Results concerning the Schutzenberger-Wallace theorem

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Results concerning the Schutzenberger-Wallace theorem
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RESULTS CONCERNING THE

SCHUTZENBERGER-WALLACE THEOREM










By
ANTHONY CONNORS SHERSHIN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE RIQL'IREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY












UNIVERSITY OF FLORIDA
August, 1967














ACKNOWLEDGMENTS


The author acknowledges his complete indebtedness and expresses

his warmest appreciation to Dr. Alexander Doniphan Wallace, Professor

of the Department of Mathematics and Chairman of the Supervisory Com-

mittee, for his guidance and patience during the preparation of this

dissertation. The author is grateful to Dr. Julio R. Bastida whose

own dissertation served as the basis and starting point for this work.

Thanks are due also to Doctors Alexander R. Bednarek, Pierre-Antoine

Grillet and Kermit N. Sigmon who helped in the critique of this manu-

script in its final phases and to all the members of the supervisory

committee for their assistance.

The author takes pleasure in thanking Dr. John E. Maxfield,

Chairman of the Department of Mathematics, for his help in complet-

ing this graduate program and Mr. Howard A. Kenyon of Autonetics,

a division of North American Aviation, Inc., who supplied financial

assistance in the form of professional employment at various times.

He is also grateful to Mrs. Thomas Larrick for her skillful typing

of this doctoral dissertation and a previous master's thesis.

Finally, the author wishes to express his appreciation to his friend,

Dr. Joseph Thomas Borrego, for many illuminating mathematical

discussions.




















TABLE OF CONTENTS


Page


Chapter

I PRELIMINARIES . .


II EXTENSIONS OF BASTIDA'S RESULTS .


III A CHARACTERIZATION OF THE CLOSED SUBGROUPS
OF THE SCHITZENBERGER GROUP .


IV ALGEBRAIC RESULTS CONCERNING ~-SLICES .


BIBLIOGRAPHY ..... .. . ...



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


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



























CHAPTER I


PRELIMINARIES



This initial chapter will present introductory material includ-


ing an important result which is due to P. Dubreil in its algebraic


setting.


1.1 Definition. A topological semigroup S is a nonnull


Hausdorff space together with a continuous associative multiplication.


Precisely, a semigroup is such a function m: S x S S that


(i) S is a nonnull Hausdorff space,


(ii) m is continuous, and


(iii) m is associative; i.e., for each x,y,z in S,


m(x,m(y,z)) = m(m(x,y),z).


At times it will be necessary to distinguish between a semi-


group and its nontopological counterpart, an algebraic semigroup.


It is common usage to say that a semigroup is compact if S is a compact


1










space and to say that a subset of S is closed if it is closed in a


topological sense.



1.2 Definitions. The empty set will be designated by l .

(-1)
If X and Y are subsets of S, then X Y = fw in S; Xw n Y pt r ,

X[-11Y = [w in S; Xw = Y], YX(-) = fw in S; wX n Y i i ), and


YX[-I = {fw in S; wX c Y).


It is noted that if X is a singleton set, then X(-)Y = X[-ljY


and Y-l = YX-.




The next result is but a fragment of a result due to


A. D. Wallace, another part of which is found in (3].



1.3 Proposition. If Y is closed, then X [-I]Y is closed.


Proof. It is easily verified that X[-]Y = a(-l)Y and
a X

since the function la: S S defined by la(s) = as is continuous,


we have that (la)-l (Y) = a(-l)Y is closed and, consequently, the


desired result follows immediately.









1.4 Definition. Letting Y be a subset of S and A be the


diagonal of Y x Y, then an equivalence relation S c Y x Y is a


closed congruence on Y if and only if A U TA c ( and e is closed


in Y x Y with respect to the relative topology.


The next two results are well known and are stated here with-


out proof. (The reader may refer to [4) and [6].)



1.5 Proposition. If S is compact or discrete, if G is a


closed congruence on S and if cp: S S/ is the canonical map,


then there is a unique continuous function yi such that the diagram




S/S x S/S --- S/


cpxp


SxS S
m



is analytic and thus S/S is a semigroup and cp is a continuous


homomorphism.








1.6 Theorem (Sierpinaki). If 0: A B and 98 A -. C are


onto functions with the property that a(a1) = (oa2) if and only if


8(a1) = 8(a2), then in the diagram there exist mutually inverse func-


tions f and g such that go? = 9 and fe = Cv. Furthermore, if


B





A C


A, B and C are semigroups and a, 9 are morphisms, then f,g are iso-


morphisms; if, in addition, A is compact, B,C are Hausdorff and 0,9


are continuous, then f,g are iseomorphisms.



1.7 Definitions. Throughout this study, and in particular the


next theorem, we will make frequent use of the following functions:


If a e S and B c S we will define ra: B S by ra(b) = ba and


la: B S by la(b) = ab. It is noted that the image of la is


aS n Sa if B = a (-(Sa), because if x is in aS n Sa, say


x = as s s'a, then there exists an element in a (-)(Sa), namely s,


such that x = as and, consequently, la maps a -1)(Sa) onto









aS n Sa. Moreover, s' e (aS)a(- and x = s'a so that ra .maps


(aS)a onto aS n Sa.



1.8 Theorem (Dubreil). If S is compact or discrete, a e S


and if we define 2(a) = f(x,y); x,y e (aS)a and xa = ya) and


3(a) = [(u,v); u,v e a (-)(Sa) and au = av}, then (a) and M(a)


are congruences on the semigroups (aS)a(-I) and a (-)(Sa), respec-


tively. If and yp are the appropriate natural homomorphisms in the


diagram, then f,g and h are such homeomorphisms that f* = ra, gcp = la

-1
and g- f = h; moreover, h is an isomorphism.



(aS)a (- /(a) h a(-) (Sa)/3(a)




GO -Or (-1)
(aS)a(-1) ---- -aS n Sa ---- a (Sa)
ra la


Proof. The sets a (-)(Sa)


because a e a(-1)(Sa) n (aS)a(-).


closed follows from (1.3), since S


is immediate that a (-l)(Sa) is an


and (aS)a are nonempty


The fact that a (-)(Sa) is


is compact and Hausdorff, and it


algebraic semigroup; in a similar










fashion (aS)a(-) is a closed semigroup. It is clear that B(a)


is an equivalence and a right congruence on a (-) (Sa). If

(-1)
u,,x,y e a (Sa) and ax = ay, then for some s s S, aux = sax 3

say = auy and, consequently, 13(a) is also a left congruence. To see


that 3(a) is closed we merely note that B(a) = a(-l)(Sa) x a (-)(Sa)) n


[(la) x (la)]-l(A), where A is the diagonal of S x S, and that


(la) x (la) is continuous. Therefore, in view of (1.5), a(-) (Sa)/B(a)


is a semigroup. Clearly, yp(x) = cp(y) if and only if la (x) = la(y).


Since la maps a(-l)(Sa) onto aS n Sa, in view of Sierpinski's


result we see that such a homeomorphism g exists.


Arguments which are dual to the preceding ones yield the other


half of the diagram and, clearly, h = g- f is a homeomorphism.


If c is in (aS)a -), it is easy to see that f(*(c)) = ca


since f* a ra and hence that g-l f((e)) = gl (ca) = gL (an) = cp(n)


for some n e a(-) (Sa), the last equality holding due to the analy-


tiCity of the right-hand side of the diagram. Now, if b,c are in


(aS)a and m,n are elements of a(-1)(Sa) such that ba = am





7



and ca = an, it follows that bca = ban = amn and therefore


g f(*(b)*(c)) = g-lf(*(bc)) g (bca) g-l (amn) = m(mn) =


cp(m)cp(n) = g -lf((b))g-l f((c)) and, consequently, h is an


iseomorphism.














CHAPTER II


EXTENSIONS OF BASTIDA'S RESULTS



The principal purpose of this chapter is to extend, dualize and


simplify results given by 3. R. Bastida in (1]. The extensions, which


culminate in (2.17), are manifold in character and include what is


termed the "relative" case (whereas Bastida treats only the "absolute"


case, namely, T = S) and, in addition, we present results of a non-


discrete type which Bastida does not consider. As to the duality,


Bastida examines one-half of the possible left-right duality and this


chapter indicates that, under certain circumstances, the structures


obtained by reversing the multiplication are topologically and alge-


braically the same. Simplicity is introduced because in the prelim-


inary propositions we isolate those properties of Wt-slices which are


truly necessary for the validity of the arguments. Lastly, it is


shown that the Schutzenberger-Wallace Theorem follows as a conse-


quence of these results.









2.1 Definition. If S is a semigroup and A and T are subsets


of S, then one defines L(A,T) = A U TA, R(A,T) = A U'AT and H(A,T) =


R(A,T) n L(A,T). When the context clearly indicates which subset T is


under consideration, then reference to T is usually omitted, that is,


we write L(A,T) = L(A), etc. Moreover, for T c S, one defines the


Relative Green (equivalence) Relations, = f(x,y); L(x) = L(y)3,


R = f(x,y); R(x) = R(y)5 and A( n R R. For x e S, we will let


H (T) denote the V(T)-class (or slice) containing x; here again refer-


ence to T is omitted if the context is clear.




In this chapter, A,B and T will denote subsets of a semi-


group S, c will be an element of S and D = c-l) A n B.



2.2 Proposition. If A c L(a) for all a e A and B c L(b)


for all b e B and if D is nonempty, then A c Sb for all b e B.


Proof. If we let b e B and d e D and if b = d, then


A c L(cd) = L(cb) c Sb; if b / d, then d e Tb and hence


A c L(cd) C Sb.








2.3 Proposition. If X and Y are subsets of S such that

3 p X c Y and if y(-l) = [-ljy, then X[-1]x c y[-l]y.

Proof. The hypothesis that X is nonempty is needed to ensure

that X[-IX c X(-l)X and so X C Y implies that X[-1l] c x (-l) c

Y(-)y = y[-l]y.


2.4 Proposition. If y e S such that y(-l)y c D[-11B n A[-lIA,

then y(-l)y c DI-1]D; moreover, if also y e yS, then D[-1]D is non-

empty.

Proof. If x e y(l)y, then Dx c B and c(Dx) = (cD)x c Ax c A

so that Dx c c(-I)A and, consequently, Dx c D, i.e., x e D[-1]D.

The second half of the result follows because y e yS if and

only if y(-)y $C .


2.5 Corollary. If A c L(a) for all a e A, if D B c L(b)

for all b e B and if B(-I)B = B[-I]B, then b(-l)b c D[-1ID for all

b e B; moreover, if, in addition, b e bS, then D[-]D is nonempty.

Proof. We will satisfy the hypothesis of the first part of

(2.4): Since B(1)B = B1[-]B, it follows that b(-l)b c b1l)B c








B (-1)B = B[-]B c D[-1]B and if D is nonempty we have A c Sb by


(2.2) so that for a e A, x e b(- b we obtain ax = (ab)x = s(bx) =


sb = a, for some s e S, and, consequently, b l) b C A-1]A. It is


noted that the conclusion also follows if D is eihpty for then

D[-I]D = S.



2.6 Proposition. If a A c R(a) and if A (-)a is nonempty,

then a e aS.


Proof. If x e A(- a we have a = a'x for some a' e A sb


that if a = a' the result is immediate and if a X a', then a' = at


for some t e T since A c R(a) and hence a = a'x = (at)x = a(tx).


It is noted that if A is nonempty the statement A (-)a p 0


is implied by the condition E l a(-I)A c fs e S; A c As], for then


DO (s e S; a e As) = A(-l)a.


2.7 Definition. For any A C S and


e(A,y) = f(u,v); u,v e A[-I A and yu = yv)

[(u,v); u,v e AA[-I] and uy = vy).


y e S, let us define


and n (A,y)=








2.8 Proposition. If A['1]A [, then S(A,y) is a congruence

on A[-I]A if and only if yu = yv implies that y'u = y'v for all

y' e y(A[-I]A).

Proof. For brevity and clarity, let 8 = S(A,y) in this proof.

If is a congruence on A[-1]A, then (AS U (S) c where A is the

diagonal of A[-1]A x A[-']A and thus, letting w e A[-IA and

(u,v) e 8, we have ywu = ywv so that y'u = y'v for all y' in

y(AI-1]A).

Conversely, if the condition holds, namely, yu = yv implies

that y'u = y'v for all y' in y(A[-']A) and if w e A[-1]A,

(u,v) e 8, then the condition implies that (yw)u = (yw)v because

yw is in y(A[-1]A) and therefore 8 is a left congruence on A[-1]A.

It is evident that 8 is a right congruence.


2.9 Corollary. If b is in B n bS, if A C L(x) for all

x in A, if 0 4 B c L(x) for all x in B, if B('1I) = B1[-1B and if

D is nonempty, then 8(D,b) is a congruence on DL-1]D and (D,b) =

S(D,b') for b' e B.









Proof. In view of (2.5) the hypothesis implies that DI-1 D


is nonempty so that in order to prove that e(D,b) is a congruence


on D [-1D it suffices to show that the condition of (2.8) is satis-


fied. In the case that B is a singleton set, say B = fb), we have


from (2.3) that b(D-1]D) cb(B[- 11B) B = sbl so that the condi-


tion is trivially satisfied. If card B > I, then it follows that


b is in Sb and thus that B c: Sb. Letting (u,v) e S(D,b) and


b' e b(D[-]D) we have b' e b(b(-B) c B c Sb since D [-1]D


b (-B so that if b' = sb we obtain b'u = (sb)u = s(bu) = s(bv) a


(sb)v = b'v and therefore the condition of (2.8) holds.


Momentarily fixing distinct elements b and b' in B and


letting (u,v) be an element of S(D,b), we use the fact that B c L(b)


to obtain b'u = (tb)u = t(bu) = t(bv) = (tb)v = b'v, where t e T, so


that (u,v) is also in (D,b'). Clearly, in a similar fashion we


have O(D,b') c S(D,b).


2.10 Proposition. A[-1 c (xA)[-](xA) for all x e S.








Proof. If y e A _]A, then (xA)y = x(Ay) c xA so that y is


in (xA) -l(xA).



2.11 Proposition. For any elements x,y and z e S, if x e Sy,


then S(A,y) C S(zA,x).


Proof. If (u,v) e S(A,y), then in view of (2.10) it remains


only to verify that xu = xv: Letting x = sy we have s(yu) =


(sy)u = xu and in a similar manner s(yv) = xv so that xu = xv.



2.12 Corollary. If A C L(a) for all a in A, if B c L(b)


for all b e B and if D is nonempty, then 3(D,b) c tS(cD,a) where


a e A and b e B.


Proof. The hypothesis is that of (2.2) so that A C Sb and


therefore in view of (2.11) the conclusion is evident.



2.13 Proposition. If A, B and D are nonempty sets such that


A c L(a) for all a e A and B c L(b) for all b e B, if A(I)A =


A[-1]A and B(-)B = B[-1B and if b e bS for some b e B, then


S(cD,a) is a congruence on (cD) -11(cD) for any a e A.








Proof. In view of (2.5) we have D1_ 1D i Ej and so (cD)1l 1(cD)


is nonempty since D-1D C (cD)[-](cD). We will now verify that the


condition of (2.8) is satisfied: In the case that A = [a) it follows


from (2.3) that a[(cD)1-l (cD)] = fal and as a result the condition is


trivially fulfilled.


If card A > 1, then we obtain a e Sa and, therefore, A c Sa.


Letting (u,v) e D(cD,a) and a' w a[(cD)1-11(cD)] we have a' in


a(A[-I]A) c Sa since (cD)[-](cD) c A[1-]A by (2.3). Consequently,


if a' = sa, then we see that a'u = sau = sav = a'v and thus the con-


dition of (2.8) holds.



The next result is well known and it is due to B. J. Pettia;


consequently, its proof is omitted.


2.14 Proposition. If a compact semigroup is algebraically a


group, then it is a topological group.



2.15 Proposition (Induced Homomorphism Theorem). If av and 0


are congruences on semigroups A and B, respectively, such that A c B









and a c: then in the diagram, where f and g are the appropriate


canonical maps and i is the inclusion map, there exists a homomor-



i*
phism i such that the diagram is analytic.


A/a -- B/B


f g


A B


Proof. This proposition follows from an evident extension of


the version of the Induced Homomorphism Theorem given in [2] and its


corollary there.


2.16 Lemma. If A(-1)A c A[-]A and if B(-G)B c B['-l B,


then D(-41 CD[-1]D and, consequently, D[-1]D = d(-1)D for all d


in D provided that D is nonempty.


Proof. First of all, it is clear that D(-1)D c B(-1)B c Bl-1]B.


Since we may assume that D D is nonempty, say t e D D, then,


using the fact that cD C A, we have that (At n A) D (cDt n cD) 3


c(Dt n D) ip o. Again noting that cD c A, it follows that cDt C At C A


so that Dt c c-1)A. As a result, DtC c (-)A n B = D and so t e D[-1]D.








The second conclusion follows from the fact that D -1D =D


n (d(-D; d e D) C dOD C U fd(-)D; d e D} = D(-I)D.



2.17 Theorem. (a) Let S be compact or discrete and A and B


be nonempty sets satisfying these three conditions:


(i) A c L(a) for all a in A,


(ii) B c L(b) n R(b) for all b e B,


(iii) A(-I)A = A[-]A and B(-1)B = B-IB.


If card B > 1 and if D is both nonempty and closed, then this dia-


gram is analytic:


D[- ]D/S(Dd)


g/ ^

D D[-llD
Id


where D[-I]D/S(D,d), for d in D, is a topological group, cp is the


canonical map and g is a homeomorphism.


(b) If, in addition, card A > I, the preceding analytic diagram


may be extended:









D[I- D ld
D1"1^-------------------------- D






DE1 ^D/S(Dd)


n y c



(cD) -I](cD)/S(cD,cd)






(cD)) -- eD
l(ed)


where (cD) -l(cD)/S(cD,cd) is a topological group, ct is a canonical


map, m is a homeomorphism and y is a continuous epimorphism.


Proof. We will consider only the case where S is compact


since the situation where S is discrete follows in a similar manner


with the topological results omitted.


Since.card B > 1 and B C R(b) for all b in B, it follows that


d e dS for each d e D and so,by (2.5), D[-1]D is nonempty.


Using (1.3), D ']D is closed and compact because D is closed and S is


compact. It is immediate from (2.9) that 8(D,d) is a congruence on









D[-I]D for any d e D and, moreover, (D,d) is closed because it is


easily verified that S(D,d) = (D[-1]D x D[-1]D) f (Id x ld)- (A)


and so D[-]D/S(D,d) is a compact topological semigroup by (1.5).


With (2.14) in mind we proceed to show that D[-l]D/(D,d) is alge-


braically a group:


We may select an element q of d ( d because d G d is


nonempty if and only if d is in dS. Then, since D C L(d) we have


dx e L(d) for x e D[-I]D and so it is easily verified that dx = dxq.


Since D[- ]D is a semigroup we have (x,xq) e g(D,d) and thus


cp(x) = cp(x)cp(q) so that cp(q) is a right unit for cp(x). If d = dx


we have d = dxx and if d ;f dx, then d = dxt for some t e T


because B c R(b) for all b e B so that in either case there is an


element x' such that d = dxx' and, consequently, dq = dxx'. In


view of (2.16), x' is in D[- ]D and therefore cp(q) = cp(x)cp(x')


indicating that cp(x) has a right inverse.


In order to show that D = D[ -iD/((D,d) we will make use of


Dubreil's result, that is, (1.8): Card B > 1 implies that









B c L(d) C Sd and thus using (2.3) we have D1'1]D c d('l)B

d(-1)(Sd). The fact that D(-1)D c D[-1]D implies that the restric-

tion of Id to D[-1D has as its image D, for if d' e D and d' = d

we know that d (l)d is nonempty and if d' $ d, d' = dt for some

t e T so that in either case d' = dt' and thus t' e D -1)D c D[-I]D;

the other inclusion is clear because D[-0]D C d(-l)D implies that

d(D[-1]D) c D. Therefore, cp(D[-lJ1) D because gp = Id where g

is the appropriate homneomorphism of (1.8).

To prove the second part of the theorem we begin by proving the

existence of a continuous epimorphism y. Since D[-1]D c (cD)[-l](cD)

and e(D,d) c 8(cD,cd), the Induced Homomorphism Theorem gives us the

existence of a function y such that Wp = Acf where i is the inclu-

sion map. cp is closed because it is continuous, D[-1]D is compact

and D[-0 D/B(D,d) is Hausdorff from a result in [5] and thus since

it is also .true that p is an onto function and rfi = 1p is continu-

ous, we have that y is continuous from another result in [5]. It is

clear that y is a homomorphism and so it remains to verify that it is








an onto function: If Y is an element of (cD)[ -(cD)/S(cD,cd) and


y is in Y, then cdy = ed' for some d' e D. If d = d' and q is


in d (-l)d, then cdy = cd = cdq and it follows that (y,q) is in


S(cD,cd) so that y(cp(q)) = Y. If d o d', then for some t e T we


see that d' = dt because B c R(d) and we note that t e D D =


D[- 1D. Then, since cdy cdt, it is true that (y,t) is in tS(cD,cd)


and it follows that y(cp(t)) = Y. We conclude, therefore, that v is


an onto function.


It is next noted that (cD)- [](cD) is nonempty because


D[-]1D c (cD)[-l](cD) and that in an analogous 'manner to the proof of


the first part of the theorem it is easy to verify that (cD) [-(cD)


is closed and compact, that (cD,cd) is a closed congruence on


(cD) -I(cD) and, hence, that (cD)-1 (cD)/(cD,cd) is a topological


semigroup. Then, since D[-]D/S(D,d) is a group and y is an epi-


morphism, it follows that (cD) -(cD)/1(cD,cd) is a topological group.


As in the proof of the first part of the theorem, we will use


Dubreil's result in order to obtain cD I (cD)[-l](cD)/A((cD,cd):








Card A > 1 implies that AC L(cd)c Scd and thus using (2.3) we

[-1] (-1) (-1)
find that (cD) (cD)C (cd) AC (cd) (Scd). If d' e D


and if d' = d, then (cd) (cd) is nonempty since d d is non-


empty and if d' X d, then d' = dt for some t e T and hence


cd' = cdt so that in either case cd' = cdt' and t' is in


D )DC D [-I]D (cD)[ (D); the other inclusion is clear because


(cD) C(D) (cd) (cD) implies that ed[(cD) (cD)] c cD.


Therefore, cp[(cD) (cD)] is homeomorphic to cD since mcp = l(cd)


where m is the appropriate homeomorphism, namely, g, of (1.8).


2.18 Proposition. Under the hypotheses of (2.17), if T is


a subsemigroup, then B, D and cD are contained in A(-slices.


Proof. T2 CT implies that V = f(x,y) e S x S; H(x) = H(y)}


and so for b e B, % = [x e S; H(x) = H(b)). It then follows that,


since B c R(b) n L(b) = H(b) for b e B, we have B c Hb. Clearly,


D C H. because D C B.


Next we notice that cD CA c L(a) for a e A and, in par-


ticular, cD c L(cd) for d e D. Also, we find that










cD c cB C cR(b) = R(cb) for b e B so that cD C R(cd) for d e D.


Consequently, cD c L(cd) n R(cd) = H(cd) for d e D and it follows


easily that cD C H d.



2.19 Theorem. Suppose S is compact or discrete and let us


define K = Hw fl n J where w e S and where H and J are nonempty


sets in S satisfying these three conditions:


(i) H c R(h) for all h e H,


(ii) J c R(x) n L(x) for all x e J,


(iii) HH(l) = HH[~] and JJ(- = Jj[-l.


If card J > 1 and if K is both nonempty and closed, then this dia-


gram is analytic:









KK-1 K
rk


where KK Y[-]/R(Kk), k e K, is a topological group, s is the canonical


map and f is a homeomorphism.








Proof. All the results preceding (2.17) may be easily "dualized"


so that this theorem may be proved in a manner analogous to the proof of


(2.17).



2.20 Theorem. If the hypotheses of part (a) of (2.17) and (2.19)


hold and if d is in D n K, then we naturally speak of the results of


(2.19) as being the "mirror image" of the results in part (a) of (2.17)


in view of this analytic extension of Dubreil's diagram,


KKC[-1/ C (dS)d(-1)/2 h- d(-1)(Sd)/ = D[-~1D/5





[-1] C (dS)d(-) ~ dS n Sd d(-(Sd) D D[-I]D
rd Id

where for brevity MI = DI (K,d), fl = 01(d), = a(d) and 8 = (9D,d)


and *' is the restriction of to KK-/lV l and similarly for cp'.


Moreover, if D = K, then in the diagram the restriction of h to


KK[- I/V is an iseomorphism with image D-]D/S.


Proof. The first part of this theorem follows easily from the


results of (1.8), (2.17) and (2.19). In addition, from (2.19) we see










that the restriction of f to KKE--ll/T is K and from (2.17) we find


that g- (D) = D[-I]D/ and from Dubreil's result we recall that

-l
h = g f is an iseomorphism,so that putting these remarks together the


conclusion follows because h(KKL[-]/T)= g-lf(KK-l]/)= g(K) =


g- (D) = D[-D/S.



In view of its position in the diagram, an iseomorphism such as


that expressed in (2.20) is known as "turning the corner."




The next theorem has been presented in its algebraic context


for T = S in [1] and it formed the cornerstone of that work. It is


in view of this last remark that we attach the author's name to the


theorem in its presentation. This result is also important to me


because it served as the prime motivation of this dissertation. Subse-


quent to this theorem it will be shown that a well-known theorem, due


originally to M. P. Schutzenberger and to A. D. Wallace in its present


formulation, follows in part as a corollary.


2.21 Lemma. H (-)H = H [-1 H for w in S.
w w w w








Proof. The reader may find the proof of this result in [4]


where it is shown to be a consequence of Green's Lemma.



2.22 Theorem (Bastida). Let S be compact or discrete, T be


a closed subset of S and D = c )H n H If card H > 1,
x y x

card H > I and D is nonempty, then this diagram is analytic:



D[- 1]D ld






D [-D/2(D,d)


n Ic

if

(cD) -(cD)/(cD,cd)






(cD) (cD)- CD
l(cd)

where d is in DD [ D/e(D,d) and (cD) -l(cD)/I(cD,cd)' are topo-


logical groups, qp and oa are canonical maps, g and m are homeomor-


phisms and y is a continuous epimorphism.









Proof. We will easily verify that the hypotheses of (2.17) are


fulfilled, where H and H will be A and B, respectively. In view of
x y

(2.21) and because H and H are W-sllces,we have that (i), (ii) and
x y


(iiI) of (2.17) hold. If S is compact and T is closed, then H and


H are closed so that c(- )H is closed and, consequently, D is
y x

closed. Therefore, it may be seen that all the hypotheses of (2.17) are


satisfied and hence this proposition now follows as an immediate corollary.



2.23 Theorem (Schutzenberger-Wallace). If S is compact or


discrete, if T is a closed subset of S and if y is an element of


S such that card H > 1, then H is homeomorphic to the topological
y y

group, y H /9(H ,y), and the groups y H y/1(H ,y) and
y Y y y

H y M? (Hy,y) are Iseomorphic.


Proof. Using the dual of (2.21) we see that card H y 1 Implies


that H H [-] is nonempty so that letting H = H in (2.22) and c
yy y x

be an element of H H -, we have D = c lH n H H because
YY y y y

H C c ()H The first part of this theorem now follows as a cor-
y y

ollary to (2.22) since we have that y(-l H H -'H from [4].
y y y





28



In a similar manner we may choose an element w In H [-IH
y y

so that the set K of (2.19) and (2.20) is H Therefore, by (2.20),


we may turn the corner and find that y (-1) (H y,y) and


H Vy-, /(H ,y) are iseomorphic.













CHAPTER III


A CHARACTERIZATION OF THE CLOSED SUBGROUPS
OF THE SCHUTZENBERGER GROUP



3.1 Proposition. If b is in S and A is a subset of S,


then with regard to the statements

(-1)
(1) b A is a semigroup,


(2) b(- )A C [b(b(-)A)][-1]A,

(-1) [-I]
(3) b A c A A,

(-1)
(4) b A c (x e s; Ax = A),


(5) A = Cb and C2b c Cb where C c S


the dependency is indicated by the diagram,


(4)


(3) ----(2) (1) .


(5)


Moreover, if A c bS, then (1) implies (3); consequently, if b e A


and b (-l)A is a semigroup, then b (-l)A = A [-I]A.










(-1)
Proof. If x and y are elements of b A, then condition (2)

(-1)
implies that b(xy) = (bx)y e A so that b A is a semigroup; con-

(-1)
versely, if x and y are elements of a semigroup b A, it then

(-1)
follows that b(xy) = (bx)y e [b(b A)]y C A and so
] ( -D ,[-1] (-1) il- ]
y e (b(b A)] A. It is easy to see that b A c A A implies

(-1)
the validity of condition (2) because b(b A) is a subset of A and,


since it is always the case that {x; Ax = A} c A A, it is clear that


condition (4) implies condition (3). If condition (5) holds and

(-1)
x e b A, then (Cb)x = C(bx)C CA = C(Cb) c Cb so that x is in

[-1]
A A and condition (4) is satisfied.

(-1) (-1) (-1) (-1)
b A is a semigroup means that (b A)(b A) c b A


so that, multiplying by b on the left and using the fact that AC bS,


we obtain A(b A) c A and, consequently, b AC: A A. If, in


addition, b is in A, then it is clear that b A = A A.




It is possible to indicate, as shown by the following examples,


that the implications among conditions (2) through (5) of (3.1) may not


be reversed:










3.2 Examples. (a) Consider the semigroup S defined by the


multiplication table, 0 1 2 and let b = 1 and A = f0,2}.
0 0 0 0
1 0 0 1
2 0 1 2

(-1)
Then b A = [0,I] which is clearly a subsemigroup so that condi-


tion (2) holds and yet b (-l) is not a subset of A -I1A which is


fO,23.


(b) Using the semigroup S defined in (a) and letting b = 1,


A = [0,1) and C = f0,3} we have that A = Cb and C 2b = Cb whereas

(-1)
b A = S and Ix e S; Ax = A) = f2} so that neither condition (3)


nor condition (5) implies condition (4).



(c) If we let a semigroup S be defined by the table


0 1 2 3 and if b = 1 and A = (0,1,2), then fx e S; Ax = A) =
0 0 0 2 2
1 1 1 3 3
2 2 2 0 0
3 3 3 1 1

(-1)
b A = (0,)3; however, the only set C such that A = Cb is A


itself and, in this case, we find that C 2b = S. Consequently, neither


condition (3) nor condition (4) implies condition (5).









3.3 Theorem. If S is such a semigroup of a compact group


that S is either open or closed, then S is a closed subgroup.


Proof. This result is well known and the reader is referred


to [7].



3.4 Theorem. Let S be a compact or discrete semigroup, T be


a closed subset of S and y be such an element of S that card H > 1.


If G is a closed topological subgroup of the Schutzenberger group,


y ( H y/(H ,y), and if w is an element of Hy, then, letting

-l
Aw = w(cp~ G) where cp is the homomorphism of Dubreil's diagram, it is


true that p-1 G = A A = A- A and A is topologically equivalent
w w w w w

to G. Conversely, if A is a nonempty closed subset of H such that


A ]A = A (-A, then for a in A the following diagram is analytic

t [-l]
and, as a result, A = cp(A A) which is a subgroup of the Schutzen-


berger group:








a (-I)A/(A, a)


/ a /Ha*H ,a)
a a a

(S Sa1) p a"


/ a U (Sa)/a(a)
9 g (p *\ \\

A C H C as n Sa la a^^CSa) M aG')H a > A1[1



where the primes and double primes indicate that those functions are

restrictions of cp and g.

Proof. We will consider only the case where S is compact

because the situation where S is discrete follows in a similar manner

with the topological results omitted.

If G is contained in the Schutzenberger group, namely,

cp(w (-H ), for any w in H then gp- G c w lH for if x is

in w(-1) (Sw) and cp(x) e G, then cp(x) = cp(y) for some y in

w(-1)H so that wx = wy e H and x e w -lH ; moreover,
y y y

Cp- G = w(-)Cw((p-G)] because if x is in w(-l)[w(rp- G)] we have

wx e w(cp G) and yp(x) e cpy((-C) = G so that x e mp- G and the reverse









set inclusion is clear. Letting A = w(cp- G) we have that cp G =
w

(-1) -I
w A and so, since cp G is a semigroup, we may use (3.1) to


obtain cp G c: [w(w-)A )] A =[-A Since w(-w is non-
w w w w


empty, it follows that cp(w (-w) is the identity of the Schutzen-


berger group, as may readily be seen by a proper modification of the

-l
proof of (2.17), so that cp G contains an element q such that wq = w.


Consequently, w e A and as a result G = A [-]A Because A <
w w w w


Hw c R(w), the restriction of the lw function of Dubreil's diagram

[-1]
to A A has as its image A and therefore, by Dubreil's result,
w w w

[-i]
G = cp(Aw A ) is homeomorphic to A It may be noted that A is
w w w w


closed because w H is compact. If x e cp G and a e A then
y w

card H > 1 implies that wx = atx = wt'tx where t e T n w(-l)H
Y Y

because H H =w H and t' e T n cp-1G because cp-1G = w(-)A .
y y y w

Therefore, p(x) = cp(t'tx) = cp(t')cp(t)cp(x) and since cp(t') and cp(x)


have group inverses in G, it follows, letting cp(t')- be the inverse


of cp(t'), that cp(t')- = p(t) e G and so t e cp- G and, consequently,


w(cp G) c a(cp G); moreover, ax = wt'x and so we have the reverse set









inclusion, namely, a(cp- G) c w(cp- G). As a result, for each a e A ,


it follows that cp-G = a(-l)[a(Cp IG)] = a(-l)[w(cp-G)] = a(-l)A and,
w
[-1] (-1)
consequently, A A = A A


Conversely, if A is a nonempty closed subset of H such that

[-1] (-i) [-I] (-1) (-1) (-1)
A A = A A, then, for a e A, A A = a A c a H c a (Sa),

[-1]
the last inclusion being true because card H > 1, so that cp(A A) r7

(-1)
cp(a H ), the Schutzenberger group, where To is the appropriate


canonical map in Dubreil's result. Since A is closed, in view of (3.3),

[-1]
we have that cp(A A) is a group. (In the discrete case, a somewhat


longer argument, similar to that used in the proof of <2.17), shows


that cp(A A) is a group.) Lastly, since A A = a A and


A c R(a), the restriction of the la function of Dubreil's diagram

[-1]
to A A has as its image A and, therefore, using Dubreil's result,

[-1]
cp(A A) is homeomorphic to A.



3.5 Proposition. If A and B are subsets of S such that A

[-1is oid and connected nd B s a component of (-1)B.
is nonvoid and connected and B is a component of S, then A B = A B.










Proof. If y is an element of A B, then Ay is connected and


so it follows that B U A(A B) is connected. Then, since B is a


component, A(A (-I)B) c B and we have that A (-)B c A B. The fact


that A is nonempty is used to ensure that A B c A B.


(-1)
3.6 Corollary. If A is a component of S, then A A =

[-l]
A A. Consequently, if S is compact or discrete, then for a set


A contained in an A(-slice having cardinality > 1 to be a component


of S it is necessary that A be homeomorphic to the topological

[-1]
group, cp(A A), where cp is the canonical map of Dubreil's diagram.


Proof. This result is immediate in view of the 3.4 and 3.5


Theorems.



If we consider a semigroup in which two distinct points a and


b are contained in a component, then by letting A = fa) it is easy to


see that the converse of (3.6) is not true. Moreover, if we look at


a totally disconnected space with cardinality > 1 which has the multi-


plication xy = x, it is easy to see that the weaker converse,


namely, A (-A = A- A implies that A is connected, is also false


because in such a semigroup equality holds for any nonempty subset.













CHAPTER IV


ALGEBRAIC RESULTS CONCERNING RI-SLICES



To determine whether a subset A in a compact semigroup is


an W(T)-slice, for a closed set T, is important because,if so, then


A is homeomorphic to a topological group according to the Schutzen-


berger-Wallace Theorem. One of the results of this chapter reduces


the investigation of ((T)-slices in commutative semigroups to those


subsets T which are subsemigroups and another result gives necessary


and sufficient algebraic conditions for a set A to be an /(T)-slice


if T is a subsemigroup. What constitutes such necessary and sufficient


topological conditions remains an open question.




It is well known that a semigroup is a group if and only if it


is an p1-slice so that, in particular, if a semigroup is not a group,


then it is not an A(-slice (for any T c S). Moreover, as the following









example indicates, the algebraic conditions on a subset, say A, with


cardinality > 1 may be relaxed further and A need not be an W-slice.



4.1 Example. Let S be a semigroup containing more than two


elements with multiplication xy = c for some fixed c e S and let


A be any subset of S containing more than one element such that c / A.


Clearly, A is not a semigroup and, since each element is its own


A(-equivalence class (for any T) in such a semigroup, A is not an


l-s 1 ice.


The previous example also shows that the conditions A A =


A A and AA = AA are not sufficient for a set A to be an


W(-slice.




In general, what constitutes necessary and sufficient condi-


tions for a subset of a semigroup to be an W(-slice for some T remains


an open question; however, if T2 c T we can specify such algebraic


conditions as indicated in the subsequent theorem:









4.2 Lemma. Let T c S and A be a nonempty subset of S and


consider the following conditions:


(1) If a and b are distinct elements of A, then b e aT n Ta.


(2) If a e A and x e S\A, then at least one of the following


four sets is empty: T n a-lx, T n xa-I, T n x-a, T ax-


Then (a) if A is contained in an /'(T)-slice, then condition (1)


holds.


(b) If condition (1) holds and T2 T, then A c H (T) for


a e A.


(c) If condition (2) holds, then H (T) c A for a P A.


(d) If H (T) c A for a e A and T2 c T, then condition (2)
a


is true.


Proof. (a) Suppose A C H (T) for a e A. If card A = 1,


then condition (1) is satisfied vacuously; if card A > i, then


condition (1) is immediate.


(b) If a and b are distinct elements of A, T2 c T, and


condition (1) holds, then L(a) = a U Ta = tb U Ttb c Tb c L(b) for








some t e T and, similarly, L(b) c L(a) and R(b) = R(a). Thus


H(a) = H(b) and, since T2 C T, (a,b) e j.


(c) If Ha(T) 9! A, i.e., there exists an x e S\A n H (T) so


that L(x) = L(a) and R(x) = R(a), then the sets in condition (2) are


all nonempty.


(d) If condition (2) is not true and T2 c T, then for some


x e S\A and for some a e A all the sets in condition (2) are nonempty


and L(a) = L(x) and R(a) = R(x). Thus H(a) = H(x) and, since


T2 c T, x e H (T) so that H (T) V A.



4.3 Theorem. Suppose T2 c T c S. A nonempty subset A of S is


an W(T)-slice if and only if the following conditions hold:


(1) If a and b are distinct elements of A, then b e aT n Ta.


(2) If a e A and x e S\A, then at least one of the following

(-1) (-1) (-1) (-1)
four sets is empty: T n a x, T rxa T n x a, T n ax .


Proof. In view of the lemma, this result is immediate.









If c is an element of a semigroup S such that xy = c for


all x,y e S, if T is a subset of S and if b / c, then fb) Hb(T)


and yet there exists no t e T such that bt = b. Hence, this example


indicates that the word distinct may not be omitted from condition (I)


in the 4.3 Theorem nor may it be removed from condition (1) as it


applies in part (a) of (4.2).



If we recall the definitions of the functions la and ra from


Chapter I, then it is possible to formulate (4.3) in functional


notation:



4.3' Theorem. Let T2 c T C S. If the domain for the func-


tions la and ra is T, then a nonempty subset A of S is an !(T)-slice


if and only if the following two conditions are satisfied:


(I') la[(la)- (A\a)] = ra[(ra)- (A\a)] = A\a for each a e A.


(2') If a e A and x e S\A, then at least one of the following


four sets is empty: (la)- (x), (ra)- (x), (lx)- (a), (rx)- (a).


Proof. It suffices to show the equivalence of the conditions


of the 4.3 and 4.3' Theorems. Since (la)-(x) = T a(-l)x, it is








evident that conditions (2) and (2') are the same because in a similar


manner qualities for the other three sets may be obtained; and so


it remains to exhibit the equivalence of conditions (1) and (I'):


If la[(la)- (A\a)] a A\a for all a e A and if b and c are


distinct elements of A, then b e A\c implies the existence of an


element t e (lc)- (A\c) such that ct a b. In a similar manner


ra[(ra)- (A\a)] = A\a for all a e A implies that b e Te.


If a and b are distinct elements of A and if b e aT, say


b = at for t e T, then t e (la)- (b) and la(t) = b so that A\a c


la[(la)- (A\a)]. Since it is always the case that la[(la)- (Aa)] c


A\a and since it is easy to see that, in a similar fashion, b e Ta


implies that ra[(ra)-1(A\a)] = A\a for all a e A, we have that


condition (1) implies condition (1').




We conclude this chapter with a result which reduces the


study of V-slices in commutative semigroups to those sets T for which


T2 c T and T = ) Consequently, (4.3) takes on added significance


since it deals with subsets T which are subsemigroups.








4.4 Lemma. If A is such a subset of a semigroup S that card


A > 1, A [-]A = A A, condition (1) of.(4.2) holds for some subset


TC S and A is normal in that T, that is, xA = Ax for all x e T, then


A is an W((T')-slice where T' is the semigroup generated by T M A [-I]A.


Proof. For distinct elements a,b e A we have [T n (ba(-1) U


a G b)] cT' so that condition (1) of (4.2) holds when we replace T


by T'. Therefore, since T' is a semigroup, it follows from part (b)


of (4.2) that AC H (T') where a e A. Now if x e H (T'), then,


because T' c A [-IA, we have that x U xT' = a II aT' c A and so


Ha(T') c A.



4.5 Theorem. If A is an W(T)-slice which is normal in T and


if card A > 1, then A is an ((T')-slice where T' is the semigroup


generated by T n A [-]A. As a result, in a commutative semigroup S


to determine if a subset A of cardinality > 1 is an V(-slice for some


T, it is sufficient to investigate the W(-slice decompositions yielded


by the subsemigroups of S.





44



Proof. This is an immediate corollary to (4.4) because the

I-'] (-
hypothesis that A is an A(T)-slice implies that A A = A A


and that condition (1) of (4.2) holds.













B IBLIOGRAPIIY


[1] Bastida, J. R. Group Homomorphisms Associated with the
Yi-Equivalence in Certain Semigroups. Doctoral Dissertation,
University of Georgia, 1963.

(2] Clifford, A. H., and Preston, G. B. The Algebraic Theory of
Semigroups. Math. Surveys 7, Amer. Math. Soc., Providence,
1961.

[3] Wallace, A. D. Relative Ideals in Semigroups. I. Colloq.
Math., 9 (1962), 55-61.

[4 ] Relative Ideals in Semigroups. II. Acta Math.,
14 (1963), 137-148.

[5] Algebraic Topology. Lecture Notes, University of
Florida, 1964-65.

[6 ___ Project Mob. Lecture Notes, University of Florida,
1965.

[7] Wright, F. B. Semigroups in Compact Groups. Proc. of Amer.
Math. Soc., 7 (1956), 309-311.



Supplementary Read ings


Bastida, J. R. Grupos y homomorfismos asociados con un semigrupo,
I, Bol. Soc. Mat. Mex. (1963), pp. 26-45.

_____ .. Grupos y homomorfismos asociados con un semigrupo, II,
Bol. Soc. Mat. Mex. (1965), pp. 7-16.

Sur quelques groups et homomorphismes de groups associes
a un demi-groupe. C.R. Acad. Sc., t. 256 (1963), 1648-1649.














BIOGRAPHICAL SKETCH


Anthony Connors Shershin was born October 16, 1939, at Clifton,

New Jersey. Having graduated from Saint Leo Preparatory School at

Saint Leo, Florida, in 1957, he then entered Georgetown University in

Washington, D.C., and in June, 1961, received the degree of Bachelor of

Arts from that institution. He enrolled in the University of Florida

in September, 1961, and completed the requirements for the degree of

Master of Science in April, 1963. After an absence of one year, Mr.

Shershin returned to the University of Florida in September, 1964, and

completed the work for the Doctor of Philosophy degree in August, 1967.

For both graduate degrees the major was Mathematics and the minor was

Physics. During his graduate studies he taught at the University of

Florida in the capacities of graduate assistant and part-time interim

instructor. From September, 1966, to June, 1967, he taught at the

University of Miami in Miami, Florida, while he worked on his doctoral

dissertation.

From May, 1963, to September, 1964, and for the summers of

1965 and 1966, Anthony Connors Shershin worked as an Operations

Research Analyst for Autonetics, a division of North American Avia-

tion, Inc., at Anaheim, California.









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.


August, 1967



Dean, College of Arts and Sciences




Dean, Graduate School


Super- isory\Comm i ttee-



Chairman







A'k K- Da au^




























































UNIVERSITY OF FLORIDA


3 1262 08556 7385