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.
TABLE OF CONTENTS
ACKNOWLEDGMENTS .. . . . ..........
I. INTRODUCTION . . . . . . . .
II. PRELIMINARY CONCEPTS . . . . . .
III. NONDUALISTIC POIPGROUPS . . . . .
IV. CONNECTIVITY . . . . . . .
V. A DECOiMPOSITION THEOREM . . . . .
VI. FINITE DIMENSIONAL POIPGBUPS . . .
VII. THE ALGEBRAIC CHARACTER OF POIPGROUPS. .
VIII. LOIPGROUPS . . . . . . . .
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 groupsthe 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 subgroupnamely 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 poset (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 updirected if and only if F # 0 for
every finite subset F of P. It is downdirected 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 poset in the requirement that F be nonvoid.
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 nonvoid 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 pogroup P (partially ordered group) is
(i) a poset, (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 1group (lattice ordered group).
If P is a simply ordered set satisfying (ii) and (iii), it is called
an ogroup (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 poipgroup P (partially ordered ideal
preserving group) is (i) a poset, (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 loipgroup (lattice ordered ideal preserving group). If
P is a simply ordered set satisfying (ii) and (iii), it is called a
soipgroup (simply ordered ideal preserving group).
Let P be a poipgroup. 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 idealcontradicting Definition 2.9. Thus if P
is of finite length, it is totally unorderedthat is, it is a
trivial poipgroup, or it is of length one and contains no isolated
elements. These statements are summarized in the following theorem:
Theorem 2.11: A nontrivial poipgroup has either infinite
length and no minimal elements or length one and no isolated elements.
Definition 2.12: A poipgroup 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 poipgroup which is not dualistic is said to
be nondualistic.
The above classifications of poipgroups will serve as a
framework for the remainder of this dissertation.
All graphical representations of poipgroups in this work are
similar to the "Hasse diagrams" (4, p. 6).
Example 2.13: Any pogroup P is a poipgroup since, for any
*+ *+ *
elements x,y(P, x + F* + y = (x + F + y) whenever F and F are
nonvoid subsets of P.
Example 2.14: The multiplicative groups of rational and real
numbers ordered according to magnitude are soipgroups.
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 poipgroups 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 nontrivial pogroups
of finite length.
Example 2.17: The direct sum I2k ; P of I2k ordered as in
Example 2.15 and any pogroup P forms a poipgroup 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 1group, 12k + P is a loipgroup 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 poipgroup which is not
the direct sum of a poipgroup of finite length and a pogroup.
Example 2.19: The additive group of rational integers
ordered as follows is a soipgroup.
. .> 2> 0> 2> . .> 3> 1> 1> 3> . .
III. NONDUALISTIC POIPGROUPS
In this section we completely characterize nondualistic
poipgroups and in so doing obtain several important theorems con
cerning dualistic poipgroups.
Definition 3.1: Elements a and b of a poset 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,
. . n1).
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 poipgroup 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 poset, (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 poipqroup.
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 + ethat 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 Cx + 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 poipgroup.
The following four lemmas on poipgroups of finite dimension
will be used in the proof of Theorem 3.10, which completely char
acterizes nondualistic poipgroups.
Lemma 3.6: In a poipgroup 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 poipgroup 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
other, this is a contradiction.
Lemma 3.8: In a poipgroup 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 poipgroup 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 iiii of Lemma 3.5 and
that it is dualistic.
Theorem 3.10: Any nondualistic poipgroup is (i) a poset
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 iiv for some normal subgroup CO is a nondualistic
poipgroup.
The latter statement will be proved first. Let P be a group
satisfying conditions iiv 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 poipgroup.
In order to show that P is nondualistic, 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 nondualistic poipgroup.
Now let P be a nondualistic poipgroup 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
dictionimplying 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 nondualistic poipgroup has finite dimension. This re
sult, along with a more useful definition of a dualistic poipgroup,
will be obtained in the remainder of this section. However, before
proceeding we give an example of a nondualistic poipgroup.
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 nondualistic poipgroup 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 poipgroup 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 poipgroup 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 Cx + (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 nonmaximal element of a poipgroup 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 poipgroup P,
and a is not maximal, then x + a and x + b are comparable.
Obviously a cx + (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 Cx + (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
poipgroup 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 poipgroup.
Suppose a is a maximal element of a poipgroup 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 Et + (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 poipgroup contains
no maximal or minimal elements.
Theorem 3.17: Comparability is preserved under addition in
an infinite dimensional poipgroupthat 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 poipgroup,
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* Zx + (x + c)+, a ex + (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 poipgroup. 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 poipgroup P is (i) a
poset, (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 iiii is a dualistic poipgroup.
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 poipgroup is
dualistic.
Theorem 3.22: A dualistic poipgroup P is (i) a poset,
(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 iiii
is a dualistic poipgroup.
The latter statement of the theorem is merely Lemma 3.5.
A dualistic poipgroup 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
poipgroup 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 nondualistic poipgroups 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 poipgroup, 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 poipgroup 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 nonvoid 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 noncommutative dualistic poipgroup.
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 leftright 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 leftright 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 poipgroup but does not necessarily have meaning for
a poset 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 50that 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 poipgroups. 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 poipgroup 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 poipgroup
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 poipgroup
is order isomorphic to the ordinal product of a certain finite di
mensional poipgroup and a pogroup.
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 poipgroup 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 connectedmeaning 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 selfdual 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 poset 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 pogroup.
To show that So is selfdual, let x correspond to x for all
x ES0. If y > z in SO, then 0 > z y, so that z > ythat is,
the correspondence is an antiisomorphism (in the order sense) and
S0 is selfdual.
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 antiisomorphic 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 poipgroup P is order isomorphic
to the ordinal product (P/SO)So of the finite dimensional dualistic
poipgroup P/SO and the pogroup 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 posets)
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 poipgroup 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 poipgroups
holds for inverted poipgroups.
Theorem 5.12: Let P be a dualistic poipgroup, 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 poipgroup
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 pogroup we need characterize only
finite dimensional poipgroups.
VI. FINITE DIMENSIONAL POIPGROUPS
The only finite dimensional poipgroups which need to be
given further consideration are the dualistic poipgroups of length
one. These systems are characterized below.
6.1: When a is a maximal element of a poipgroup of length one,
Ta will represent the set x EP / x < a) and when a is
minimal, it will denote the set IxP / x > a .
Theorem 6.2: An upright dualistic poipgroup 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 nonvoid subset To satisfying condi
tions iiv. 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 poipgroup 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 poipgroup.
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 xkl = 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 iiv. 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 eventhat is,
that either m and n are both even or both odd. Thus conditions v
and vi establish a welldefined partial ordering for G. Under this
ordering it is clear that G is a poset 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 iiv 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 poipgroup
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 POIPGROUPS
It has been shown that many of the properties of P, in
particular its structure as a poset, 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 pogroup, 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 pogroups So and S coincide and the poipgroups
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 pogroup, 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 poset, and we want to show that under
this ordering G is a dualistic poipgroup.
First observe that if x yEO then yEx + 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 poipgroup P. It is easily shown that SO
and P/SO coincide with S and G/S, respectively.
Now consider an upright dualistic poipgroup 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 pogroup, 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 poipgroup P such that the pogroups 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 poset under any one of these orderings
is similar to that in the preceding theorem. We now show that G is
a dualistic poipgroup 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 poipgroup 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 poipgroup
are: (1) G contains a normal subgroup S #f 0) which can be ordered
so that it forms a weakly connected pogroup, 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 poipgroup. 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 pogroup 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
pogroup by defining a > b in S to mean a bC R.
It is obvious that a pogroup 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
polpgroup 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 poipgroups from a given dualistic poipgroup
of finite dimension and a given weakly connected pogroup. 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 poipgroup G and a weakly connected pogroup S. Then P
is a dualistic poipgroup 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 pogroup 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 poipgroup consisting of the
group 12 ordered by setting 0 > 1, and let S be the pogroup 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 poipgroup 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 poipgroup 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
poipgroup. In addition, we have exhibited the relationships be
tween the algebraic properties of a dualistic poipgroup 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. LOIPGROUPS
The preceding results are much simpler to state if it is
assumed that the poipgroup itself is weakly connected.
A weakly connected dualistic poipgroup P is a pogroup if
and only if P/S0 is of length zerothat 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 poipgroup 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 pogroup 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 iiv are strengthened as shown in the corollaries below.
Corollary 8.2: If P is an upright dualistic poipgroup
in which every pair of elements has either an upper or a lower bound,
it satisfies conditions iiii, and (iv)' every element of C1 is
0
greater than every element of C4.
Clearly P satisfies conditions iiv. 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 cC1 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 poipgroup
which is either updirected or downdirected, 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 loipgroup, it satisfies
(i), (ii), (iv)', (v), and (iii)' under the relativized ordering C1
is an 1group and C4 is order isomorphic to it.
By condition iii, CO is a pogroup 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
1group with a maximal element is the group of order one. Thus
a bCCO, and C1 is a loipgroup.
As a result of this corollary many of the results concerning
1groups are easily extended to loipgroups. We note several
examples. In an 1group L, x + (a U b) = (x + a) U (x + b) for all
a,b,x EL. In a loipgroup 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 1group and every loipgroup is a distributive lattice.
In (4, p. 222) it is shown that the congruence relations on an
1group L are the partitions of L into the costs of its different 1
ideals. On the other hand, the congruence relations on a loipgroup P
are the partitions of P into the costs of its kideals, where the
1k
kideals are P and its normal subgroups which are 1ideals 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 nontrivial 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. 22334.
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
Dean, Graduate School
SUPERVISORY COMMITTEE:
Chairman
CoChairman
^^.^c02
