduate Council,

and was approved as partial fulfillment of the requirements for the

degree of Doctor of Philosophy.

Date June 3.1958

Dean, College of Arts and Sciences

SUPERVISORY COMMITTEE:

Chairman

Co-Chairman

^^.^c02

Full Text

PAGE 1

PARTIALLY ORDERED IDEAL PRESERVING GROUPS By JAN FREDERICK ANDRUS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA June, 1958

PAGE 3

ACKNOWLEDGMENTS The author wishes to thank Dr. Alton T. Butson, A.ssistant Professor of Mathematics, for suggesting the problem treated in this dissertation and for his encouragement and guidance in working towards its solution. He is also grateful to his wife and to George B. Morgan for their invaluable work in preparation and reproduction of the manuscript. li

PAGE 4

T;'\BLE OF CONTENTS Page ACKNOWLEDG^ENTÂ£ ii I. INTRODUCTION 1 II. PRELIMINAf?y CONCEPTS 3 III. NON-DUALISTIC POIP-GROUPS 9 IV. CONNECTIVITY 25 V. A DECOivIPOEITION THEOREM 35 VI. FINITE DIMENSIONAL POIP-GiÂ©UPÂ£ 44 VII. THE ALGEBRAIC CHASiR.CTER OF POIF-GROUPS 49 VIII. LOIP-GROUPS 58 IX. SUI-W/.RY 61 LITERATURE CITED 62 BIOGRAPHICAL SKETCH 63 iii

PAGE 5

I. INTRODUCTION In recent years there has been considerable study of algebraic systems in which there is a relation of simple or partial ordering which is closely related to the algebraic operations of the system. Examples of such systems are partially ordered groups and semigroups, ordered fields and rings, vector lattices, and partially ordered linear spaces. It has been usual in studying such systems to assume a very strict connection between the operations and the ordering, namely that one or all of the operations should preserve the order relation. In a recent paper Frink (1) proposed a definition of an ideal in a partially ordered set and suggested the possibility of generalizing an ordered algebraic system by requiring the algebraic operations to preserve ideals (determined by the order relation), rather than the order relation itself. Butson (2) obtained a structure theorem for simply ordered ideal preserving groups--the suggested generalization of simply ordered groups. This dissertation is concerned with partially ordered ideal preserving groups. The principal results show how a general partially ordered ideal preserving group may be decomposed into a partially ordered group and a trivial partially ordered ideal preserving group. Consequently, many of the results concerning partially ordered groups are easily extended to partially ordered ideal preserving groups.

PAGE 6

The multiplicative group of real numbers ordered according to magnitude is a significant example of a system which is a partially ordered ideal preserving group (actually a simply ordered ideal preserving group) but is not a partially ordered group. This system contains a maximal partially ordered subgroup--namely the positive real numbers ordered as above. Orderwise, the set of negative real numbers can be considered as the dual of the set of positive real numbers so that the structure of this particular partially ordered ideal preserving group as an ordered system is completely determined by the maximal partially ordered subgroup. Actually any lattice ordered ideal preserving group is the ordinal sum of a maximal lattice ordered subgroup and its dual. However, the structure of a general partially ordered ideal preserving group is more complicated.

PAGE 7

II. PRELIMINARY CONCEPTS The basic concepts of group theory may be found in (3) and those of partially ordered systems in (4). Some of the less widelyknovm concepts are presented in this section. Let P be a po-set (partially ordered set). If F is a nonvoid subset of P, F* will denote the set (xÂ€? / x> f for every fÂ€F} and F* the set {"xCP / X < f for every f Â€ f} . The sets (F ) and (F ) will be denoted by F * and F* , respectively. It can be shown that FCZF**, FCF''*, (F**)* = F*, and (F**)"" = F*. Definition 2.1 ; P is up-directed if and only if F* ^ (? for every finite subset F of P. It is down-directed if and only if ? i ^ for every finite subset F of P. Definition 2.2: A subset J of P is an order ideal if and only if F f ^ and F ^CJ for every finite subset F of J. A subset J of P is a dual order ideal if and only if ?"" i ^ and F**Cr J for every finite subset F of J. The above definition generalizes the concept of a lattice ideal. It differs from the definition suggested by Frink (1) for an ideal in a po-set in the requirement that F* be non-void. In the remainder of this thesis order ideals and dual order Ideals will be referred to more briefly as ideals and dual ideals, respectively. No ambiguity will result since algebraic ideals will not be considered.

PAGE 8

The following properties of P are immediate consequences of Definition 2.2. 2,3 ; If J is an order ideal of P, x* C J for every x^ b implies that x + a + y> x + b + y for all x,yÂ«P. If P is a lattice satisfying (ii) and (iii), it is called an 1-group (lattice ordered group) If P is a simply ordered set satisfying (ii) and (iii), it is called an o-group (simply ordered group).

PAGE 9

The following definition generalizes the above concepts in the manner suggested by Frink. This dissertation will be primarily concerned with these generalizations. Definition 2.9 : A poip-group P (partially ordered ideal preserving group) is (i) a po-set, (ii) a group, in which (iii) if J is an ideal, x + J + y is either an ideal or a dual ideal for all x,y
PAGE 10

t Xo = Xq. Since Xq is isolated, jt + Xj = Xg, t + x,i = (j5 and ft + Xg = Xq, t + x,! = (p. By Definition 2.2 t + Xg* is neither an ideal nor a dual ideal--contradicting Definition 2.9. Thus if P is of finite lengrth, it is totally unordered Â— that is, it is a trivial poip-group, or it is of length one and contains no isolated elements. These statements are summarized in the following theorem; Theorem 2.11 ; A non-trivial poip-group has either infinite length and no minimal elements or length one and no isolated elements, Definition 2.12 ; A poip-group P is dualistic if x + J + y is either an ideal or a dual ideal for every dual ideal J of P and all elements x,y Â€P. A poip-group which is not dualistic is said to be non-dual istic . The above classifications of poip-groups will serve as a framework for the remainder of this dissertation. All graphical representations of poip-groups in this vrork are similar to the "Hasse diagrams" (4, p. 6). Example 2.13 ; Any po-group P is a poip-group since, for any *+ *+ Â» elements x,y CP, x + F +y=(x + F + y) whenever F and F are non-void subsets of P. Example 2.14 ; The multiplicative groups of rational and real numbers ordered according to magnitude are soip-groups. The additive group of rational integers and the group of integers modulo p will be represented by I and Ip, respectively.

PAGE 11

Example 2.15 ; I and I2J, are poip-groups of length one when ordered so that m > n if and only if ra is even and n = m J^ 1. The diagram of I ordered in this manner is shown in Figure 2.16. -4-2 2 4 Fig. 2.16 It should be noted that there are no nontrivial po-groups of finite length. Example 2.17 ; The direct sum l2k "t" P of Igj, ordered as in Example 2,15 and any po-group P forms a poip-group when ordered as follows: (i) if m y n, (m, x) > (n, y) if and only if ra > n, (ii) if m = n and ra is even, (m, x) > (n, y) if and only if x > y, and (iii) if m = n and ra is odd, {m, x) > (n, y) if and only if x < y. When k = 1 and ? is an 1-group, Igj, + P is a loip-group under the above ordering. Figure 2.18 is the diagram for the case in which k = 2 and P is the additive group of rational integers in their natural order.

PAGE 12

(p.' 2)

PAGE 13

III. NON-DUALISTIC POIP-GROUPS In this section we completely characterize non-dualist ic poip-groups and in so doing obtain several important theorems concerning dualistic poip-groups. Definition 3.1 : Elements a and b of a po-set P are said to be weakly connected if and only if there exists a finite sequence a = x^, X2Â» . . . , x^ = b of elements of P such that Xj^ is comparable to Xj^+j^ (i = 1, 2, . . . , n-1) . Definition 3.2 ; A set C is said to be weakly connected if a is weakly connected to b for every pair of elements a,b^C. A maximal weakly connected set is called a weak component , and C will denote the component containing a. Weak connectivity is obviously an equivalence relation. It will be discussed in greater detail in Section IV but is applied in several theorems of this section. Unless otherwise stated, all discussion will be concerned with the elements of a poip-group P. The results concerning addition of an element on the left of elements of P are true when restated in terras of addition on the right. Lemma 3.3 ; If a is weakly connected to b, x + a is weakly connected to x + b.

PAGE 14

10 Suppose that a is weakly connected to b. Then there exists a sequence a = x-^, X20 . . . , x^ = b such that x^ is comparable to x^^.^. The elements x. and x.^^ are contained in either x^ or xt^^. By Definition 2.9 either Cx + x., ^ " ^i+lj ^ "^ Â°^ /x + x^, X + x.^J i (f. Hence x + x^ is weakly connected to x + x^^-j^. It follows from the transitivity of weak connectivity that x + a is weakly connected to x + b. Theorem 3.4 ; The weak component Cg is a normal subgroup of P whose cosets are the weak components of P. Let c and c' be elements of Cg. Then c and c' are weakly connected to 0. By Lemma 3.3 c + c' is weakly connected to c + = c. By transitivity c + c' is weakly connected to 0, implying that c + c'CCg. Again by Lemma 3.3 -c + c = is weakly connected to -c + = -c. Hence -cÂ€Cg. Thus Cg is a siibgroup of P. By Lemma 3.3 -a + Cg is a weakly connected set, all the elements of v*\ich are weakly connected to 0. Hence -a + C^CCg and Cg da + Cg. Likewise, a + Cg is a weakly connected set, all the elements of vAich are weakly connected to a. Thus C^IDa + Cg so that Cg = a + Cg. Similarly, Cg + a = Cg. Lemma 3.5 : Let P be a system vd\ich is (i) a po-set, (ii) a group, in which (iii) for each weak component C of P and each pair of elements x,yCP, either x + a + y_> x + b + y for all pairs a,bÂ€C

PAGE 15

11 such that a > b or x + a + y_< x + b + y for all pairs a,b Â€C such that a > b. Then P is a dualistic poip-group. Let J be an ideal of P. Since {a, bj ^ >^ for all a,b(J, J is a subset of C for some weak component C of P. Let X be any element of ? and F any finite subset of J. Consider the case in vd\ich x preserves order when added to the left of elements of C (that is, x + a > x + b for all a,bÂ€C such that a> b). It can easily be shown that -x preserves order when added to the left of elements of x + C. Let x + d be any element of x + F ^nd x + e Â« any element of (x + F) . ?ince x + e is over every element of x + F, e is over every element of F. Hence e^F . Therefore, d < e so that *+ + *+ X + d< X + e--that is, x + de(x + ?) . Hence x + F ( (x -^ F) * + Now let X + g be any member of (x + F) . It is under every element of (x + F) . If we let h be any element of F , it is over all members of F so that X + h is over all members of x + F. Thus x + h Â£{x + F) and X + F CZ (x + F) ; in fact, it can be shown that the sets are equal. Therefore, x + g is under all elements of x + F --that is, g * Â»Â• *+ is under every element of F and g^F . This means x + gÂ€x + F so that X + F '"'iDlx + F) *. Hence x + F'^=(x + F)"^. It can be shown in a similar manner that if x reverses *+ order when added to the left of elements of C, then x + F = +* *+ *+ +Â• (x -^ F) . Thus X + F is ecrual to either (x + F) or (x + F)

PAGE 16

12 Suppose X + F = (x + F) . Since x + F C x + J, (x + F) C_. X + J. Now any finite subset of x + J can be expressed in the form x + F where F is a finite subset of J. Therefore x + J is an ideal. On the other hand, it can be shown that if x + F = +* (x + F) , X + J is a dual ideal. Hence for any ideal J of P and x 6 ?, x + J is either an ideal or a dual ideal. Dually, x + J is either an ideal or a dual ideal if J is a dual ideal of P. Clearly x + J + y is an ideal or a dual ideal for any ideal or dual ideal J of P and any pair x,yc P. Thus P is a dualist ic poip-group. The following four lemmas on poip-groups of finite dimension will be used in the proof of Theorem 3.10, which completely characterizes non-dual istic poip-groups. Lemma 3.6 : In a poip-group P of length one the figures a c x+a x+b b d . x+d x+c cannot exist simultaneously. Assume that the figures do exist in P. By (2.4) (a, cj is an ideal containing b and d. The elements x+b and x+d are in X + fa, c\ . Therefore x + /a, c] is not an ideal since X + c Â€ (x + b) but c^ |a, c] , and it is not a dual ideal since X + a f (x + d) but af. \a, cj . This is a contradiction since X + a* must be either an ideal or a dual ideal.

PAGE 17

13 Lemma 3.7 : In a poip-group P of length one the figures a c x+d b d x+c cannot exist simultaneously if x + a and x + b are maximal. Assume that the figures do exist simultaneously and that X + a and x + b are maximal. The ideal (x + d) contains x + c and x+d but contains neither x + a nor x + b. Hence -x + (x + d) contains c and d but does not contain a and b. By (2.3) -x + (x + d)* is neither an ideal nor a dual ideal. Since it must be one or the other, this is a contradiction. Lemma 3.8 : In a poip-group P of length one x + a and x + b are not both maximal if a > b. Assume that a > b and that x + a and x + b are maximal. Since x * a is either an ideal or a dual ideal and contains x + a and X + b, there exists an element x + c6 {x + a, x + bj *. Obviously (x + a)"*" and (x + b)* are ideals containing x + c. Since -x + (x + a) is either an ideal or a dual ideal, either -) + * + [a, cj ^ if or (a, c] f (p. Also -x + (x + b) is either an ideal or a dual ideal, so that either [b, cj * ^ :? or [b, cj ' =/ vS. Clearly this is impossible unless either c > b or c < a. Case i; f^uppose c > b. The set -x + (x + a)"^ contains a and c but not b since x + b is maximal. By (2.3) and the fact that b(a*, -X + (x + a)* is a dual ideal. If (bj = (a, c] *, then

PAGE 18

14 hi [a, c] . By the definition of a dual ideal {bj = [a, cj implies be -x + (x + a) . Hence (b]^ /a, cj, so that there must {> + + a, cj distinct from b. The set x + a contains X + a and x + d, so that either /x + a, x + d] i (^ or * + /x + a, x + dji (p. Since x + c contains x + c and x + d, either jx c, x + dj i (f or /x + c, x + d] i ^. The above conditions can occur only ifx+ dx + c. However, these relations contradict Lenraas 3.6 and 3.7, respectively. Case ii : Suppose c < a. The set -x + (x + b) contains b and c but not a since x + a is maximal. By an argument similar to that in Case i, there exists an element d such that d > b, d > c, and d y a. Since x + d contains x + b, x + c, and x + d, either [x + b, X + c, X + dj "*" ^ x + c. As before, these conditions contradict Lemmas 3.6 and 3.7, respectively. This establishes the lemma. Lemma 3.9 : If P is a poip-group of length one in vi^ich a_> b implies x + a + y and x + b + y are comparable for all x,y e PÂ» then P is dualist ic. Let c and c* be maximal elements in a weak component C of P. Then there exists a secfuence c = X]^, X2, . . . , Xj^ = c' of distinct elements of C such that x, is comparable to x.^-j^. We note that x^ is maximal if and only if i is odd and in particular

PAGE 19

15 that n is odd. The sequence x+c=x+x,, x+ xÂ„, . . . , X + xÂ„ = X + c' is also a sequence of distinct elements such that x + x. is comparable to X + Xj^i. If X + c is maximal (minimal), then x + x^ is maximal if and only if i is odd (even). Since n is oddx + c' = x + x is n maximal if and only if x + c is maximal. Thus if c' is some maximal element of C and if x + c' is maximal (minimal), then x + c is maximal (minimal) for all maximal elements cfC. Since every element of ? is either maximal or minimal, X preserves (reverses) order when added on the left of the elements of C. It follows that P satisfies conditions i-iii of Lemma 3.5 and that it is dualistic. Theorem 3.10 : Any non-dualistic poip-group is (i) a po-set of length one, in which (ii) the wealc components are the cosets of Cg, (iii) every weak component is a principal ideal, and ( iv) Cq contains more than two distinct elements. Conversely, any group P satisfying conditions i-iv for some normal subgroup Cq is a non-dualistic FÂ»ip-group. The latter statement will be proved first. Let P be a group satisfying conditions i-iv for some normal subgroup Cg of P, and let J be an ideal of P. By Definition 2.2 J is contained in some weak component C, which by condition iii is a principal ideal, say, c . If J is an ideal (jj consisting of a single element, then x + J + y = {x + j + yj is, according to (i), either an ideal or a dual ideal. If J contains two distinct elements j and j', by Definition 2.9 it

PAGE 20

16 contains Ij, j'j . Thus c f J so that Jl^c* = C. Hence J = C. Since x + J + y = x-Â»-C + y = x + C +y = Gx+c+y, x + J + y is a principal ideal. Therefore, P is a poip-group. In order to show that P is non-dualistic, consider a weak component c and distinct elements c-^ and C2, which are properly under c. That such elements exist follows from condition iv. Now there is an x such that x + Cg = c. bince c, c-,, and Cg are distinct, X + c =?< c and x + c, ^^ c. Thus x + c and x + c-, are minimal elements such that [x + c, X + c-^ = (x + C2} = [cj . Hence fc} ^ *+ Â» |x + c, X + c^\ but (c]i [x + c, X + c-^'l = X + cj^ . This Â» implies that x * c-^ is not an ideal. It also is not a dual ideal since [x + c, x + c,] = ?. Thus P is a non-dualistic poip-group. Now let P be a non-dualistic poip-group of finite dimension. By the contraposit ive of Lemma 3.9 there exist elements a,b,xÂ€P such that a > b but x + a and x + b are not comparable. By Lemma 3.8 and the fact that fx + a, x + b} CZ. x + a , there exists an element X + c properly over x + a and x + b. The elements a, b, and c are members of -x + (x + c) , so that either /a, b, cj ^ ^^ or [a, b, c) i ^. This implies either c > b or c < a. If c > b, then a and c are maximal and x + c > x + a, which contradicts Lemma 3.8. So we must have c < a. The set x + a is not a dual ideal since it contains two minimal elements, x + a and x + b. Thus it is an ideal. It cannot contain a maximal element distinct from x + c or a minimal element

PAGE 21

17 not under x + c. Hence x + a*CI(x + c)*. Clearly x + a 21>(x + c) , so that X + a = (x + c) . Suppose there exists an element e such that e > c. By Lerama 3.8, x + e is minimal. Since |_ x + c, x + ^jCZ. x + e , X + c > X + e. Thus x + eC(x + c) =x + a,so that e_< a. Since e is maximal, e = a. This shows that e = [a, ej . Now assume that there is an element f such that f > b. There exists y such that y + b=c. So [y + a, y + b = c, y + cjCIy*^Â» which implies that either | y + a, y + b = c, y + cj i ^ or ly + a, y + b = c, y + cj i ^Now c is minimal and c = |^a, cj . This means that ly + a, y + b=c, y + c^ ='?so that [a ) = ^y + a, y + b = c, y + cj . Hence af/y + a, y + b = c, y+cj , meaning a^y + a . It follows from Definition 2.9 and the equality
PAGE 22

18 such that c^ > cq and C2 > cg . There exists an element z such that z + Cj = a. By Definition 2.9 either |z + Cq, z + c .\ i :f or (z + CO, z + c^j * ^ i? (i = 1, 2). So {z + Cq, z + cj^, z + C2\Cla*. Consequently, we have z+c b in a poip-group P, then x + a^ x + b when x + b* is an ideal, and x + a_^ x + b when x + b* is a dual ideal. Assume x + b"^ is an ideal and x + a < x + b. Then x + aÂ£x + b which implies a_< b. Hence x + a^x * b.

PAGE 23

19 Suppose X + b is a dual ideal and x + a > x + b. Then X + ae X + b which means a < b. Hence x + aj^ x + b. Lemma 3. 13 ; If x + b is an ideal of a poip-group P and b is not maximal, then x + b = (x + b) . Since x + bÂ€x + b, obviously (x + b) clUx + b . So -X + (x + b) dlh . If -X + (x + b) were a dual ideal, we would have b C^-x + (x + h)'*'czh'^. But b CI b* is possible only if b is + a maximal element. Thus -x + (x + b) is an ideal. Since be-x+ (x + b) , h CZ -X + (x + b) and x + b CI (x + b) . Therefore, X + b = (x + b) . Lemma 3.14 : Let a be a non-maximal element of a poip-group P. If a > b, -x + (x + a) is a dual ideal, and x+b is an ideal, then x + a > X + b. Since a is not maximal, there is an element c > a in P. The dual ideal -x + (x + a)* contains a and hence c. So x + c f (x + a)"*" which gives x + c b in an infinite dimensional poip-group P, and a is not maximal, then x + a and x+b are comparable.

PAGE 24

20 Obviously a Â€-x + (x + a) . If -x + (x + a) is an ideal, it contains a and hence b. Thus if -x + (x + a)* is an ideal, X + b Â€ (x + a) and x + b X + b. Case ii ; Suppose x + b is maximal. If x + b"^ is an ideal, then by Lemma 3.14 x + a and x + b are comparable. Therefore, assume X + b is a dual ideal. Since P is of infinite length, it follows from (2.10) that it has no minimal element. So there must exist an element d< b. Now -x + (x + b)"^ is an ideal containing b and hence d. Thus x + d Â£(x + b)"^ and x + d < x + b. Now a > d where a and x + d are not maximal. Replacing b by d in the proof of Case i, it follows that x + a is comparable to x + d. If x + a < X + d, then x+a x + d. Since d fib"", X + d ex + b*. Then (x + d)*CZ. x + b*
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21 In Section II it was shown that an infinite dimensional poipgroup P contains no minimal elements. If we can show that such a poip-group contains no maximal elements either, we shall be able to conclude from the above lemma that comparability is preserved under addition in an infinite dimensional poip-group. Suppose a is a maximal element of a poip-group P of infinite length. Since P has no minimal elements, there exist a chain a > e > f > g and elements t,b,c eP such that t + a = f, t + b = e, and t + c = g. The above chain now becomes a>t+b>t + a>t + c. Since t + b > t + a, it follows from Lemma 3.15 that a and b are comparable. Moreover, a > b because a is maximal. By Lemma 3.12, -t + (t + a)"^ is a dual ideal (replace x by -t, a by t + b, and b by t + a). Now t + c 6 (t + a)*, so that c e-t + (t + a)"^. Thus c d-t + (t + a)* and t + c*CZ(t + a)*. * + If b ec , then t + b 6(t + a) , which is impossible since t + b> J * t + a. Kence b f c . However, by Lemma 3.15 b and c are comparable, so b < c. The elements a and c are also comparable because t + a > t + c and t + a is not maximal. Since a is maximal, a > c. From Lemma 3,12 and the relations t + a > t Â» c and a > c, it follows that -t + (t + c)* is an ideal. By Lemma 3.13 -t + (t + c)* = c*. Since b Cc , t + b e (t + c)* and t + b
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22 Theo rem 3.16 ; An infinite dimensional poip-group contains no maximal or minimal elements. Theorem 3.17 ; Comparability is preserved under addition in an infinite dimensional poip-group-that is, if a and b are comparable, then X + a + y and x + b + y are comparable for all x,y t P. Lemma 3.18 : If a > b in an infinite dimensional poip-group, then (i) X + a is an ideal if and only if x + a > x + b, (ii) x + a* is a dual ideal if and only if x + a < x + b, (iii) x + b is an ideal if and only if x + a > x + b, and ( iv) x + b"*" is a dual ideal if and only if X + a < X + b. Conditions iii and iv follow from Theorem 3.17 and Lemma 3.12. Suppose X + a is an ideal. By Lemma 3.13x+a ={x+a) . Since bÂ€a,x + bÂ€(x+a) and x + b < x + a. Now suppose that x + a"^ is a dual ideal and x + a > x + b. By Lemma 3,12 x + b is an ideal. It follows from Theorem 3.16 that there is an element c > a. Again by Lemma 3.12x + c X + b. Hence x + a>x + c>x + b. The inequalities x + a>x + c, c>a, and Lemraa 3.12 imply that -x + (x + c)* is a dual ideal. Now x + b e (x + c)"^, so that b ( -x + (x + c)"^. Then b CL'X + (x + c)*, at-x + (x + c)*, and x + at(x + c)*. This implies x+a x + b, and if x + a* is a dual ideal, x + a < x + b. These statements and their contrapositives give conditions i and ii.

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23 An immediate consequence of this lemma is the following result. Lemma 3.19 : Let a and b be comparable elements of an infinite dimensional poip-group. Then x + a is an ideal if and only if X + b* is an ideal, and x + a* is a dual ideal if and only if X + b is a dual ideal. Lemma 3.20 : An infinite dimensional poip-group P is (i) a po-set, (ii) a group, in which (iii) for each weak component C of P and each pair of elements x,y eP either x + a + y_> x + b + y for all pairs a,bÂ€:C such that a_> borx+ a + y b. Conversely, any system P satisfying conditions i-iii is a dualistic pxjip-group. Let X be any element and C any weak component of P. First suppose that x + Cq is an ideal for some cq 6 C. For any other c Â€C, there exists a sequence Cq = ^1.^2 ^n = = + such that x^ and x.^j^ are comparable. If x + x. is an ideal, by Lemma 3.19 x + x,^, must be an ideal. Hence x + c is an ideal. Similarly, if x + Cq is a dual ideal for some c^ Â€ C, x + c is a dual ideal for all c fcC. That ? satisfies condition iii now follows from Lemma 3.18 and the extension of the above argument to the case in which an arbitrary element y is added on the right of elements of C.

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24 This lemma and Lemma 3.5 enable us to conclude the following result which we noted before was necessary to complete the proof of Theorem 3.10. Theorem 3.21 : Every infinite dimensional poip-group is dualistic. Theorem 3.22 ; A dualistic poip-group P is (i) a po-set, (ii) a group, in vd\ich (iii) for each weak component C of P and each pair of elements x,y Â€ P either x + a + y_> x + b + y for all pairs a.beC such that a_> b or x + a + y< x + b y for all pairs a.bcC such that a_> b. Conversely, any system P satisfying conditions i-iii is a dualistic poip-group. The latter statement of the theorem is merely Lemma 3.5. A dualistic poip-group P obviously satisfies conditions i and ii. By Lemma 3.20, if it is of infinite dimension, it also satisfies condition iii. Hence it remains to be shown that any dualistic poip-group of dimension one satisfies condition iii. So assume that P has length one. Consider elements a,b Â€ P such that a > b and suppose x + a and x + b are not comparable for some X tP. Now [x + a, x + bj d x + a , so that either {x + a, X + b] * ^Â« (? or [x + a, x + b} "^ ^ (?. Since x + a and x + b are not comparable, there exists an element which is either properly over them or is properly under them. Hence x + a and x + b are either both maximal or both minimal. By Lemma 3.8 and its dual, this is impossible. Therefore x + a and x + b are comparable.

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25 In the proof of Lemma 3.9 it was shown that if P is a poipgroup of length one in which a > b implies x + a + y and x + b + y are comparable for all x,yt?, then P satisfies condition iii. This completes the proof. Since non-dualistic poip-groups were described completely in Theorem 3.10, only dualistic ones will be considered in the sequel. Also, the above theorem provides us with a more useful definition of a dualistic poip-group, which will be employed in the following sections.

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IV. CONNECTIVITY The immediately preceding theorem suggests the following definitions, which will lead to an even finer partitioning of a dualistic poip-group than that determined by the weak components. Definition 4.1 ; An element x is said to be order preserving on the left relative to the weakly connected set C^ if x + a_> x + b for all pairs a > b in C. It is said to be order reversing on the left relative toCifx + a b in C. By Theorem 3.22 any element of P is either order preserving or order reversing on the left relative to any given weak component. Henceforth there will be little occasion to consider ideals. However, it should be noted that if J is an ideal contained in the weak component C and if x is order preserving (reversing) on the left relative to C, then x + J is an ideal (dual ideal). Also, if F is a non-void subset of the weak component C, then x + F = (x + F) + * or X + F = (x + F) depending on vdiether x is order preserving or order reversing on the left relative to C. These results are contained in the proof of Lemma 3.5. Definition 4.2 ; An element x is of order preserving type 1 relative to the weakly connected set C if it is order preserving on the left and on the right relative to C. It is of order preserving type 2_ relative to C if it is order preserving on the left and order reversing on the right relative to C. It is of order preserving 26

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27 type _3 relative to C_ if it is order reversing on the left and order preserving on the right relative to C. It is of order preserving type 4 relative to C if it is order reversing on the left and on the' right relative to C. An element of order preserving type 1 (type 2) relative to C^ and an element of order preserving type 4 (type 3) relative to C^ are said to be of opposite order preserving types relative to Cg. The set of all elements of C^ which are of order preserving type i relative to C^ will be denoted by CÂ„(C, ) (i = 1, 2, 3, 4). i j We note that C^(.C^) and C^(Cg), i ^ j, are disjoint. Any weakly connected set partitions P into from one to four classes of element types. If it is commutative, all elements of P are either of order preserving type 1 or order preserving type 4 relative to any given weakly connected set. Example 4.3 ; Let P be generated as a group by a and x under the conditions x + x = and x + a = -a + x. '.vlien it is ordered as shown in Figure 4.4, P is a noncommutative dualistic poip-group. It can be verified that a is in Cq (CÂ„) and Cq (C ) and that x is a 2 2 member of Cx (Cg) and Cx (C^^) Â• 1 2a

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28 Lemma 4.5 : If x is order preserving (reversing) on the left relative to C^, -x is order preserving (reversing) on the left relative to C^+aLet a^^ and a2 be elements of C^ such that aj > ^2 ^"^"^ suppose that X is order preserving on the left relative to C^, so that X > 3]^ > X + ag. The element -x is order preserving on the left relative to C^+a because -x + (x + aj^) = a2^_> a2 = -x + (x + 82 ) . The remaining case can be proved in a similar manner. Theorem 4.6 ; Cg = C^ (Cq)UCq^ (Cq). For any element y Â€ Cg there exists a sequence = X]^, Xg, . . . , Xj^ = y of distinct elements of Cq such that x. is comparable to Xj^n. 1 Clearly 6CQ(Cg), so to complete the proof it is necessary to show only that xi (Cq[Cq)UCq{Cq) implies x^^.;^ 6 Co(Go)^Co(Co) . 12 1 Assume Xi6Cg(Cg) and x^^j^ f Cq(Co) . By Lemma 4.5 -x-f Cq(Cq) 2 and 'X^^i eCQ(CQ). Case i: Suppose Xj^ > Xj^+i. It follows that Xj^ xj+]^ < 0. However, it is also true that > Xj^-^ x^ whence -x^+j^ > -x^ and x^ " ^i+1 > 0Â» a contradiction. Case ii ; Suppose x^ < x^+j^. Obviously x^ x^+j^ > 0. But < x^^j^ x^, so that -Xj+2^ < -xand Xj x^^j^ < 0, which is impossible. 1 2 This proves that x^ eCg{Cg) implies Xj^+j^ ^Cq(Cq) . Continuing 1 / / 4 with this procedure it can be shown that x. 6 CQ(CQ)l7Cg(Cg) implies

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29 2ll3 1 I I A that x^^-^(Cq{Cq)\JCq{Cq) and, therefore, that x^^.j^ 6 Cq(Cq)C/Cq(Cq) . The theorem follows. Theorem 4.7: For all e 6 P, c];(C_) = 0^(0 ) and '00 e CqCCq) = Co(C^). Let 6]^, Â©2 ^ '-'e ^ such that ei > e2 and consider an element Case _i: Suppose e, is order preserving on the right relative to Cq. By Lemma 4.5 -e^ is order preserving on the right relative to C so that > e, e,. Therefore, s > s + (e^ e, ) = (s + eg) -e, and s + e-^> s * e2' Case ij ! Suppose e^ is order reversing on the right relative to Cq. By Lemma 4,5 0< e^ e, . Thus s< s + (e^ e-i) = ^^ * Â®2^ ~ Â®1 and s + ej^ > X + eg . These arguments show that s is order preserving on the left 1 relative to Cg . By left-right symmetry, stCgCCg). 4 4 Similarly, s(Cq(Cq) implies s^CqCC ), enabling us to conclude that cJ(Cg)ClcJ(C^) and C^(Cq)CZCq[C^) . However, by Theorem 4.6 Cg = cJ(Cg)L/Cg(CQ) . Since cJ(Cg)/lCg(Cg) = (?, C^^O^ =^l^V Â«ndc5(Co) =cJ(C^). As a result of this theorem, no ambiguity will arise if 14 14 C-(C ) and Cg(C ) are denoted by Cand C^, respectively. 1 4 Theorem 4,8 : The set Cg is a normal subgroup of P, and Cg either is empty or is one of its cosets. For any e Â€P,

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30 Cg = (e + cj) U (e + Cq). Furthermore, for any f 6 P, e + cj = C^iC^) 4 j and e + Cg = Cg(Cf) where i is the order preserving type of e relative to Cf and j is the opposite order preserving type relative to C^. Obviously Cg = e + Cq = e + (Cq UCq) = (e + cj) U (e + Cq) and Cg = Cq + e = (cj UCq) + e = (cj + e) (J (cj + e). Now let f;^ and f 2 be elements of Cf such that f i > f2 and let e + s be any member of e + Cq. Since s ^ cj = C^(C^), s + f-j^ > s + fg. Clearly s + f;i^ Â€ Cq + fi = C^, so that (e + s) + f ^ > (e + s) + fg if e is order preserving on the left relative to Cf, and (e + s) + f j^ < (e + s) + f2 if e is order reversing on the left relative to Cr. In other words, e + s is order preserving (reversing) on the left relative to C^ when e is order preserving (reversing) on the left relative to C^. Similarly, it is order preserving (reversing) on the right relative to C, when e is order preserving (reversing) on the right relative to C^. This shows that if i is the order preserving type of e relative to Cf, then e + CoCi:Ce(CfK By left-right symmetry, Cq + eClCg(Cf). It may be proved in much the same manner that if i is the order preserving type of e relative to C^, then e + Cq ClCg(Cf ) and Cq + eCIC^(C^) where i and j are of opposite order preserving types relative to C^. Since (e + Cj) (J (e + C^) = C^ and Ci(Cj) fl c|(C^) = (p. it is true that e + cj = C^(Cf) and e + Cq = C^(Cf:).

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31 Likewise ci + e = cMc.) and ci + e = C^(C,). It follows that u e r u e r 11 4 4 e * Cq = Cq + e and e + Gq = Cq + e. If s and t are any members of Cq, then s + tÂ£s + CQ = C^{C_) = CL By Lerniaa 4.5 s Â€ C^ implies -s Â€ C^. Hence C^ is a normal subgroup of P. If Ci is not empty, there exists an r^ci. Then r + Cg = r + C1(C ) = C4(C_), so that C^ is a coset of cL This completes the proof. Corollary 4.9 ; For any weak component C and any f Â€ P, either C = G^(C^)Uc*(C^) or C = c2(C^)(jc3( C^). The decomposition of P into the cosets of Cg yields little information concerning the structure of P, but it motivates the investigation of another decomposition determined by the maximal weakly connected set in Cg. Definition 4.10 : The element a_ _is strongly connected to b if and only if there exists a sequence a = x-j^, X2, . . . , x^ = b such that X. (C^(Cg) for some j and i = 1, 2, . . . , n and such that x^ is comparable to Xj+i for i = 1, 2, . . . , n 1. Obviously if two elements are strongly connected, they are weakly connected. Note that strong connectivity is an equivalence relation on a poip-group but does not necessarily have meaning for a po-set per se.

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32 Definition 4.11 ; A set S is strongly connected if and only if every two elements of S are strongly connected. A maximal strongly connected set is called a strong component of _P, and Sg will denote the strong component containing a. Any strongly connected set consists of elements of only one order preserving type relative to any weakly connected set. Obviously SqCICqCUCq. Lemma 4.12 ; If a is strongly connected to b, then x + a is strongly connected to x + b. There exists a sequence a = x^, x^, . . . , x^ = h of elements of S such that x, is comparable to x.^^. Now Â£^CICJ(Cq) for some j so that x + S^Clx + C^^Cq) = C^+a(Cg) for some k. In particular, the members of the sequence x + a = X + Xj^, X + X2, . . . , X + Xj^ = x + b are contained in C^+qC^q). Furthermore, by Theorem 3.22 x + x. is comparable to x + ^c.^-, . Hence x + a is strongly connected to x + b. Let a and b be elements of Â£g. Then a and b are strongly connected to 0. By Lemma 4.12 a + b is strongly connected to a + = a, so that by transitivity it is strongly connected to and is therefore in S . Again by Lemma 4.12 -a + a = is strongly connected to -a + = -a whence -afSQ. Thus Â£q is a subgroup of P.

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33 The set a + SÂ„ is contained in i: because a ^ E^ and all eleU a a ments of a + S^^ are strongly connected to a according to Lertima 4.12. Likewise, -a + Sg is contained in EQ--that is, Sg is contained in a + Sand, therefore, a + S. = S . Similarly, S_ + a = S so that ' ' a a S^ is a normal subgroup of ?, proving the following theorem. Theo rem 4.13 : S is a normal subgroup of P, and the strong components of P are the cosets of Sg. In the next section the strong components are used to characterize the order structure of dualistic poip-groups. This section is concluded with the following example illustrating the relationships among the cosets of Â£q, CQ(Cg), and Cg. Example 4.14 : Let P be the poip-group composed of the additive group of integers ordered as shown in Figure 4.15. Any block of integers in the diagram is a class of integers modulo 12. In this poip-group SgUS^USg = Cj(Cg), Â£2USgUSj^Q= Cg(Cg), S^USyUSiicJ(Co), s^UsjUSg = c|(Gg), cJ(Co)UcJ(Co) = Cg, Cj(Cg)L/C^(Cg) = C^.

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34 1

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V. A DECOMPOSITION THEOREM In this section it will be shovm that a dualistic poip-group is order isomorphic to the ordinal product of a certain finite dimensional poip-group and a po-group. Lemma 5.1 : If S^CH^i^^o^ ^"^ -y^cJ^^Q), where i f j, and if xi > Yi for some x^" in Sj^ and some y^ in Sy, then x* > y' for every x' in S and every y' in S . A y Consider the case in which i = 1 or 2 and j = 3 or 4. Let xÂ„ be a member of S vAiich is comparable to x, . If xÂ„_> x^ , then X2 > yj^. So assume that ^2 < xj^. Case i; Suppose Xj^^C^lCg). Then -x-^ is order preserving on the right relative to C^, and x-^ > y-^ implies that > y^^ x-^. Now x^^-C^iC^) , so that Xg > (y-i x,) + Xj. On the other hand, x-^> X2 and the fact that -x-j^ is order preserving on the left relative to C imply that 0_> -Xi + X2. Since y^ is order reversingon the left relative to Cq, we have Yi^ (Yt x-,) + X2 and, therefore, Yi < X2. 2 Case ii : Suppose X]^eCjj(Go). Then -xj^ is order reversing on the right relative to C^^, and x-^ > y^ implies that < y-^ x-^. 2 Now X2 Â€Cjj(Cq), so that X2 > (y^ x-^) + X2. However, xj^_> X2 and 2 -Xj^C^CC^) give >_ -Xi + X2. Now since y is order reversing on 35

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36 the left relative to Cq, it is true that Yi <_ ivi X2) + X2 and hence that y^^ < X2. The above argument proves that every element of S^^ which is comparable to Xi is greater than y,. It follows that any x' in S , being strongly connected to x-^, is also greater than y^^. Using a similar procedure it can be shown that any y' in S is less than x^^. So each y' (_ S is under x, and hence under every x' Â€ S . This y i X establishes the cases in which i = 1 or 2 and j = 3 or 4. The dual argument establishes the remaining cases. Definition 5.2 : Order the elements of P/Sg as follows: S^ >^ S if and only if Â£ = S or every x' in Â£ is greater than every y' in Sy. This ordering will be called the natural ordering of P/Sq . The natural ordering of P/Sq is obviously a partial ordering, and henceforth the symbol ?/Â£q will represent the group P/Sg ordered in this manner. Theorem 5.3 : The mapping h given by h(x) = S^ is a group horaoraorphism of P onto P/Â£q such that x_> y implies h(x) >_ h(y) and h(x) > h(y) implies x > y. A set C is a weak component of P if and only if h(C) is a weak component of P/Â£q. Furthermore, when some weak component of P/Sg contains at least two distinct elements, every weak component of P/Sg contains at least two elements; for any a,xÂ€ P X is of order preserving type i relative to C if and only if h(x) is of order preserving type i relative to h(C^) (i = 1, 2, 3, 4).

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37 Clearly h is a group horaoraorphisra. Consider elements x^ y i" "^x "^ ^x^^O^ ^ <^x^^0^^^ ^ ^"^ y ^^Â® ^Â°^^ elements of cJ^{Co) or both elements of C^(Go), then they are members of the same strong component i^, so that B^ = Sy. However, if x CC^lCo) and yeCxCCg), where i t^ j , then according to the preceding lemma every element of Sjj is over every element of C , so that S^ > 3 . This shows that X > y implies h(x) >_ h(y) . Suppose now that h(x) = S^ > S^ = h(y). Then S^^ i Sy. By Definition 5.2 every x' in S^^ is greater than every y' in Sy, so that in particular x > y. It follows from the isotone property of h (that is, the property that x_> y implies h(x) > h(y) ) that if C is a weak component of P, then h(C) is a weak component of P/Sg. Conversely, since h(x) = S^_> S = h(y) implies that x is connected to y, it is clear that if h(C) is a weak component of P/Sg, then C must be a weak component of P. By Theorem 4.8 if some weak component of P/Sg contains at least two distinct elements, then all of them contain at least two elements. Suppose h(a) > h(b) . V/hen h(x) + h(a) > h{x) + h(b), it is true that h(x + a) > h (x + b) and x + a > x + b. On the other hand, h(x) + h(a) < h(x) + h(b) implies h(x + a) < h(x < h) and X + a < X + b. Thus when h(x) is of order preserving type i relative to a weak component h(C ) having at least two elements, xfC^^CCg) Similarly, if x^C^CC^), then h(x) is of order preserving type i relative to hCC^).

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38 Corollary 5.4 : The system P/Sg is a dualistic poip-group of finite dimension. If each weak component of P/Sq consists of merely a single element, P/Sq is totally unordered and is obviously a dualistic poipgroup. Otherwise, the result follows from Theorems 3.22 and 5.3. Suppose that S^ > Ey > S^, in P/Sq. Then by Theorem 5.3, X > y > z. However, x, y, and z are all in C^^ = C^{Cq) U C^ICq), so that at least two of them are members of either C^(Co) or C^{Cq). These two elements are thus strongly connected--meaning that they are in the same strong component. This is impossible since ^x '^ ^y ^ ^z ^ ^x' ^^"^^s there is no chain of length greater than one in P/Sq* Corollary 5.5 : Let x and y be weakly connected elements of P. If P/Sg is of length one, x and y are of the same order preserving type relative to any given weak component of P if and only if S^ and S are both maximal or both minimal elements of P/Sg. This corollary follows from the observation that when Sjj f Sy , then S^ is comparable to Â£y only if x and y are of different order preserving types relative to any given weak component of P and the fact that any weak component of ? contains elements of only two order preserving types relative to any other weak component. Corollary 5.6 : If P/Sg is of length one, the elements of E^ have the same order preserving properties in P as S has in P/Sg--

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39 that is, X is of order preserving type i relative to C if and only if S is of order preserving type i relative to h(C). Corollary 5.7 : If P/Sq is commutative and of length one, then C^ = ci(n ) U cJ(C^) for all a, x 6 P. A X d X d Theorem 5.8 ; The strong component Eg is a self-dual pogroup under the relativized ordering. Furthermore, every coset of Sq is order isomorphic to S g , and if a is any element of the coset S of Sg, such an isomorphism may be described as follows: (i) if a is order preserving on the left relative to Cg, then bÂ€S corresponds to -a + bCEg, and (ii) if a is order reversing on the left relative to Cg, then bÂ€Â£ corresponds to -b + agSg. Obviously Â£q is a po-set under the relativized ordering. Moreover, SgCICg(Cg), so that every element of Sg is order preserving on the left and right relative to Sg. Hence Â£g is a po-group. To show that Sg is self-dual, let x correspond to -x for all xÂ€Sq. If y > z in Sg, then > z y, so that -z> -y Â— that is, the correspondence is an antiisomorphism (in the order sense) and Sg is self-dual. Now let S be some coset of Sg and a some element in S, Suppose that the members of S = a + Sg are order preserving on the left relative to Cg and let bÂ€S correspond to x = -a + bÂ€Sg (denote this by b < > x) . The correspondence is clearly one to one. If y and z are members of Sg such that y_> z where c < > y

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40 and d< > z, then -a + c = y_> z = -a + d and, since a is order preserving on the left relative to Cg, c_> d. Now if c > d in Lq, where c < ^ y and d < > z, then y = -a + c_>-a + d=z since -a is order preserving on the left relative to Cg = C^,. Therefore, the correspondence is an order isomorphism. If the elements of Â£ are order reversing on the left relative to Â£q, a procedure similar to that above shows that Â£g is antiisomorphic to Â£ under the correspondence which carries b6Â£ into -a + b6Â£g. Since Â£q is ant iisomorphic to itself under the correspondence which carries every element of Sg into its inverse, this indicates that Â£ is order isomorphic to Sg under the correspondence which carries bÂ£E into -(-a + b) = -b + aÂ£Â£^. Theorem 5.9 : A dualistic poip-group P is order isomorphic to the ordinal product {P/Â£g)Â»Â£g of the finite dimensional dualistic poip-group P/Â£g and the po-group Sg. If A is a set of coset representatives, then the isomorphism may be described as follows: (i) if b Â€ P is order preserving on the left relative to Sg, then b corresponds to (Sj^, -a + b) where a is in Sj^D A, and (ii) if b^P is order reversing on the left relative to Sg, then b corresponds to {S)^, -b + a) where a is in S^, D A. When b6P and (Sj^, y) correspond in the manner described in (i) and (ii), denote this correspondence by b< ^> (Sj^, x). Suppose b< ^> (Sj-,, y) and c< ^> (Â£b, z). Then b and c are both members of Â£^ . It follows from Theorem 5.8 that b > c if

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41 and only if y > z. Hence b> c if and only if {L.-^, y) >_ {%, z) since (according to the definition of the ordinal product of two po-sets) when u = V, (u, y) > (v, z) if and only if y_> z. Suppose now that b< > {Â£},, y) and c< Â•> (S^, z), where S^ ^ t^. Then b ^ c. If b > c, it is implied by Definition 5.2 that S^> S^. Hence if b > c, {l^, y) > (Â£^, z). When {S^, y) > (Â£^, z) , it follows that %,> %Â• Thus by Definition 5.2, (S^, y) > (Â£(,, z) implies that b > c. Therefore, b > c if and only if (Sj^, y) > i^^Q, z), completing the proof. The above theorem is illustrated by Example 4.14 in which P/Â£g is composed of the integers modulo 12. When it is ordered in the natural manner, P/Sg may be represented by the diagram below. The elements of the figure are, of course, added modulo 12. 4 8 3 7 11 10 Fig. 5.10 It is easily verified that when P is of finite length, ^0 ~ 1^3' This enables us to give the following definition. Definition 5.11 : A poip-group P is said to be upright if P/Sg is of length one and tg is maximal in P/Â£g. It is said to be inverted if P/Â£g is of length one and Sg is minimal in ?/Sg. The dual of any theorem concerning upright poip-groups holds for inverted poip-groups.

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42 Theo rem 5.12 ; Let P be a dualistic poip-group, P/Sq be of length one, and S^ be a maximal (minimal) element of ?/Sg. Then x is order preserving on the left relative to C^ if and only if x + S^ is maximal (minimal), and x is order reversing on the left relative to Cg if and only if x + S^ is minimal (maximal). As before, let h be the mapping defined by h(x) = S^. Consider P/Eg = h(P) where Â£g = h(a) is maximal (minimal). Then S^ = h(x) ^ P/Cg is order preserving (reversing) on the left relative to h(Ca) if and only if ^x+a ~ ^^^ "^ ^^ ^^ maximal (minimal), and it is order reversing on the left relative to h(Cg) if and only if x + a is minimal (maximal). The theorem now follows from Corollary 5.6. Corollary 5.13 ; Let P be an upright dualistic poip-group such that P/S is of length one. An x6P is of order preserving type 1 (order preserving type 4) relative to Cg if and only if t-^ is a maximal (minimal) element of P/Sg. Many of the results of this section have been concerned only with the case in which P/Sq is of length one. When it is of length zero, P has a simpler form since then Sj^ = Cj^ for all xÂ€'P. This means that the elements of any weak component of P are all of the same order preserving type relative to any other weak component of P. Furthermore, every weak component is order isomorphic to the pogroup Cq.

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43 It was shown in this section that when P/Sq is of length one, many of its properties are carried over to P. Also, the order structure of P can be determined if the order structure of Cq and P/Sg are known. Since Sg is a po-group we need characterize only finite dimensional poip-groups.

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VI. FINITE DII-ENSIONAL POIP-GSOUPS The only finite dimensional poip-groups which need to be given further consideration are the dualistic poip-groups of length one. These systems are characterized below. 6a: ;>fTien a is a maximal element of a poip-group of length one, Tg will represent the set [xÂ€P / x< a] , and when a is minimal, it will denote the set [xÂ€P / x > al . Theorem 6.2 ; An upright dualistic poip-group P satisfies the following conditions: (i) if teT^, then -tCT^^; (ii) for any element s of s' + Cg, every element of s' + Cq can be written in the form s t^ . t2 . . . t^, t-fTg (i = 1, 2 n), where the elements t^ are not necessarily distinct; (iii) if ^1 * ^2 * Â• Â• Â• * *n = Â°' ^i^^O (i = 1' 2 n), then n is an even integer; (iv) a + Tq = Tq + a for all aÂ€P; (v) a + Tq = T for all aÂ€P; and (vi) if s is maximal, then s + t, + . . . + t 1 n' tj^eTg (i = 1, 2, . . . , n), is maximal (minimal) if and only if n is even (odd). Conversely, let G be a group and Cq a normal subgroup of G which contains a non-void subset Tq satisfying conditions i-iv. Define an ordering on 6 using conditions v and vi, first assigning some representative element sCs' + C (let 44

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45 be the representative element of Cg) the role of a maximal element. Under this ordering G is an upright dualistic poip-group in which Cg is the set of all elements weakly connected to and !Â„ is the set of elements of G which are less than 0. Let P be an upright dualistic poip-group. If t < 0, then -t + t = is comparable to -t, so that, since is maximal, -t < 0. This establishes condition i. If b Â€T , by (6.1) b is comparable to a. Hence -a + b < and -a + b = t for some tCT . Consequently, b = a + t, so that b is in a + Tg, proving that T da + Tg. Now let t be a member of Tg, whence t < 0. Therefore a + t is comparable to a but a + t f a-that is, a + t is in T^. This implies that T^IDa + Tg and thus Tg = a + Tq. Similarly, T^ = Tg + a, proving conditions iv and v. Next suppose that a is an element of Cg not of the form t-i + tg + . . . + t , t.Â£Tg. There exists a sequence = X]^, X2, . . . , Xj^ = a such that Xj^ is comparable to Xj^+]^. Assume that k is the minimum integer such that Xj, is not of the form t-]^ + tg "*Â•Â•. + tj^, t, f T-. Then x, , = t, + tÂ„ + . . . + t for t . Â€ TÂ„ and x, , is 1 k-1 12 n 1 k-1 comparable to x,. By condition v^ x, = Xr^_-, + t for some tÂ€Tg, so that Xj, = ti + tg + Â• . . + t + t, a contradiction. Hence every element of s + C^ can be expressed as in condition ii.

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46 Suppose a = s + tj^ + t2 Â•...+ t where s is maximal and t.^Tpj, Now t][< means s + t, is comparable to s. Since s is maximal, s + t^ < s, s + tj^ + tg > s + t-]^, ^ ' h " "^2 " *3 < = " *1 " *2' ^ " h " ^2 * ^3 ' ^4 > ^ ' H * ^2 * *3' and 80 on. Thus s + t-j^ + t2 + . . . + tj^ > s + t^ + t2 + . . . + t i if n is even and s + t, + to + . . . + t_ < s + t, + t^ + . . . -^ t , 1 z n 1 z n-i if n is odd. Hence a is maximal (minimal) if and only if n is even (odd), establishing condition vi from which condition iii follows immediately. Conversely, let G be a group containing a normal subgroup Cq which has a subset Tq satisfying conditions i-iv, Suppose that s + t-j^ + tg + . . . + tj^ = s + t|'+ tg' + . . . + t^' where *i' ^j' Â€^0 (i = 1Â» 2, . . . , m; j = 1, 2, . . . , n). Then t-j^ + tg + . . . + tjjj tj^' t^_-^' ... t]^' = 0. This equation, along with conditions i and iii, implies that m + n is even Â— that is, that either m and n are both even or both odd. Thus conditions v and vi establish a well-defined partial ordering for G. Under this ordering it is clear that G is a po-set of length one with no isolated elements.

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47 Let t-| + to . . . + t , t.Â€Tg, be an element of Cg. By (i) and (v) , t^ + tÂ„ + . . . + t is comparable to c + t, + tÂ„ + . . . + t , ; t, + tÂ„ + . . . + t , is comparable to 1 I n-i 1 I n-i c + t, + tr, + . . . + t Â„i t, + to + . . . + t Â„ is comparable to iii n-z'iz n-^ c+t, +to+...+t Â„. Continuing in this manner, it follows that ti + to + . . . + t is weakly connected to 0. Similarly, any element weakly connected to is of the form ti + t2 + . . . + t , t 6 Tq. Hence C^ is a weak component of G. Let X be some element of G. If a > b, then x + a is comparable to X + b since a = b + t for some t Â€ Tq implies x + a = (x + b) + t. The remainder of the proof is that of Theorem 3.9. Example 6.3 : Let G be a group which contains a commutative normal subgroup Cq with basis elements q-^i <3'2Â» Â• Â• Â»
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48 As a particular example, consider a group G in which Cq is generated by a and b under the conditions a + b = b + a, 2a = 0, and 4b = 0. Let Tq = {a, h, 3bj . Then when Cq is ordered according to conditions v and vi of Theorem 6,2, Cq becomes the poip-group represented by the following diagram. a + b a Â»Â• 3b 2b Fig. 6.4 Assign s + a C s Â•Â• Cq , sdCr,, the role of a maximal element of G and order s + C according to Theorem 6.2. Then s + C is as indicated below. s + a s + b s + 3b s + (a + 2b) s + (a + b) s + (a + 3b) 6 + 2b Fig. 6.5

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VII. THE ALGEBR.'IIC Cftf.RACTER OF POIP-GROUPS It has been shown that many of the properties of P, in particular its structure as a po-set, can be determined when Sq and P/Sg are known. In this section more is said about the relationships between the latter systems and the algebraic nature of P. Theorem 7.1 ; Let G be a group satisfying (i) G contains a normal subgroup S ordered so as to form a connected po-group, such that (ii) G/S is ordered so as to form an upright dualistic poip* Â» group of length one, and (iii) a + =0 + a for all aÂ€G. Then G can be ordered in one and only one way to form a dualistic poipgroup such that the po-groups Eq and S coincide and the poip-groups P/Sq and G/S coincide. This ordering is described as follows: (1) if a + S ^f b + S, then a > b if and only ifa + Â£>b+Â£in G/Â£, (2) if a + Â£ = b + Â£ is maximal in G/Â£, a_> b if and only if a b (0 , and (3) if a + Â£ = b + Â£ is minimal in G/Â£, a > b if and only if b a Â€0 . The ordering defined on G is obviously a reflexive and antisymmetric relation. In order to show that it is transitive, consider a > b and b > c in G. Case _i : Suppose a + Â£ = b+S = c + Sis maximal in G/S. Then a b > and b c > 0. Â£ince Â£ is a po-group, we have a c (a b) + (b c) > (a b) + = a b > 0, implying that a c C and a > c. 49

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50 Case ii ; Suppose a+S=b+S=c+Sis minimal in G/Â£. Then b a > and c b > 0. Therefore, c a == (c b) + (b a) > (c b) + = c b > 0, implying c a 60* and a > c. Case iii : Suppose a+S=b+Â£is maximal and c + S is minimal in G/Â£. Then a+S = b+S>c + S, proving that a > c. Case iv ; Suppose a + S is maximal and b+S=c+Sis minimal in G/Â£. Then a + S>b+S = c + S, giving a > c. Therefore, G is a po-set, and we want to show that under this ordering G is a dualistic poip-group. First observe that if x yÂ€0 , then -y6-x + =0 -x, Â« implying that -y + xCO . Now consider any element xÂ€G and a > b in a weak component C, Obviously a + Â£ > b + S. Suppose X + S is of order preserving type 1 relative to the weak component of G/S which contains a + S. It will be shown that X itself preserves order when added to the left and to the right of elements of C. Case i: Suppose a and b are ordered according to (1). Then a + S > b + S and (x + a) + Â£ = (x + S) + (a + Â£) > (x + Â£) + (b + S) (x + b) + S. Hence x Â•Â• a > x + b. Similarly, a + x > b x. Case ii : Suppose a and b are ordered according to (2). This means a b60 and a + S = b+Â£is maximal in G/S. Then -(x + b) + (x + a) = (-b x) + (x + a) = -b + a CO . Therefore,

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51 (x + a) (x + b) CO . Since x + S is order preserving on the left relative to the weak cxamponent of G/Â£ containing a + S, and a + S is maximal, then (x + a) + S = (x + S) + (a + Â£) must be maximal. Thus X + a > x + b. On the other hand, (a+x)-(b+x)= {a x) + (-X b) = a bfO . However, (a + x) + S is maximal in G/S, so that a + X > b + X. Case iii ! Suppose a and b are ordered according to (3). The proof that x + a > x + b and a + x>b + xis sinilar to that of the preceding case. Continuing in this manner, it can be shown that the order preserving properties of x 6 G are the same as those of x + S in G/S. It follows from Theorem 3.22 that G under the ordering described by (l)-(3) is a dualistic poip-group P. It is easily shown that Sq and P/Sg coincide with S and G/S, respectively. Now consider an upright dualistic poip-group P. Obviously it satisfies (1). If a > b, where a+S=b+Sis maximal in P/Sq, then bCC^(Cg) and -bÂ€C_J(Cj^). Hence a b > and a b CO*. Â» liareover, a bÂ€ implies a b > 0. Thus if b + S is maximal, a bÂ€0 if and only if a_>^b. We have proved that P satisfies (2), Similarly, it can be shown that P satisfies (3). This establishes the uniqueness of the ordering of G described by (l)-(3). Theorem 7.2 : Let G be a group such that (i) G contains a normal subgroup S ordered so as to form a connected po-group, and (ii) for each coset S' of S either a + =0 + a for every a 6 S'

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52 ora+0 =0 +a for every a Â€ B' . Then G can be ordered to form a dualistic poip-group P such that the po-groups tig and S coincide and P/Sq is totally unordered. All such orderings are effected by leaving elements not in the same coset of S unordered and ordering each coset Â£' f S in either of the follo'- b if and only if a bÂ€0*, or (2) for all a.bCS', a > b if and only if b a Â£ . The proof that G is a po-set under any one of these orderings is similar to that in the preceding theorem. V.'e now show that G is a dualistic poip-group under any one of them. Suppose a + =0 + a and consider a + s where sCO . There exists t (G such that a + s = t + a. Then a s = -t + a and, since -s^O , -tCO*. It follows that tf 0* and a + 0"*"CZO* + a. + * + Likewise, a + ID + a, so that a + 3 =0 + a. A similar argument will show that a + =0 + a Lmplies a + 0* = 0* + a. * Suppose a> b according to (1). Then a bÂ€0 . Clearly # afO + b and a + xÂ€0 +(b+x) for any x. This means (a + x) (b + x)Â€0 , which implies that x is either order preserving or order reversing on the right relative to any coset of S. Case i: Suppose b + =0 + b. Thert aÂ€0 +b = b+0 and X + aÂ€" (x + b) + 0*. If (x + b) + 0* = 0* + (x + b), then (x a) (x + b)Â€0*. If (x + b) + 0* = O" + (x + b), then (x + a) (x > b)Â€0* and (x + b) (x + a)Â€0*.

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53 Case jLi: Suppose b + 0* = O"^ + b. Then aÂ€0 +b = b+0'^. Clearly x + a f (x + b) + O"*" . If (x + b) + O"^ = + (x + b), then (x + a) (x + b)eo*. If (x + b) + 0* = 0* + (x * b), then + * (x + a) (x + b) f whence (x + b) (x + a) CO . The proofs of the above cases imply that x is either order preserving or order reversing on the left relative to any coset of Â£. The case in which a > b according to (2) can be proved in a similar manner. By Theorem 3.22, G is a dualistic poip-group under any one of the orderings described in (1) and (2). We now state an important corollary of the above theorems. Corollary 7.3 : Necessary and sufficient conditions that a group G can be ordered to form an infinite dimensional poip-group are: (1) G contains a normal subgroup Â£ ^ \^J ''^hich can be ordered so that it forms a weakly connected po-group, such that (ii) for Â» each coset 3*^5, either a + =0 + a for every a a, so that Â» + -a + b> and b a <_ 0. Hence bÂ€a + and bÂ€0 + a, implying that a CT a + and a C 0^ + a. If c Â€ and dÂ€0 , then c_> and d<_0, so that a + c> a and d + a> a. Thus a + cf a and Â» Â» Â« + d + afa , giving a = a + and a =0 + a.

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54 Similarly, aecl(CQ), aec^CCg), and afc4(CQ) imply Â»Â» Â« + + + a + =a =0 +a, a + =a =0 +a, and a + =a =0 +a, respectively. Since SgCC^tCQ) for some i, the result follows. The following theorem occurs in (4, p. 214). Theoreni 7.4 : Any po-group S is determined to within isomorphism by the set i? = , since a_> b, a bÂ€i?, and -b + aÂ€R are equivalent conditions. liareover, (i) OCR, (ii) if a,bÂ€R, then a + bCH, (iii) if a,bÂ€"S and a + b = 0, then a = b = 0, (iv) for all af S, a + R = S + a. Conversely, if S is any group, and R is a subset of S satisfying {l)-(iv), then S can be ordered to form a po-group by defining a_> b in S to mean a bÂ£R. It is obvious that a po-group S is weakly connected if and only if, for every pair a,bf S, there exists a sequence a = x-j^, Xg, . . . , Xj^ = b of elements of S such that either x^ x^^+j^^R or x^+j^ x^fR (i = 1, 2, . . . , n 1). The previous corollary can now be stated in purely algebraic terms. Theorem 7.5 ; Necessary and sufficient conditions that a group G can be ordered to form an infinite dimensional dualistic poip-group are: (i) G has a normal subgroup S =)< { j which contains a subset R satisfying (ii) OCR; (iii) if a,bCR, then a + bfR; (iv) if a,bCR and a + b = 0, then a = b = 0; (v) for all aCS,

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55 a + R=R+a; (vi) for every pair a,bÂ€Â£, there exists a sequence a = x^, X2Â» . Â• Â• Â» '^n ~ of elements of Â£ such that either x. x.^, CR or x.^-, x. C R ( i = 1, 2, . . . , n 1) ; and (vii) for each coset Â£' i S, either a + R = R + a for all aÂ€Â£' or a + R = -R + a for all a C S' . Theorems 7,1 and 7.2 also lead to methods for constructing infinite dimensional poip-groups from a given dualistic poip-group of finite dimension and a given weakly connected po-group. These are described below. In the discussion of the direct sum of two groups A and E, the symbol (a, B' ) , where aÂ€A and B' is a subset of B, represents the set of all pairs (a, b' ) such that b' Â€ B' . The symbol (A', b), where A' is a subset of A and bCB, is defined accordingly. Theorem 7.6 : Let P be the direct s\xra G Â• Â£ of an upright dualistic poip-group G and a weakly connected po-group S. Then P is a dualistic poip-group when it is ordered as follows: (i) if g-i i ^2' ^^1Â» ^1^ -^ ^'^2' ^2^ ^^ ^"*^ only if g-i > ^o ^" ^Â» ^^^^ ^^ g, = ^2 is maximal, (g^, s,) > (gÂ„, s_ ) if and only if s. > s^ in Â£; and (iii) if g, = g2 is minimal, (g,, s,) > (gg, Sg) if and only if E^ < s in Â£. Furthermore, Â£ coincides with (0, Â£), and P/Â£Â« coincides with (G, 0). The theorem follows directly from Theorem 7.1 once we have * Â« shown that (g, s) + (0, 0) = (0, 0) + (g, s) for every element

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56 (g, s) ^G Â•Â»Â• S. Here it is understood that (0, Â£) is ordered such that (0, s^) > (0, sg) if and only if si> S2, and that (G, 0) is ordered such that (g^, 0) _> (g2, 0) if and only if g2^_> g2 . But * * Â» (g, s) + (0, 0) = (g, s) + (0, ) = (g, s + ) = {g, + s) = (0. 0*) + (g, s) = (0, 0)* + (g, s). Theorem 7.7 : Let P be the direct sum G S of a group G and a weakly connected po-group S. Then P is a dualistic poipgroup when it is ordered in any one of the ways indicated below: (i) Qi i ^2 implies (g^, s-j^) and (g2, S2) are not comparable; (ii) (0, sj^) > (0, S2) if and only if s^ > S2; g ^ implies either (iii) for all s-j^, S2 (g, Sg) if and only if s-,^ > Sg or (iv) for all s^, s^ Â€S, (g, s^) > (g, s^) if and only if S]^ < S2. i^breover, for any one of these orderings, Â£q coincides with (0, t>) and P/Â£q = (G, 0) is totally unordered. The proof of the theorem is essentially that of Theorem 7.6. Example 7.8 : Let G be the poip-group consisting of the group I2 ordered by setting > 1, and let S be the po-group consisting of the even integers in their natural order. When G + S is ordered according to Theorem 7.5, we have . . . > (0, 2) > (0, 0) > (0, -2) > . , . > (1, -2) > (1, 0) > (1, 2) > . . . Comparing this example to the poip-group P of Example 2.19 which is not the direct sum of P/Sq and Sq/ even though P/Sg and Sg

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57 are order isomorphic and group isomorphic (under the same correspondence) to G and S, respectively, shows that the algebraic structure of a poip-group P is not completely determined by the systems S^ and P/S^., In this section we have given necessary and sufficient conditions that a group can be ordered to form an infinite dimensional poip-group. In addition, we have exhibited the relationships between the algebraic properties of a dualist ic poip-group P and the structures of Sq ^"<^ -^/Sg* and have shown that the latter two systems do not always completely determine P as a group.

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VIII. LOIP-GROUPS The preceding results are much simpler to state if it is assumed that the poip-gxDup itself is weakly connected. A weakly connected dualistic poip-group P is a po-group if and only if P/Sg is of length zero Â— that is, if and only if P = S . 1 4 Otherwise, P is the union of Gq and its coset Cg. Moreover, it can easily be shown that, under the relativized ordering, ci is a pogroup and Cg is order isomorphic to Cg under the correspondence carrying a^Cg into -a + rÂ€Cl, where r is the coset representative of C^. As a consequence of Corollary 5.5, no member of ci is less than an element of Cg when P is upright. These results are summarized in the following theorem. Theo rem 8.1 ; If P is an upright dualistic poip-group which is weakly connected, then (i) P = cj U cl, where (ii) C^ is a normal subgroup of P with cosets C^ and C^, (iii) under the relativized 4 1 ordering Cg is order isomorphic to the po-group Cg, and (iv) no element of Cg is under an elsnent of C^. When every pair of elements of P is bounded in some manner, conditions i-iv are strengthened as shown in the corollaries below. 58

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59 Corollary 8.2 : If P is an upright dualistic poip-group in which every pair of elements has either an upper or a lower bound, it satisfies conditions i-iii, and (iv)' every element of Cq is 4 greater than every element of Gg. Clearly P satisfies conditions i-iv. Now let a and b be members of ci and Cg, respectively, and suppose that they have an upper bound c. By condition iv, c^C^, so that cfC^ and cC^^ . However, c > b and S^ = 5^ > Sj^ whence a > b. Similarly, if a and b have a lower bound, a > b. Corollary 8.3 : If P is an upright dualistic poip-group which is either up-directed or down-directed, it satisfies (i), (ii), (iii), (iv)', and (v) C^ and C^ are strong components of P. Corollary 8.4 : If ? is an upright loip-group, it satisfies (i), (ii), (iv)', (v), and (iii)' under the relativized ordering C^ is an 1-group and Cq is order isomorphic to it. By condition iii, Cg is a po-group under the relativized ordering. If a and b are elements of Cq, by ( iv) a U bCCg. Now 4 4 assume a fl b^Cg, so that every member of Cq must be less than a n b, in which case aH b is a maximal element of C^. Condition iii thus implies that Cq has a maximal element. However, the only 1-group with a maximal element is the group of order one. Thus aDbCcJ, and Ci is a loip-group.

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60 As a result of this corollary many of the results concerning 1 -groups are easily extended to loip-groups. We note several examples. In an l-group L, x + (a U b) = (x + a) U (x + b) for all a,b,x ^L. In a loip-group P, this becomes x + (a U b) = (x + a) U {x + b) if xÂ€cj and x + (aU b) = (x + a) (x + b) if xCCg. Every l-group and every loip-group is a distributive lattice. In (4, p. 222) it is shown that the congruence relations on an l-group L are the partitions of L into the cosets of its different 1ideals. On the other hand, the congruence relations on a loip-group P are the partitions of P into the cosets of its k-ideals, where the k-ideals are P and its normal subgroups which are 1ideals of Cg. These results are all easily established.

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IX. sm^^RY The order structure of partially ordered ideal preserving groups was deteimined in terms of partially ordered groups; many results concerning these latter systems can easily be extended to the more general partially ordered ideal preserving groups. Necessary and sufficient conditions that a group can be ordered to form a non-trivial partially ordered ideal preserving group were obtained. The principal concepts employed in this dissertation are the equivalence relations called weak and strong connectivity. The investigation of partially ordered ideal preserving rings and fields is suggested as a problem for further research. 61

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LITERATURE CITED 1. 0, Frink, Ideals In partially ordered sets , Amer. Math. Monthly, vol. 61 (1954) pp. 223-34. 2, A. T. Butson, "Simply ordered ideal preserving groups." Paper read before the meeting of the American Mathematical Society, Rochester, N. Y., December, 1956. 3, C. C. MacDuffee, An introduction to abstract algebra . New York, J. Wiley and Sons, 1940. 4. G, Birkhoff, Lattice theory , /^jnerican Mathematical Society Colloquium Publications, vol. XXV. New York, American Mathematical Society, 1948. 62

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BIOGPJiJ^HICAL SKETCH Jan Frederick Andrus was bom September 17, 1932, in Washington, D. C. He was graduated from the College of Charleston in Charleston, South Carolina, in May, 1954, with the degree Bachelor qf Science. Work for the degree Master of Arts was undertaken at Emory University, Emory University, Georgia, and the degree was granted in August, 1955. The author's undergraduate major was in chemistry with a minor in mathematics; his graduate major has been mathematics, and his minor work has been in the field of physics. During his studies the author has held assistantships in chemistry at Emory University and in mathematics at the University of Florida. He was employed in the summer of 1957 as an aeronautical research engineer with the National Advisory Committee for Aeronautics at Langley Field, Virginia. While pursuing undergraduate study, the author was elected to membership in Sigma Alpha Ph i, College of Charleston honorary society. He was awarded the S. Keith Johnson medal for outstanding achievement in science and a similar award for the highest achievement in mathematics. He was elected to membership i n Alpha Sigma Phi, national physics honorary, vd\ile a graduate student at Emory University. 63

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory conmittee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dat e June 9 .1958 -^Â» .^ Â£A^ Â— Â• Dean, College of Arts and Sciences Dean, Graduate School SUPE.WISORY Ca'^ITTEE ; Chairman ^ Co-Chairraan ^/.'i^.y^T^m.o Wt.

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