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RESULTS CONCERNING THE SCHUTZENBERGERWALLACE 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, PierreAntoine 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 Yl = 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 gl 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 righthand 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)) = glf(*(bc)) g (bca) gl (amn) = m(mn) = cp(m)cp(n) = g lf((b))gl 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 onehalf of the possible leftright 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 Wtslices which are truly necessary for the validity of the arguments. Lastly, it is shown that the SchutzenbergerWallace 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 = cl) 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 DI1]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 A1]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(AI1]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 DL1]D and (D,b) = S(D,b') for b' e B. Proof. In view of (2.5) the hypothesis implies that DI1 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(D1]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 D1D 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)1l (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)111(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 Bl1]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 c1)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 = BIB. 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: KK1 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 KKEll/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)= glf(KKl]/)= 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 wellknown 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 Wsllces,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 (SchutzenbergerWallace). 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 p1 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((pG)] 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 cp1G because cp1G = 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 cpG = a(l)[a(Cp IG)] = a(l)[w(cpG)] = 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 RISLICES 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 bergerWallace 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 p1slice 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 Wslice. 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 ls 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 alx, T n xaI, T n xa, 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 Vslices 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 YiEquivalence 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), 5561. [4 ] Relative Ideals in Semigroups. II. Acta Math., 14 (1963), 137148. [5] Algebraic Topology. Lecture Notes, University of Florida, 196465. [6 ___ Project Mob. Lecture Notes, University of Florida, 1965. [7] Wright, F. B. Semigroups in Compact Groups. Proc. of Amer. Math. Soc., 7 (1956), 309311. Supplementary Read ings Bastida, J. R. Grupos y homomorfismos asociados con un semigrupo, I, Bol. Soc. Mat. Mex. (1963), pp. 2645. _____ .. Grupos y homomorfismos asociados con un semigrupo, II, Bol. Soc. Mat. Mex. (1965), pp. 716. Sur quelques groups et homomorphismes de groups associes a un demigroupe. C.R. Acad. Sc., t. 256 (1963), 16481649. 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 parttime 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 