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TOPOLOGICAL mGROUPS By ROBERT LEE RICHARDSON A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1966 Digitized by [ie Inlernel Archive T62 l&iggr.fding from University of Florida. George A. SmaIners Libraries wi[n support from LYRASIS anrid [ne Sloan Foundation http://www.archive.org/details/topologicalmgrou00rich ACKNOWLEDGEMENTS I would like to thank Dr. F. M. Sioson, the Chairman of my Supervisory Committee, for his assistance and patience throughout the preparation of this disser tation. I am grateful to Dr. John E. Maxfield and the University of Florida for providing the necessary finan cial assistance. I would also like to thank Mr. Charles Wright whose initial encouragement and assistance was crucial to me. Finally, I would like to thank my wife, Dee, with out whose constant encouragement this work would not have been possible. TABLE OF CONTENTS DEDICATION .. . .. ACKNOWLEDGEMENTS . . INTRODUCTION . . . Chapter I. SOME PROPERTIES OF ALGEBRAIC mGROUPS II. TOPOLOGICAL mGROUPS . . III. AN EMBEDDING THEOREM . . IV. THE UNIVERSAL COVERING mGROUP . BIBLIOGRAPHY . . . BIOGRAPHICAL SKETCH . . Page ii . iii . 1 . 26 . 37 . 51 . 63 a . . . INTRODUCTION In this paper some of the theory of topological mgroups will be developed with some new contributions to the theory of algebraic mgroups. The main result is that any topological mgroup can be considered as the coset of an ordinary topological group. While almost no work has been done in the field of topological mgroups, an exten sive theory of mgroups has been developed through the years. In 1928, W Dornte [4] introduced the concept of an mgroup as an extension of a group or 2group which has as basic operation one that is polyadic instead of dyadic. Previous attempts had been made at this, notably by E. Kas ner in an unpublished paper, but his work indicated that he still considered the system to have a basic dyadic op eration of which the polyadic operation was merely an ex tension. Dornte was the first to publish a paper consider ing an algebraic system in which the basic operation was polyadic with no underlying dyadic one. In 1932, D. H. Lehmer [10] introduced the concept of a triplex, which is an abelian 3group in Dornte's ter minology, apparently without knowledge of Dornte's work and proceeded to develop a theory of these triplexes. In 1935, G. A. Miller [13] obtained a result for finite polyadic groups stating that every finite mgroup is the coset of an invariant subgroup of some ordinary group. Unfortunately, he makes the tacit assumption that the set of elements in the mgroup comes from an ordinary 2group initially. A major advancement in the field of mgroup theory was achieved by E. L. Post [15] in 1940 when he proved that any mgroup (finite or infinite) is the coset of an invari ant subgroup of an ordinary group. In addition, he proves in his paper that most of the 2group concepts, with the notable exception of Sylow's theorem, can be extended to mgroups. In 1952, H. Tvermoes [25] introduced the concept of an msemigroup and did a little work on them, but his main interest was again in mgroups. In 196365, F. M. Sioson introduced the concepts of topological msemigroups and topological mgroups. In pa pers by him [17], [18], [19], [20], [21], [22], [23], var ious generalizations of many theorems in 2semigroups to msemigroups have been achieved. In a paper with J. D. Monk [24], it is shown that any msemigroup can be embedded in an ordinary semigroup in such a way that the operations in the msemigroup reduce to those in the containing semi group. A representation theorem for msemigroups then re sults. 3 The general theory of such algebraic systems has also been the subject of study in papers by L. M. Gluskin [6], [7] and D. Boccioni [2]. CHAPTER I SOME PROPERTIES OF ALGEBRAIC mGROUPS In this chapter we will develop some elementary ideas concerning algebraic mgroups including a new proof of the Post Coset Theorem [15]. With the exception of [16], very few results on mgroups with idempotents have been published; however, some new theorems concerning mgroups with idempotents are obtained in this chapter. One of the important problems arising in the study of mgroups is the question of which congruences determine a submgroup. In this chapter a set of necessary and suf ficient conditions will be given. Notation 1.0: In the sequel, a sequence of juxta i+k posed elements x. ... x will be denoted by x. and if k+1 x. = x = ... = xi+k = x simply as x k+1 denoting the number of times the element x occurs. Definition 1.1: An msemigroup is a set A and a function f: Am> A such that for all al, ..., a2m1 E A, f(f(al, ..., am), am+l, ..., a2m1) = f(al, f(a2, ..., am+ ), am+2, *., a2m1) ... = f(al, a2, ..., am1, f(am' ."' a 2m1)). Following customary usage, we shall write [a1 ... am] = [am] for f(al, ..., am) and (A, []) for (A,f). Proposition 1.2: For any k = 1, 2, ..., (A, []) is also a k(ml)+l semigroup (A, ()), where (ak(m)+l = [m .. ] k(ml)+l [[...[a ]...a(k1) (m1)+1 + For example, a 2semigroup A is a k(21)+l = k+l semigroup for any k = 1, 2, ... If k = 2, then A becomes a 3semigroup by defining (ala2a3) = [[a a2]a3]. Definition 1.3: An msemigroup (A, []) is an m group iff for each i and for all al, ..., ai1, ai+1, ..., am, b E A there exists uniquely an x E A such that [a xam I = b. The following are :ome examples of mgroups. Example 1.4: Let R be the set of negative real numbers and define []:(R) 3> R by [xyz] = xy.z, the ordinary product of x, y, and z. Then (R []) forms a 3group. Example 1.5: Let SI, 2, ..., S m be any m1 sets of the same cardinality. Let F(S1,..., Sm1 ) be the family of all bijective and subjective functions f: U1 S. U m S. such that f(S.) = S where p is any fixed per i= 1 1 p(i) mutation of 1, 2, ..., ml. Let []: Fm> F be defined as the composition of m functions, i.e. [f ...... fm (x) = (f..... (fm(x))). (F,[]) is an mgroup since it is clear that [] is massociative and if x E S., f (x) E Sp(i), f1 (fm(x)) E S (p(i)),...... (f "' (fm(x))) which is an element of S where pm(i) is the mth permutation of i. Pm (i) mi th p (i) is i, so the m permutation of i is p(i). Hence, [f1 ... f : S> Sp(i) so that [f ...1 f ] E F. Unique solvability follows from the fact that the functions are bijective and surjective. Example 1.6: Let Z be the integers. Then (Z, []) forms an mgroup when []: Zm> Z is defined by (xl ... x ] = x + ... + x + h for any h E I. Example 1.7: More generally, if (G,.) is an ordi nary group and if h is in the center of G, (G,[]) will be an mgroup if []:Gm> is defined as [x ... x] = xl'*x2 **..m.h Example 1.8: Let 2, 2m1 = I be any complex (m1)th roots of unity. Define []:m > t by [a1 ... am] = m1 aI + a2 + ... + a m (C,[]) forms an mgroup for it is clear that all conditions except massociativity are satis fied, and that condition being satisfied is apparent if the following two expansions are studied: [[a ]a 2m1] = [am] + m i+ I am+l + ... + a21ml = a1 + a2t + ... + ai + ai+1 + m1 i+1 m1 ... + a+ 1+ + + a m+i+l + ... + a m . iim 1i1 i+m i [air i+m 2m] = a + a + ... + ai + [a i+mi l i+1 i+m+l 1 2 + i+1 a i+m+l + + ... + aml 1 = aI + a2 + ... + aiI + i+m+i1 i+m+li m ai+(mi)i+mi1 + ai+(m+li) + ... + a2ml  i+m+li1_ Equality is apparent by noting that a i+(m+li) = a E1m = a M+ 1 ?= a F m+1 r+1 am+l. Example 1.9: Let Zodd be the odd integers under the operation [x1 x2 x3] = xl+ x2 + x3. Then Zodd is a 3group. *Example 1.10: Let V1, V2, ..., Vm1 be finite dimensional vector spaces of the same dimension n. Let L(V1, ..., V m1) be the set of all (m1)tuples of non singular linear transformations (Al, A2, .... Am 1) where A.:V > Vp(,) for the permutation p = (12...ml). L is 1 1 2 an mgroup under the operation [(A1, ..., A _)(A2, ..., 2 m m 1 2 m 1 2 m2 Am1...(A, ..., Am1] = (A1 A2 ...A1, A2 A3 ...Am1 m m 1 1 2 3 mn A1 A2 ... ,A A2 A ... Am ). Associativity is clear, 1 2 m1 1 2 m1 and unique solvability follows from the condition that the linear transformations be nonsingular. Definition 1.11: An (ml)tuple (el, e2, ..., e 1) of elements e. E A is a left (right) madic identity iff mm1 for all x E A, [em' x] = x ([xe1 ] = x). When (el, e2, ..., em1) is both a left and right madic iden tity it is simply called an madic identity. Proposition 1.12: For any a, el, e2, ..., em1 E A, if [el a] = a([ael1] = a), then (el, e2, ..., em_) is a left (right) madic identity. Proof. Let x E A be arbitrary and [el a] = a. Choose a2, ..., am E A such that [aam] = x. Then x = [aa2] = [[el a]a2] = [e1 x]. The other part is proved in a similar fashion. Proposition 1.13: Every left madic identity in an mgroup is also a right madic identity and conversely. Proof. Let (el, e2, ..., e m_) be a left madic identity. Note that [ae e2[e el]] = [a[eml1 e e J1] 1 m1 1 1 1 2 = [aeml]. From the definition of an mgroup it then fol lows that [e em11 = e which by Proposition 1.12 im mi 1 m1 plies that (el, e2, ..., em_ ) is also a right madic iden tity. Proposition 1.14: If (e,, e2, ..., em_1) is an  adic identity, then for each i, (el, ..., em1, e1, ..., e _I) is also an madic identity in the mgroup. Proof. If (el, e2, ..., em_) is an madic identity, then [e1 e2...em1 e11 = eI and hence by Propositions 1.12 and 1.13 it is also true that (e2, ..., e 1, el) is an m adic identity. By a repetition of this argument, the re sult follows. Definition 1.15: An element x in the mgroup will be called an idempotent if and only if [xm ] = x. Proposition 1.16: If x is an idempotent in an m group A, then (x, x, ..., x) (m1) times is both a left and right madic identity. Proof. (x, x, ..., x) (ml) times is a left and right identity on x; hence, by Proposition 1.12, for all z E A. By Proposition 1.13, (x, ..., x) (m1) times is both a left and right madic identity. Definition 1.17: The inverse of an (m2)tuple (xl, x2, ..., Xm2) of elements from an mgroup is the unique element x m1 also denoted by (x x2, ..., x m_2) such that (x x2, ..., xmI1) is an madic identity. We note that such an element always exists by the definition of an mgroup and Proposition 1.12. Definition 1.18: Let S1, ..., Sm be any (m1) sets. An madic function on S, ..., Sm1 is a function m1 m1 f: U S. > U S. i=l 1 i=l 1 such that f(S S (i) where a = (12...m1). Proposition 1.19: The family of all surjective and bijective madic functions on sets Sl, ..., S m1 of the same cardinality forms an mgroup. Proof. See Example 1.5. Definition 1.20: Two ktuples (k < m) of elements from an mgroup A are equivalent, i.e., (a i+, ..., ai+k) (bi+1 '.., bi+k), iff for all x ..., x, +k+, ..., xm E A (0 < i, i+k < m), [i i+k m ] r i i+k m m x1ai+1xi+k+1 = [xli+i i+k+1 Note that by the above definition (al, ..., am) m (bl ..., bm) if and only if [am] = [b ]. Proposition 1.21: (ai+1, ..., ai+k) (bi+l' . b i+k) iff there exists c ..., ci, c++. ..., c E A such that i i+k m i i+k m [clai+lci+k+l] = c i+ici+k+1 Proof. Let di+, ..., dm, d2, ..., di+k 6 A such that (di+1, ..., dm_1, cl, ..., c ) and (ci+k+l, ..., cm' d ..., di+ ) are madic identities of A. Then for each xsl, ., x xi+k+l', , xm E A, [xi i+k m xlai+ i+k+1 1 i m i i+k1 m i+k m = i+clai+ i+2 i+kCi+k+l2 xi+k+l r i[mlr i i+k m rri+k m I = [xI di+1 clai+lci+k+ilEd2 xi+k+1 Sim1 i i+klr m i+k = [xtdi b d]bi+k1 [b+cm di+k xNM S i+l i+l i+2 i+k i+k+12 +k+l Sii+k m = LX1bi+iXi+k+l Proposition 1.22: a is an equivalence relation. Proof. That a is reflexive and symmetric is clear. Suppose (ai+1, ..., ai+k) (bi+1 **..., bi+k) and (bi+1' ..., bi+k) a (ci+l, ..., 'i+). By Definition 1.20, for i i+k m any xl, ..., xi, xi+k+l ..., xm t A, [x a i+xi+k+l Sib i+k m i r i+k m i xlbi+ixi+k+l = [xlci+xii+k+I1. Hence, (ai+, ..., ai+k) S(ci+l, ..., Ci+k). Let S. = Ai /. Note S = A. Theorem 1.23: Any mgroup is isomorphic to an m group of bijective and surjective madic functions on dis joint sets S1, ..., Sm_. Proof. Let F(S, ..., S 1) be the family of all surjective and bijective madic functions on Si = A /a, m1 m1 i = 1, ..., ml. For each a E A, define L : U Si> U S. i=l i=1 such that La((Xl, ..., x )/a) = (a, xl, ..., xi)/2, i = 1, ..., m2, and La((xl, ..., xm_1)/*) = [ax 1 a 1 ix i This is well defined since if (xl, ..., xi) a (y' ..., yi), then also (a, xl, ..., xi) u (a, yI, ..., yi). Suppose La((xlp, ..., x)/) = La((Y "... Yi)/) so that (a, xl, ..., xi) a (a, y ', ..."' yi) i m1 i m1 and hence [ax ai+1] = [aylai+1] for some ai+, ..., a 1 E A. Thus, by Proposition 1.21, (xl, ..., xi)/E = (y' ... yi)/e; that is to say, La is bijective. Let (y,l ..., Yi+)/ E Si+I. Then for ai+2, ..., a E A, there exists uniquely (by definition of an mgroup) an x E A such that i m i+1 m [ay2xai+2 = [y1 ai+2 ]. This means that La((y2, ... yi, x)/a = (yl' ...," yi+)/'" Thus La is also surjective. Define f: A > F(SI, ..., Si ) such that f(a) = La Note that ([am], xl, ..., xi) a (al, [amxl], ..., x ) [a2x...., ..., x. ) (a2, [a.3x ], ..., xi) ([axI ], x) (ai, [a x ]), ([ai+1xl]) a (ai+ ..., am, xI ...o, x ). Thus, L[am]((xI, ..., xi)/s) = ([am], xl, ..., x )/a =La([ax], x2, ... x)/s) = ... = L a2 ..L ((x ..., xi)/). 1 2 ^m Whence f([am]) = L = LL ...L = f(al)f(a2)...f(am [al ] 1 2 m f is also onetoone. For, if La = f(a) = f(b) = Lb, then La((xl, ..., xm_)/m) = Lb((xl, ..., Xm_)/a) and therefore [ax ] = [bxl ] or a = b by Definition 1.3. Next we shall prove the Post Coset Theorem. Other proofs may be found in Bruck [3] and Post [15]. The ana logue of this theorem for msemigroups has been proved by Los [ll] for m = 3 and later for arbitrary m by Sioson and Monk [24]. Theorem 1.24: (Post Coset Theorem). Let (A,[]) be an mgroup. Then there exists a group (G,) and normal subgroup N of G such that G/N is cyclic and A = xN with [am] = a *a2* *an for all a,,a2,...,am E A. In fact, G/N = tA,...,A 1), N = Am1 and the order of G/N divides m1. Proof. By the representation theorem for mgroups, every mgroup is isomorphic to an msubgroup of the mgroup of surjective and bijective madic functions on Si,..., Sm_. Let G be the group generated by A. Note that for a fixed a E A and any element b = al...am1 E Am there exist uniquely x, y E A such that Sm1 i [aI x] = a1...a x = a [yam ] = yal...am = a. Thus every element of Am1 can be expressed as ax or y a for any fixed a E A and x, y E A. Thus, if b1 = y1 a, b2 = y2 a are any two elements of A then blb2 = 1 1 1 I 1 1 (yl a)(y2 a) = yl1 aa y2 = yl y2 is also an element of A This means A is a subgroup of G. Note also that rM1 rm1 l rn1 ,i for each a E A, aA m1= A = A a. Thus a A a = Am = aA a for each a E A. Since A generates G, then any g E G may be written as g = all...a in for a. E A and i = n im 1 or 1. Then g A g = A and A is a normal sub ri la mi 1 group of G. From aA = A we obtain a aA = a A = A m. Thus a Aml = (a A)Am2 = A m2. Similarly, for l 1 m1 l m2 nm3 al, a2 E A, a1 a2 A = a A = A and so on. Thus G/Am1 = A U A2 U A3 U ... U Am1. Some of the Ai's may be equal. In any case, the order of G/Am1 is a divisor of m1. Definition 1,25: G will be called the containing group of A and A m the associated group of A. If the order of G/A is exactly m1, we shall say that G is a covering group of A. E. L. Post [15] has shown that this can always be achieved for any mgroup by considering a free group generated by the elements of the mgroup. Tne following theorem due to Sioson [20] will prove useful in the sequel. Theorem 1.26: The following conditions for an m semigroup are equivalent: (1) A is an mgroup; (2) For all i = 1, ..., m, for al, ..., ai1, ai+l, ..., am, b E A, there exists an x E A such that [al xa i+ = b, i.e., [ai1Aa = A; (3) For some i between 1 and m, for al, ..., a b E A, there exists an x E A such that [a xa. ] = i 1 i+l b, i.e. [a i Aa+] = A. (4) For each al, a2, ..., am_1 E A, [am A] = A = [Aa1]; (5) For each a E A, [aAm1] = A = [Amla]; (6) For all a., ..., am2 E A, there exists a 1 E A such that (al, ..., am1) is an madic identity. Proof. (1) implies (2) implies (3) implies (4) im plies (5) are obvious. (2) implies (1) by definition of m i1 m iI m group; for, if x, y E A such that [aI xai+l] = [aI Yai+l' then for some elements b., ..., bm_ c2, ..., c. E A, i m1 21 (bi ..., bmn_l, al, .., ail), (ai+1, _., am, c2, ..., ci) are madic identities and hence x = y. (5) implies (2). Let (5) hold and al, a2, ..., a E A. Noting A = A, then Si1 m i1 2m1 m m [a Aai+l = [a A ai+ Si2 rmli rmi m = [a1 [a A ]A[A a+ 4a ] ] = ie1ai+l i+2] = [a2 A a+2] = ... = Am = A. (2) implies (6) as we have already seen. (6) implies (4). We shall show the existence of an x such that [xa ] = b for any a2, ..., am, b E A. Since such thtxa [Am] = A, then we may write the above equation as rrr 1 1 1b 22 b2 m2 m2 i.nm2 [[...[[xb b ...bm]bb2...b m1...]b b2 ...b ] = b. By applying (6), then m2 m2 m2 1 2 2 2 I 1 x = [b[bl ,b2 ,...b m_1.. .[b,b2,...b m_1[bl,b21... b1 . mi Definition 1.27: A subset S of an mgroup (A, []) is called a submgroup iff S is closed under the same operation [] in A and for each xl, x2, ..., Xm2 E S, (xl, x2, ..., Xm2) E S. Proposition 1.28: If S is a submgroup of an mgroup A, then H = S is a subgroup of the containing group, in fact, of the associated group of A. Proof. Let A be an mgroup with containing group G. 2 M1 As before, we may assume A c G = A U A u .... U Am Let S be a submgroup and H = Smi c G. Since S is a subm group of A, for each x x2, ..., m2 S, there exists (x x, ..., x ) = x 1 E S such that (x x2, ..., x m) is an madic identity. Hence x x2 ... Xm1 as an element of G is the identity and is also an element of H. Next, let x1, x2, ..., xm1 be m1 elements of S. Then for any y, y2l ...' Y m2 6 S there is a ym1 E S such that ([xmy]' Y2' ... Ym_) is an madic identit" which im plies then that (xlx2 ... Xm1 (y12 .. Ym_1) is the identity of H. Hence yly2 ... ym1 is the inverse of x x2 ... x 1 an arbitrary element of H. Also note that H.H = Sm1 Sm1 = Sm Sm2 = Sm1 = H. Thus H is closed both under the binary operation in G and inversion and hence is a subgroup of G and hence of A m, the associated group of A. Definition 1.29: A submgroup S of A is called in variant iff [a' S ai+.] = S for each madic identity (al, a2, ..., am_1) of A and each i = 1, 2, ..., m2 and also [a1 [ a 2]] = S for each a a2, ..., a2_r2 such that ([am], a ..., a ) is an madic identity. Definition 1.30: Let S be a submgroup of A. If the associated subgroup of S in the containing group G of A is invariant in G, then S is called semiinvariant. Proposition 1.31: Every invariant submgroup is semiinvariant. Proof. Let S be an invariant submgroup of A with the associated subgroup H = S in the containing group G = A U A2 U .... U Am1 of A. Let x C G so that x = aa2...a. for some i < m1 and al, a2, ..., a, E A. If i < m1, then there are a.i+, ..., a m E A such that i (a,, a2, ..., am_1) is an madic identity. Thus x = a. .... a i is the inverse of x in G. S is invariant, i mi so aI S a i+ = S. Since the group operation gives the m Srm1 1 mi Iml group operation, then H = S = (a S a )i+ nm1 i m2 (a a .... a)(S a1 a)2 S(ai+ .... am) = rm1 i i x S x = x H x If i = m1, then there exist a ..., a E A such that ([am], am+,, ..., a2m_2) is an madic identity and hence also (a2m2, [am], a +, ..., a2m). Note also that then 2m2a1 m+1 2m2 ([a2m2am]am, ..., a2m3) and (am, ..., a23 [a am1 ]) are also madic identities. Then I x = a a ... a_, x = a a m+ .... a 2m2 and Sm1 m 1 2m2 ml ( H = S = (arI (S am )) = (a1 a .... ar) " ( 2m3 m2 smi 1 (Sam (a a )) S (a... ) = xS x Sm 2m2 1 am 2m2) = l x H x Thus in both cases H is invariant in G. Proposition 1.32: A submgroup S of an mgroup A is semiinvariant iff [a Sm1] = [Sml a] for all a E A. Proof. Suppose [a Sm 1 = [Sm1 a] for all a E A. Let H = S m1 be the associated subgroup of S and G the l containing group of A. Thus in G we have a H a = H. Since a E A, then a = (a2, a3, ..., a m1) such that (a, a2, ..., a ) is an madic identity. Since G is generated by A, then H is normal in G and the result fol lows. Theorem 1.33: If S is a semiinvariant submgroup of A, A/S exists and is an mgroup. Proof. Let H be the associated subgroup of S and G the containing group of A. First note that A = U(y yExH]. Consider the set [xH xEA). Since H is normal in G, if x 1 x2, either x 1H = x2H or they are disjoint. An mary operation may now be defined on the set (xHIxEA) by defin ing (x1H x2H ... xmH) = [xm]H. This operation is associ ative for ((x H ... xmH)x m+H ... x2m1H) = ([x ]H x H ... x H) = [[x] x2m1 ]H = [x [x i+] 1 m+1 2m1 1 m+1 1 i+x [x [x i+mxm 1H = (x H ... x i[x+mH ... x H) Xi+l ]xi+m+r"l' xiLxi~1H +*i+m+H X2m1H) = (x1H ... x H(xi+1H ... xi+mH)x i+m+H ... x2m1H). Next, it is necessary to show that for any xEA there exists aEA such that for x,, x2, ..., xmEA, (x H ... xH a H xi+H ... x H) = xH or [xi a X+l ]H = xH. Since A is an mgroup, this can clearly be done. The set (xHIxEA) is A/S. As opposed to a 2group, a congruence relation on an mgroup need not determine a submgroup. Consider the con gruence relation of equality. This will determine a subm group if and only if the mgroup has an idempotent. The problem will now be formalized and a set of necessary and sufficient conditions given for a congruence relation to determine a submgroup. Definition 1.34: A relation R on A is a subset of A X A. Definition 1.35: The domain TT,(R) of a relation R on A is the set nl(R) = (x: (x,y) E R for some y E A) and its range is the set "2(R) = [y: (x,y) E R for some x E A). Notation 1.36: If U is a subset of A, denote RU = n (R n (A x U)) UR = 12(R n (U X A)). Definition 1.37: A congruence relation R on an m group A is a relation which is reflexive, symmetric, and transitive, and such that if (xl,yl) E R, ..., (x ,ym) E R then ([x1,[y1]) E R. Theorem 1.38: Let R be a congruence relation on an mgroup A. Then R* = (A x A) U [(x,y) x,yEA for some i = 2, ..., m such that x = e ...e. x', y = e ...e. y' implies (x',y') E R) is a congruence in the covering group of A. Proof. Let (el2,e2, ..., e m_) be an madic identity of A. Note that each x E G is either an element of A or can be uniquely written in the form x = e1e2...ex' for some x' 6 A and i = 1, 2, ..., m2. For each x, y E G define (x,y) E R* if and only if either x, y E A or x, y E A for some i = 2, ..., m1 such that x = ele2...e ix' and y = e1e2 ...eily' with (x', y') E R. Reflexivity, sym metry, and transitivity of the relation R* are clear. Sup pose (xt, yt) E R* (t = 1, 2) so that either xl, y, 6 A or x = ele2 ...ei 1x Yl = eie2...e yl with (x', Yl) E R and either x2, y2 E A or x2 = le2e...e ejx2, Y2 ele2...e jlY with (x,' y) E R. Obviously, x x2, y1y2 Ak for some k = 1, 2, ..., m1. Thus it suffices to consider the case when k is greater than 1. Let x lx2 = ele2...ek_x' and yly2 = e e2. *.ekly. We shall show (x', y') E R. Ob serve x [m1 ki ,1 m1 m1 i1 j1 x = [ekel x ] = [ek x1x2 = [ek el xel x ] m r o1 k1 m1 m1 i1 j* and y' = [e el y'] = [ek 1Y2] = [ek el'lyeJly]. Since (xi, y) E R, (x2, y2) E R and R contains the diagonal of R, then ([emk ei x'e x2], [ee ei. Ye1 Y12]) E R. Whence (x', y') E R; and (x x2, y1y2) E R*. Lastly, we also show that (x, y) E R* implies (x y ) E R*. Suppose (x, y) E R* so that either x, y E A or 4 = e ...e. ix' and y = el...eilY' with (x', y') E R. We shall consider two cases. In case i = m2, so that x = ele2...em3x' and y = ele2...em3y' with (x', y') E R, l 1 then x = x' E A and y = y' A. Thus m1 ,1 1 m3 ,y x = [e xo] = [[e xoJe y'y ] m1 mlx m3 Y = [e ly'] = [e [xem x yo]]. Since (x', y') E R and R contains the diagonal of A, then r mi m3 mx1 m3 1 1 ([eI xoel x yo], [el xe Y'y o) E R. Whence (x y ) = (x', y) E R. Otherwise, in case i A m2, let x = 1 ee ...e *xm and y = e e ..e As in the pre e2.emi_2xO ad ye12 m2i_2 vious case, from the fact that m1 mi2 i1 mi2 ,1 r m1 mi2 ii mi2 ([emi e ye x '], [emilel Ye y1 x) mnil 1 o1 1 o 'e ]mi) o 1 1 o E R and S m1 mi2 iix mi2x] and Yo emile [ye1 'el1 xo = emi mi2'] = em remi2 ily e mi2 S emilel o =emi1 iel Yoel y'el~e2 xo rmI mi2 ii, mii2 S[emi le 2oel y1 e1 xo]], it follows that l 1 (xo', y) E R and hence (x, y ) E R*. Since el ..em1 is the identity in G and R* is a congruence in the group G, then e e2. .e 1R* is a normal subgroup of G. Observe also that e e2...e 1R* = (ele2...em2)(em1R). If x E (ele2...em2)(em1R) so that x = ele2...em2x' where (x', em_) E R, then (x, e e2...er1) E R*. Thus, the right side of the above equality is contained in the left. If (x, ele2...e m1) E R*, then x = ee2...em2x' for some x' E A with (x', em1) E R and hence also x E (ele2...e _m2)(em1R). The above equal ity is thus demonstrated. Theorem 1.39: Let R be a congruence relation on an mqroup A. The congruence class em1R of R is a subm qroup of A if and only if there are elements el, e2, ..., em_2 E e mR such that (el, e2, ..., em_,) is an madic identity. Proof. If emlR is a submgroup of A, then obvious ly it contains elements el, e2, ..., em_2 such that (el, e2, ..., em) is an madic identity. Conversely, suppose em,R contains elements el, e2, .., em2 such that (el, e2, ..., eml) is an madic identi ty. By the preceding theorem, R and (el, e2, ..., em,) determine a congruence relation R* on the covering group G of A whose congruence class of the identity e e2*...em1R* = (ee2...e m2)*(e m1R) = N is a normal subgroup of G. We shall show that em_R = (emle...e m2). (e mR) = e mN is a submgroup of A. It is clearly a subset of A. The closure of e ,R = e ,N is also clear, for, if x. E e1 R for i = 1, 2, ..., m, so that (xi, em,1) E R, then ([xm], [(e )m]) E R. Since also (e., e ) E R (i = 1, 1 ml i m1 2, ..., m1), then (e [(e )m]) = ([emle S ml 1 m1 [(e )m ]) E R. Hence ([xm], e ) E R and therefore m1 1 m1 [xm E e R. Finally, let x, x2, ..., x m e R 1 mI 1 2m ..., m) a m e 1N. Then x = e1 n (i = 1, ..., ml) and [(em_ ln) (em1n2)...(emlnml)(em1N)] = emln lem n2 e n e N = m[(e )m]nn ^.n N for some n', n, ., n' E N. From the steps above recall (e ,, [(e m_)m]) R or [(em_)m] E em1R = e1N so that [(e )m] = e 1n for some n E N. Hence [X1X2...xm1(em1N)] = [(emln)(em1n2) ...(emi nml)(e mN)] = [(eml )m]nn'...n' m = e m (nni'n ...n' ,N) = e mN. In an analogous manner m1 1 2 m1 m1i [(e _1N)x 2...x _] = e N. This completes the proof that e 1N = em LR is'a submgroup of A. Corollary 1.40: If R is a congruence relation on an mgroup A with idempotent e, then eR is a submqroup of A. Proof. The proof follows immediately from 1.38. Under certain circumstances it may be possible to reduce, say, a 5group to a 3group immediately by re defining the operation. E. L. Post [15] summarized this possibility in the following. Similar work has also been done by Hosszu [9]. Definition 1.41: An ni tuple (a,, a2, ..., an1) will be said to be commutative with an element x if and only if (x, al, a2, ..., anl)s(al, a2, ..., an1, x). Proposition 1.42: An mgroup is reducible to an n group, m = k(n1) + 1, k a positive integer, if and only if there exists an ni tuple (al, a2, ..., an_1) which com mutes with every element of the mqroup and such that (al, a2, ..., an_, al, a 2, ..., an_, ..., a, a2, ..., an), with the n1 tuple (al, a2, ..., an_) repeated k times, is an madic identity. The nadic operation may be defined as (x1.. .x) = ni ni [xl...x al .. aI ] where the righthand side of the above equation is the ordinary mgroup operation, and an occurs ki times. (kl) (nl)+n = knn+1k+n = knk+l = k(nl)+l so the operation is properly defined. Proof. If there exists an ni tuple (a,, a2, ..., an) satisfying the stated properties, Condition 2 of Theorem 1.26 is immediately satisfied since the mgroup has this property. Furthermore, the operation is associa n 2nl n) 2n1 n1 nl tive for ((x ) x = [(x) x+ a ...a ] = n n1 ni 2n1 n1 nl [ [Xn+1 n1 nl L[x a ...a1 ] x1 a ...a ] = Lx[x a ...a 1 1 1 n+1 1 1 1 2 1 1 2n1 ni n1 n+l 2n1 "n1 xn+2 al ...a = (X(X2 ) n+2 = = [2nl n1 nl n1 nli n1 2nl [x a ...a a ...a = (x (x ) n i 1 1 i n Next, if the mgroup is reducible to an ngroup, the ngroup has an nadic identity. (al, a2, ..., an, al, a2, ... an1, a a2, an_, ..., al, a2, ..., anl) k times will then be an madic identity and that (al, a2, ..., an,) commutes with every element of A is clear by Proposition 1.13 and the fact that A reduces to an ngroup. Corollary 1,43: Every commutative mgroup with idem potent reduces to a 2group. Proof. It is immediately apparent that the condi tions of the preceding theorem are satisfied. Example 1.6 is an example of an mgroup, m > 2, that Pages Missing or Unavailable CHAPTER II TOPOLOGICAL mGROUPS In this chapter we derive some of the properties of topological mgroups and quotient mgroups. The following definition is due to F. M. Sioson. Definition 2.1: A topological mgroup (A, [], .) is an mgroup (A, []) together with a topology T on A under which the functions f and g defined by f(x1, x2, ..., xm) = [xm] and g(x x2, ..., Xm2) = Xr where (x1, x2, ..., Xm_ ) is an madic identity, are continuous. Proposition 2.2: If (A, [], 7) is a topological m group, then for any k = 1, 2, ..., (A, [], T) is also a topological k(ml)+l group (A, (), 7) when we define (ak(m1)+l) = [[.*[am]*.]ak(l)m1)+l 1 1 (k1)(m)+l1 Proof. The proof is clear. Example 2.3: Let R be the set of negative real numbers with []:(R)3> R defined by [xyz] = xy.z, the ordinary product of x, y, and z. Let the topology on R be the usual topology T on R restricted to R. (R, T) is a topological 3group. Example 2.4: Let S' = (zlz = ex 0 A x < 2r ). Define []:(S')m> S' as follows: If zl, ..., zm E S' and ix ix z = e ..., = em then [zl] = e .(X1+ If 1 1 T is the usual topology on S', then (S', [], T) is a topo logical mgroup. Example 2.5: If (G, *, 7) is an ordinary topologi cal group, and if h is in the center of G, (G, [], 7) will be a topological mgroup if []: Gm> G is defined as [xT] = X *x2 Xm h. 1 2 m Theorem 2.6: The function h: A A defined by h(x) = [ai1xa+ ] is a homeomorphism for each choice of i = 1, ..., m and elements al, ..., am E A (where by con vention [a i xam is [xam] when i = 1 and [amlx] when I xai+l 2 i = m). Proof. The function h is the restriction of the map f to the subset [al)x...x(ail)X A X (ai+]x...X(a m of A x A X...X A (m times) and hence is continuous. Let bl, b2, ..., bm_1 E A such that (al, a2, ..., ail, bi, ..., b _) and (bl, b2, ..., bil, ai+l, ..., am) are madic Sii ml identities and define k(x) = [b1 xb 1]. As before, k is continuous. Note however, that hk = identity = kh. Thus h and k are inverses of each other and are both bijective. Whence, h is a homeomorphism. Corollary 2.7: Every topological mgroup is homo genous. Proof. If a and b are any two elements of a topo logical mgroup, then for elements al, ..., am in the m group such that [al 1aai+l] = b, the map h(x) = [al xai+] is a homeomorphism that takes a to b. Proposition 2.8: Let A be a topological mgroup and let Al, A2, ..., A be any m subsets of A. If A. is open for some i, then [A,, A2, ..., Am] is open. If A1, .., Am are compact, then [A1...Am] is compact. Proof. If Ai is open, then by Theorem 2.6 [al..a.i Ai Ai+l *am] is open, and [A.**.Am = U[[al''ai_ A ai+l' am]lai Am) is open. since f:(x1, ..., xm)  [x] is continuous, if each Ai is com pact, A1 x A2 x..x Am is compact, so [A1A2...A ] being the continuous image of a compact set is compact. Proposition 2.9: Let A be an mgroup with idempo tent e. Then for any neighborhood U of e there exists a neighborhood V of e such that [Vm] c U. Proof. Since f:(e,...,e) >e is continuous if U is any neighborhood of e, there exists U...UUm neighbor hoods of e with [UI...Um] c U. m Let V = n U. and we see that [Vm] c U. i=1 Proposition 2.10: Let A be a topological mgroup with S a submgroup of A. Then S with the relative topo logy is a topological mgroup. Proof. The mappings f and g in Definition 2.1 are continuous and hence the restrictions of f and g to S are continuous. Proposition 2.11: Let A be a topological mqroup and let A1, A2, ..., A be subsets of A. han (i) [Al...A] C [A1...A] S1 1 (ii) (A', ., A 2) c ((A1, ..., Am2) Sii  (X ..AT [Xi1 A. X.' (iii) [x ..A x ] = [x1 A x 1 j 1+1 1 3 1+1 Proof. (i) It is known that for any continuous function f, f(A) c f(A). (ii) Same as (i). ii m (iii) By Theorem 2.6 f:x  [xI x Xi+1] is a homeomorphism. Proposition 2.12: If H is a submsemigroup, subm group, or semiinvariant submgroup of a topological m group A, then H is, respectively, each of these. Proof. Let H be a submsemigroup of A. Then [H] c H and by Proposition 2.11 (i) [(H)m] c [Hm] c H. Next, let H be a submgroup of A. Then, as in the first part of this proof, [H"] c H. Since (H, H, ... H)1 c H, by Proposition 2.11 (ii) (H H ... H )1 c ((H, H, ... H) 1) c H. H is defined to be an invariant submgroup iff for each madic identity (el, e2, ..., em_),[e H e ] H and by Proposition 2.11(iii) [e H e +] = [e H e = H The proof is similar for semiinvariant submgroups. Proposition 2.13: A submgroup H of a topological mgroup A is open iff its interior is not empty. Every open subnqroup is closed. Proof. Suppose the interior is not empty and let em1 be an interior point of H. Then there exists an open neighborhood V of em1 with V c H. Let e, ..., em2 be a collection of m2 elements from H such that (el, ..., em) is an madic identity. (Such a collection exists since H is a submgroup.) Hence for any h E H, [h m1] = h, so [Hm1 V] = H and by Proposition 2.8, H is open. By Theorem 2.6, [xm1 H] is open for any choice of xl, ..., xm_ E A. Hence, U ([x1m H]lh 4 [x1 H] for any h E H) is open and is A\H. Hence, H is closed. Proposition 2.14: A submgroup H of a topoloqical mqroup A is discrete iff it has an isolated point. Proof. Suppose H has an isolated point x. Then there exists an open set U c A such that U n H = (x). Let y E H. Since H is a submgroup of A, there exists xl, x2, ..., xm1 E H such that [x1~ x] = y. Since U is open about x, [xm~ U] is open about y by Theorem 2.6. Since U n H = (x), (y) 6 [x1l U] n H. If y $ y E [x1 U] n H, y E [xI U] which implies that for some x E U, Y = [x1 x. Now x, x2, ..., Xm E H, y E H and H being a submgroup implies that xo E H, and therefore, that x E U n H. But x 0 x by uniqueness and the choice of y 0 Yl,a contradiction. Hence, [xm1I U] n H = [y) and H is discrete. The converse is clear, for if it is discrete, all of its points are isolated. Definition 2.15: A relation R on a topological space A is lower semicontinuous iff UR is open for every open set U in A. Definition 2,16: A relation R on a topological space A is said to be closed iff it is a closed subset of A X A under its product topologv. Definition 2.17: A relation R on a topological space A is upper (lower) semiclosed iff xR = [x)R (Rx = R(x)) is closed for every choice of x E A. The following theorem has been proved for general algebraic systems in which the congruences commute by Mal'oev [12, p. 136]. It can be shown that the congru ences in an mgroup commute but we will prove the theo rem directly. Theorem 2.18: Any congruence R of a topological mgroup A is lower semicontinuous. Proof. Let U be any open subset of A and suppose UR is not open; that is to say, there is an x E UR such that for any neighborhood V of x, we have V n (A\UR) 0]. Since x E UR, then there is a y E U such that (x, y) E R. Let x2, x3, ..., Xm1 be elements in A such that (x, x2, ..., Xml) is an madic identity so that [xxmly] E U. By m1 2y the continuity of the mary operation on A, then there exists an open set V containing x such that [vxly] c U. By hypothesis, since UR is not open, then there is a v E V such that v ( UR. Since (x,y) E R, (v,v) E R, and (x, x i) E R for all i = 2, ..., ml, then ([vxmy [vxmx]) R. However, (x2, ..., Xml, x), being a cyclic permutation of an madic identity, is also an madic identity so that ([vx y], v) E R. Note [vxmly] E U and hence ([vx2 y], v) E R n (U X A). This implies that n2([vx y, v) = v E UR, which is a contradiction. There fore, UR must be open. Definition 2.19: Let (A, [], 7) be a topological m group and let R be an equivalence relation on A. Define n: A  A/R by n(a) = aR for each a E A. n will be called the natural map. Let U be the family of subsets of A/R defined by U E 21 iff nl (U) is open in A. Remark 2.20: If R is a congruence and U E 2 then U may be expressed as (xRlx E T E T) for if U E 2 set 1 T n (U). Conversely, if T is open in A, n(T) E 9 for 2. n (n(T)) = TR which is open in A since R is lower semi continuous. Theorem 2.21: The family of sets z in Definition 2J19 is a topology for A/R. The mapping n is continuous and 2 is the strongest topology on A/R under which n is continuous. Proof. Let (uRju E T X TX E T be an arbitrary collection of sets in S. Then U (uRlu E T ) = XEA (uRju E U T ) E since U T is open in A. If XEA XEA (uRju E Til.= Ti E T is a finite collection of members of n n s, n (uRju E Ti] = (uR u E n Ti) which is in U since i=l e=l n n is open in A. Hence 1 is a topology on A/R. It is i=l clear that n is continuous. Next, let a be another topolo gy on A/R such that s c a. Let F E B be an Iopen set which is not %open and suppose n is continuous under the topology 8. Then n (F) is open in A,so suppose n(F) = T E T. Then n(T) is an element of 9 by the remark, so n(T) is Xopen and n(T) = n(n(F)) = F, i.e., F is aopen, a contradiction. So n is not continuous under the topology B and is the strongest topology under which n is continuous. Proposition 2.22: The natural mapping of A onto A/R is open. Proof. By Remark 2.20, if T is open in A, n(T) is open in A/R. Proposition 2.23: Let A be a topological mqroup with congruence relation R on it. If aR is compact for some a E A, xR is compact for all x E A. Proof. Let x E A and choose a, ..., am_1 E A such that [am1 a] = x. Observing that [am1 (aR)] = m1 = [al a]R = xR we see that xR is compact by Theorem 2.6 being the continuous image of a compact set. Proposition 2.24: Let A be a topological mgroup with a congruence relation R on A. Then A/R is discrete if and only if aR is open in A for some a E A. Proof. Suppose aR is open for some a E A. For any x E A, let a,, ..., am_l be chosen so that [aI1 a] = x. Then [am1 (aR)] = xR which is open by Theorem 2.6. Hence, if aR is open for some a E A, xR is open for any x E A. By Proposition 2.22, (yRly E xR) = xR E A/R is open. Hence, if xR is an open subset of A, (xR) is open as an element of A/R, so A/R is discrete. If A/R is discrete, (aR] is 1 open in A/R, so n1 aR) = aR is open in A. Proposition 2.25: If an equivalence relation R on a. T topological space A is closed, then A/R is a T, too logical space under its quotient topology. Proof. From Wallace's Algebraic Topology Notes [26, Cor. 3, p. 8] we know that if Y is compact, then the pro jection map n: A x Y > Y is closed. If R is closed, then R n (A x (y)) is also closed for any y E A. Hence, Tl(R n (A x (y])) is closed and A/R is T . Theorem 2.26: If h: A  A is a topological (con tinuous) homomorphism between two topoloqical mgroups and 1 A is T then the congruence relation R = hh is lower semicontinuous, lower and upper semiclosed, and closed. Proof. Lower semicontinuity follows by a previous theorem. If A denotes the diagonal relation on A, then note that hh1 = (h x h)1 (). A being T implies it is in fact T2 and hence A is closed in A X A. Hence, hh1 is also closed. By the same reasoning as in the proof of Pro position 2.25, Ry = nl((h X h)1 (W) n (A x (y))) and yR = r2((h x h)1 (A) n ((y] x A)) are closed for each y E A and hence R is both lower and upper semiclosed. Theorem 2.27: If h: A A is an open continuous epimorphism of T0 topological mgroups, then A/R where R = h*hl is iseomorphic to A under the natural mapping. __ ~_ Proof. The algebraic isomorphism between A/R and A follows from general algebra. A/R and A possess precisely the same open sets which are images of open sets in A. Hence, they are homeomorphic. Theorem 2.28: Let A be a compact (locally compact) topological mgroup. If R is a congruence relation on A, then A/R is compact (locally compact). Proof. Since n is continuous, A/R is the continuous image of a compact set and hence compact. Next, let (xR) E A/R and let U be an open neighborhood of x in A such that U is compact. Then n(U) is open about (xR) in A/R and n(U) is compact since n is continuous [8, 3.13]. Definition 2.29: A topological space X has the fixed point property if and only if for each continuous map h: X > X there exists x E X such that h(x) = x. Theorem 2.30: Let A be a topological mgroup. Then A does not have the fixed point property. Proof. Define a function h: A > A by choosing x1, ..., Xm E A such that (x ..., Xm ) is not an madic identity and letting h(a) = [xm1 a]. By Corollary 2.7, h is continuous and suppose h(a) = a for some a E A. Then [xT1 a] = a, so by Propositions 1.12 and 1.13, (xl, ..., Xml) is an madic identity contradicting the choice of (xl, ...;, Xml) Remark 2.31: If A is a topological mgroup, then A is not homeomorphic to [0,1] x...x [0,1] = [0,1]n for any n, nor is A homeomorphic to the Tychonoff cube since both have the fixed point property (5, p. 301]. Theorem 2.32: Let A be a topological mqroup, and for some x E A, let C be the component of x. Then, if [x1 x] = y, [xlm C] is the component of y. Proof. Let x and y be given as in the statement of the theorem and let xl, ..., Xm1 E A be chosen so that [x11 x] = y. Then y E [xl1 C] and [xm1 C] is connected and closed by Corollary 2.7. [xm1 C] is the component of m1 y, for if not, let K be the component of y. Then [x1I C] c K. Let x i, ... i,m2 E A be chosen such that (x, ..., Xim2, xi) is an madic identity, i = 1, ..., m1. Then [Xm1,1l.''ml,m2 1'Xl,1 '''l,m2 X1 2"...m1 C] = C X,1l... Xml,m2...xl, 1.. .xl,m2 K] which is closed and connected, contradicting the assumption that C is the com ponent of x. CHAPTER III AN EMBEDDING THEOREM In this chapter we will prove the topological ver sion of the Post Coset Theorem by showing that each topo logical mgroup can be considered as the coset of a normal (in the group sense), open, and closed subgroup of an ordi nary topological group. Several consequences of this theo rem will also be exhibited. Proposition 3.1: Let (A, []) be an mgroup with m1 A as its associated group according to the Post Coset Theorem. If (el, e2, ..., em_1) is any madic identity of A so that el, e2, ..., em E A, then for each x E A1 and any fixed index i = 1, ..., mi, there exists an ele ment a E A such that el...eilaei+...em1 = x. Proof. Since x E Am1, then x = ala2...a m for some al, a2, ..., am_1 E A. For any am E A, there exists, by definition of an mgroup, a unique a E A such that [e lae +am] = [al]. Considered in the containing group of A, the equality above reduces to el. ..eilaei+1...emlam = ala2...a a = xa and hence, e ...e. aei+. ...e = x. Proposition 3.2: Let (el, e2, ..., em1) be an madic identity of a topological mgroup (A, [], T) with S(e _) a local open basis at em_1. Then S(ele2...e ) = (ele2...em2U U E6 (emI)) is a local open basis of the identity e = ele2...em 1 of the containing group of A under some topology. Proof. It is sufficient to show closure under inter section. If U1, U2 E 1(eml), then clearly we have ele2...em2U1 n ee2...em2U2 = ele2...em2(1 n U2). Remarks. Preparatory to the demonstration that 2(ele2.. .em ) defines a topology that converts the con taining group of A into a topological group, we shall prove a series of Lemmata. Lemma 3.3: Let (el, e2, ..., em_1) be an madic identity of a topological mqroup (A, [], 7). For any i = 1, ..., m2 and each open set V containing ei, there exists a basis element U E 9(em1 ) such that ee2*...em_2U c el. 1 emi Vei+l ... eml1 Proof. For each v E V and fixed element x1 E A, there exists, by definition of an mgroup, an element w E A such that m2 i1 m1 [e1 wxl] = [el vei+lxl] i1 m1 Define two functions h and k by h(v) = [e1 vei+xl] and k(w) = [e2wxl]. Let (xl2,x, ..., Xm ) be an madic identity. Then m2 ii mi W = (w [e wxl] = [el ve'i+x ] for some v E V) = ii m1 mi (wI [eml [e ve i+x ]x2 = w for some v E V) = i1 mi m1 m1 .1 [emi Ve x x] = [e h(V)x ] = kh(v) m1 1 i+X1 12 m1 2 is an open subset of A containing em1 since both h and k are homeomorphisms by Theorem 2.6. Since 21(em_ ) is a local basis at em_1, then there i1 mi exists a U E 1(e ) such that U c W. Whenc e [ei Ve" xl] = Si+m 1I [e2WxI] D [em2 Ux1]. Considered in the containing group this gives e ...e. iVe. +...e xml e ...e m2Ux1 and hence e 1...ei1Vei+l...em_1 3 e...em2U. Notation. In Lemmas 3.4 and 3.5 let f and g be the functions of Definition 2.1. Lemma 3.4: For each U E N(em1), there exists a V E (em_l) such that (e e2...e _2V)(ele2...em_2V) c ele2 erm2U Proof. Let U E 9(e 1) and define a function m2 h: A x A > A by h(x,y) = [xel y]. Being a restriction of the continuous function f on A X (e1) X...x (em_2) X A, l h is therefore also continuous. h (U) is thus an open subset of A X A. Since the first and second projection maps Vl and V2 are open maps, then V = nlh (U) n T2h (U) is also open. It is also nonempty, since h(em_,em,) = [emle1] = em E U and therefore em E V. We now claim (le2.2 for, if 2 E V, then (ee2...e m2V)2 c ee2...e m2U; for, if vl, V2 E V, then h(v1,V2) E U and hence (ee2.. .em2V1) (ele2 ...em2v2) = ee2 ...e [lel 2] = ele2..em2 h(v, v2) E e e2..em2U. Lemma 3.5: For each U E 1(eml ), there exists a l V ( M(e_ ) such that (e e2...e mV) c ee2...e _U. m1 2 m2 1 2 m2 Proof. By definition of a topological mgroup, 1 g (U) is open in A X A X...X A (m2 times). Since U 6 5(em ) and therefore em_1 E U, then (el, e2, ..., em2) E g (U). Thus, there exist W1 E 2l(el), W2 E S(e2), ..., l Wm_2 E f(em2) such that W1 W2 X... W 2 g (U). Let 1 W = W W ..W Observe that W c U in the containing 1 2* m2 group. Thus 1 1 W 1 (We2...e )1 c (We ) = eW = e "e ...e U. 1 2 m1 m1 mL 1 2 m2 By Lemma 3.3, there exists a V E 9(em_) such that ele2...em2V c Wle2...e 1 and therefore (ele2.. .em2V)1 ce ele2..em2U. Lemma 3.6: For each U E N(eml ) and x E ele2.* .em2U, there exists a V E W(em_ ) such that x(ele2...em2V) c ele2" .em2U. Proof. Let x = ele2...em_2u for u E U. Define the function h such that h(v) = [ue2 v]. Since this is a homeomorphism by Theorem 2.6, then hl(U) = W is an open and nonempty subset of A. Inasmuch as em_1 E W, then there is an element V E %(e ) contained in W. Hence x(ele2...em2V) c x(ele2...em2W) = (ele2...em2u). (ee2 ...e W) = ele2...e_2 ue mW = ee ...e h(W) = 1 2 m2 1 2 m2 1 1 2 m2 ele2" .em2U. Lemma 3.7: For each U E (em ) and any element x in the containing group of A, there exists a V E M(em1) ' such that x(e e2...em_2V)x c ele2...e mU. Proof. Recall that the containing group of A is given by A U A U...U Aml. We consider two cases: Case I. Suppose x A1 so that x = a a2 ...ai. Let a i+, ..., am E A such that aa2 ...aiai+l...am_ is 1 the identity element in the containing group. Denote x = ai+. ..am_ For any fixed element a E A, let W i i e m2 ] m1 m2 1W = w [e wai+1 ai+2a] E [e Ua]J. Since [e 2Ua] is an open subset in A (in fact a basis ele ment at the point a E A), then, being the inverse under a homeomorphism, W is also open. Since em1 E W, there exists a V E U(e 1) such that W contains V and hence r m2 m1 i m2 m1 m2 [al[e vai1 ]ai+2a] c [al[e1 Wai+]ai+a] = [el Ua]. Thus, after simplication in the containing group of A, we 1 obtain x(ee2...e m2V)x = (a ...a.)(e e2...e V)* (ai+l...am1) c ele2...em2U. Case II. Suppose x E Am so that x = a a2...am1 for al, ..., am 1 E A. Let am, .., a2m2 E A such that ([a], am+l, ..., a2m2) is an madic identity and denote x = a a ....a 2. For any fixed element a E A, let Smi m2 2m2 m2 W w[[ai [e wa]] am+1 a] E [e Ua]. The rest of the proof proceeds as in Case I. Theorem 3.8: Any topological mgroup is the coset of a topological group G by a normal subgroup N with the property that G/N is a finite cyclic and discrete topolo gical group. If m > 2 and A does not reduce to a 2group. then G is disconnected. Proof. Lemmas 3.4 to 3.7 showed that 8l(ele2...em_) is an open basis at the identity of the containing group G of A which converts G into a topological group [8, 4.5] with basis (xU: x E G, U E W(ele2...em_)) or (Ux: x E G, U E (ele2 ...em1). Since A in the containing group is clopen, then each coset of G by N is also clopen (being homeomorphs of A). If m > 2, then G is disconnected and G/N is discrete under its quo tient topology. It thus remains to show that the topology defined in G by )(ele2...em1 ) gives rise to the same topology on A when restricted or relativized. By homogeneity of A, it suffices to consider the basis at any point, say em_1 E A. If U is any basis element of A containing em_1, then el ele2...em2U is a basis element in G of e = ele2...e m2em . Hence, eml(ele2...em2U) = U is also a basis element of G at e Thus, every basis element of A at e 1 (and hence at any point) results from a basis element of G. This means that the topology induced on A by the topology in G coincides with the original topology on A. Corollary 3.9: A_ T topological mqroup is always completely regular and hence Hausdorff. Proof. By Theorem 3.8, any topological mgroup A is the coset of a topological group modulo a normal sub group N which is homeomorphic to A. Since A is To, then N is T and hence completely regular. Thus A is also completely regular. Theorem 3.10: Any compact T topological mqroup A is homeomorphically representable as an mgroup of madic homeomorphisms. Proof. Observe that the topological mgroup is in fact completely regular and hence Hausdorff (Corollary 3.9). By the representation theorem for algebraic m groups, the given mgroup A is isomorphic to a submgroup of the mgroup of madic functions F(S1,S2, ..., Sm1) on the sets S. = A x...x A/m (i times). Since A is compact Hausdorff, then A x...x A (i times) is compact Hausdorff and hence also Si under its natural quotient topology. m1 Thus U Si is also compact Hausdorff under its sum topology. i=1 m1 Any bijective continuous function on U S. is also i=l a homeomorphism and thus the collection of all such func m1 tions H( U Si) is a group (of homeomorphisms). By a result i=l m1 m1 of Arens [1, p. 597], since U Si is compact, then H( U Si) i=l i=l is a topological group under the compact open topology and the operation of composition. Since F(S, S2, .,Sm_) c m1 H( U Si), then for any fl f2' .*.. f E F(Sl, S2', .' Sm_), i=1 the functions (fl f2' fm) fl f2"''fm [f= ] and 1 1 1 1 (fl' f2' "'.. fr2)  (fl.f2*f...m_2) 2 2 3"f "m2 1^2* m2m2 m3*f 1 are continuous. Thus F(S1, .*, S1 ) is a topological m group under the compact open topology relativizedd). Consider now the regular representation m1 h: A > F(S1,S2, ..., S) c H( U S) i=1 defined by h(a) = La. The following Lemma will be needed. mi m1 Lemma 3.11: The function k: U S > U A = G such i=l i=l that k((al, a2, ..., ai)/a) = ala2...ai is a continuous open map onto the containing group G of the topological m group A. Proof. Let U be any open subset in G such that ala2...ai E U. By the continuity of the multiplication in G, then there exist open subsets in G, a1 E VI, a2 ( V2' ..., a E Vi such that k(V1 X V2 X...X V /a) = VIV2.Vi c U. Thus, h is continuous. That h is open is obvious. Let L E (C, U) or a {(a,al, ..., ai)/aJ for some i = 1, ..., m1 and (al, ..., a )/m E C3 c U. Applying k on both sides, we have (aala2...ail for some i = 1, ..., mi and (a1, ..., ai)/u E C) c k(U). By Lemma 3.11 k(C) is compact and k(U) is open. Thus, for any fixed index i = 1, ..., m1 and (al, ..., ai)/! E C, by the continuity of the multiplica tion in G, there exist open subsets a E Va ...a a V, a2 E Va, ..., ai E Va such that aa .ai Vala2 .aVa Vl c k(U). Thus, a V ...Vai k(C). a, ..., ai E C By compactness of k(C), then there exists a finite number 1 1 n n of ituples, (al, ..., a ), ...,(aa, ..., a) such that n U V k k...V k 3 k(C). k=l aI a2 ai n Let U = n v k k'** k which is nonempty since it con k=1 ala2...a tains a. Then n n U UV kV k'V k = U( U V kV k...V k) c k(U) k=l aI a2 ai k=l al a2 ai and hence n U U X V k X...x V k / U. k=l a1 a. This means, n U(C) C L( U V k X...X V k / a) c U, k=l a ai in other words, h(U) = LU c (C, U). Whence h is continuous and the final result follows. Corollary 3.12: Let (A, [], t) be a topological m group, F a compact subset of A, and U an open subset of A containing F. Then for each madic identity (el, ..., em1) of A there exists open subsets U1, ..., Umr1 with ei E Ui, i = 1, ..., m1 such that [FUm1] U [U1F] c U. If A is locally compact, then Ul, ..., Um 1 may be chosen so that ([FUml] U [fU1F]) is compact. Proof. Let G be the containing group of A. By a result in Hewitt and Ross [8, 4.10] for topological groups, for F, U as above there exists an open set V containing the identity of G such that FV U VF c U. Then e = el...em1 E V and by the continuity of the operation in G there exist open subsets U1, ..., Um1 with i E Ui, i = 1, ..., m1 such that U ...Ul c V. Thus [FUm1] U [Um1] = S...Um1 1 1 (FU ...U M) U (U...U mF) c FV U VF c U. If A is locally compact so that its containing group is locally compact, the second part of [8, 4.10] states that V may be chosen so that (FV U VF) is compact. Hence, ([FUm1] U [Um1F]) as a closed subset of a compact space is compact. Proposition 3.13: Let Al and A2 be two mgroups such that A1 is iseomorphic to A2. Then their covering groups G1 and G2 are iseomorphic. Proof. Let f: Al1 A2 be an iseomorphism and G = m1 m1 Al U...U AL G2 = A2 U...U A2 . Define g: G1 G2 as follows. If x E G1, x = x ... xi with x1, ..., xi E A1 let g(x) = f(xl)...f(x ). Then if y E G1 and y = yl...Yk with yl, ..". yk E A1, we have g(xy) = g(xl...x iy...yk) = f(xl)...f(xi)f(yl)...f(yk) = g(x)g(y) so g is a homomorphism. Next, suppose g(x) = g(y). Then f(x1)...f(xi) = f(yl)...f(yk) so i = k and if Zi+l,. zm E A1 we have [f(x1)...f(xi)f(zi+) ...f(zm)] = [f(yl)... f(yi)f(zi+l)...f(zm)] and f[x zi+1] = fly i+l]. f is bi izm izm jective, so [x 1 i+] = [yz i+l] which implies that xl...xi = Y'..Yip i.e., x = y and g is bijective. Now let U be an open neighborhood of el the identi ty of G1. Then e1 = X1 ...xM1 with (x1, ..., xm ) an m adic identity, xi E A i = 1, ..., m1. U can be expressed as U = x ...X2 U' with U' open in A1 and x m U'. g(U) = g(x1...xm2 U') = f(x1)...f(xm_2)f(U') and f(U') is open in A2 so f(x1)...f(xm2)f(U') = g(U) is open in G2 and g is an open function. In a similar manner it can be shown that g is continuous,so g is an iseomorphism. Theorem 3.14: If a topological mgroup A is compact, locally compact, acompact, or locally countably compact, then its containing group G is. respectively, each of these. Proof. Since G is a topological group, multipli cation is continuous, compactness, local compactness, and local countable compactness are clear. Since the finite union of a countable number of sets is countable and there are m1 costs of G, F compactness is clear. Proposition 3.15: Let A be a topological mgroup and let Al, A ..., A be subsets of A. If A is T and [xP'mj) = [xT] for any permutation p and any choice of xi E A, then [xP(m)] = [xm] for any choice of xi E Ai. i i t p(1) 1 Proof. Let e be the identity of the containing group G of the mgroup A. Let 1 1 H = ((al, ..., am) E G x...X Gal.amap( )..ap(m) = fell. Since A is To, G is T and (e) is closed. H is the in verse image of a closed set under a continuous function and hence is closed in G X...x G. Now A is closed as a subset of G, so H n (A X...X A) is closed. It is clear that A1 x...X A c H. Hence (A X...X A )c A X...X A H. 1 m 1 m 1 m Proposition 3.16: If A is T topological mqroup and H is an abelian submsemigroup or submqroup of A, then H is (respectively). Proof. By Proposition 2.12 H is a submsemigroup or submgroup if H is. By Proposition 3.15, H is abelian. Lemma 3.17: Let A be a topological mgroup with congruence relation R on it, If aR is compact for some a E R, xR is compact for all x E A. Proof: Let x E A and choose al, ..., am1 E A m1 m1 such that [a 1a] = x. Then [al (aR)] = xR is compact since by Theorem 2.6 it is the continuous image of a com pact set. Lemma 3.18: Let Al, ...Am be a collection of sets such that A1, ...Am_1 are compact and Am is closed. Then [A1...A] is closed. Proof. Since in a group the product of compact sets is compact, we have A ...Am1 compact in the containing group so [A1...Am] is closed in A [8, 4.4]. Theorem 3.19: Let A be a topological mgroup with a congruence relation R on A. If A/R and aR are compact for some a E A, then A is compact. Proof. By Proposition 2.23, since aR is compact, xR is compact for any x E A. Let el, ..., e 1 E A such that (el, ..., em1) is an madic identity. As shown in Theorem 1.37, N = el..em2(em_1R) is a normal subgroup of the covering group G. Since G is a topological group,and em1R is compact, N is compact. We next show that G/N is compact. Since A/R is compact and A/R = (xNlx E A), (xNlx E A) is compact. Hence, in G, y.[xNlx E A) is com pact for any y E A and y.xNlx E A) = (yxNlx E A) = (zNjz E A ). Continuing this process, we see that for any i = 1, ..., m1, (xNlx E A ) is compact so m1 i G/N = U (xNIx E A ) being a finite union of compact sets i=l is compact. Thus N and G/N are compact, so G is compact [8, 5.25]. A being a closed subset of G is also compact. Theorem 3.20: Let A be a locally compact, acompact topological mqroup. Let f be a continuous homomorphism of A onto a locally countably compact To topological mgroup A'. Then f is an open mapping. Proof. Let G be the covering group of A and G' of A'. It is shown in the proof of Proposition 3.13 that f can be extended to a continuous open homomorphism f' be tween G and G'. Since A is locally compact and acompact, G also is. Since A' is locally countably compact and To, so is G'. Thus f is an open mapping from G onto G' [8, 5.29]. Since A and A' are open subsets of G and G', respectively, the restriction of f' to A (which is f) is open. CHAPTER IV THE UNIVERSAL COVERING mGROUP In this chapter it will be shown that each arcwise connected, locally arcwise connected, and locally simply connected topological mgroup with idempotent has an arc wise connected, locally arcwise connected and simply con nected universal covering mgroup [14, p. 232]. Definition 4.1: A regular, To, and second countable space A is arcwise connected (locally arcwise connected) if and only if for each pair a, b E A there exists a continu ous function c: [0,1]  A such that cp(0) = a and t(1) = b (for each a E A and every neighborhood U of a there exists a neighborhood V of a contained in U such that for all x E V there is a continuous function :p [0,1] > U such that C(0) = a and (Il) = x). Definition 4.2: A space A is simply connected (lo cally simply connected) if and only if for each a E A (for each neighborhood U of a there is a neighborhood V of a contained in U) such that for any continuous function cp: [0,1] > A (p: [0,11 > V) such that p(0) = c(l), then p is homotopic to 0 in A (in U). Lemma 4.3: If f: [0,1]  A is homotopic to zero and cp: [0,1] > A is an arbitrary continuous function. then e*f is homotopic to cp, where c*f is defined as fol lows: Scp(2t) for 0 < t < (Op*f)t) =  f(2t1) for < t < 1. Proof: Note f(0) = p(l). Then the homotopy is effected by the following function: I 2t for 0 t < l+s 9(o +s 2 F(s,t) = f(0) for 1 s t 1. Theorem 4.4: For each arcwise connected, locally arcwise connected, and locally simply connected topological mgroup (A, [...], T) with an idempotent element e, there exists an arcwise connected, locally arcwise connected, and simply connected universal covering mgroup (A, [...], T) which is locally iseomorphic to (A, [...], T) and such that if 8: A > A is the covering homomorphism, then A/e8e1 is iseomorphic to A. Proof: Let A be the family of homotopy classes of continuous functions p: [0,11 > A such that p(0) = e. The homotopy class containing p will be denoted by $. Consider any 9 E 6 E A with cp(l) p E A. If V is an open basis of the topology T on A, then for each U con taining p in 8, let U = ($? f: [0,1] > U such that f(0) = p) and I = (I" U E 8. It is easy to see that if c is re placed by any q E U E U, then U and 4 will determine ex actly the same 0 [14, p. 221]. (1) i is an open basis for some topology i on A. If U, V E E, so that U, V E Z, then U n V E U. It is not difficult to show that U n V = 0 n V. (2) (A, ;) is a To topological space. Consider any pair p1, c2 E A such that pl 2. Let cpi E pi (i = 1, 2) such that cpi(l) = Pi. If p. 0 P2' then since A is To there exists a neigh borhood U of pl such that p2 L U. In this case, l E ri but P2 P U and hence A is also T. If p, = p2, then since A is locally simply connected, there is a neighborhood U of p, = p2 such that every con tinuous f: [0,1] > U with f(0) = f(l) = p, is homotopic to zero. If U = ([~  f: [0,1]  U with f(0) = pl), then ;2 P U; for, if c2 E U, then there is an f: [0,1] > U with f(0) = p, such that cpl*f E ,2' where *f, (2t) for 0 Note: cp(1) = f(0) = = (Cpl*f)(l) = f(l). Thus f is homotopic to zero and l = cp*f = 2, which is a con tradiction. (3) If for ci E Pi E A (i = i, ..., m) one de fines [m~2...cpm](t) = [cl(t)c2(t) ... pm(t)] and [152"'..m] = [pl2''...], then (A, [...], l) is a topological mgroup. Since [cPlP2...pm](0) = [cpl(0)2(0)...cp(0)] = eem] = e and [cpP2...p m] is a continuous function on [0,1] to A, the above operation on A is clearly well defined. Associativity follows from the following relations M 2m1 r [ilm1+ ( [m[+ ](t)' m ] =[[CPl(t)2(t). .. m (t) ]m+l (t) ... cp2ml (t)] = [C(t)... icp t) [0Pi(t).. Pi+ml(t)] cpi+m (t)'...2mI(t) = [ l (t) tpil(t) [pi(t) .. pi+ml (t) ]Pi+m(t) .. .2ml (t)] = [ t) i(t) +m(t)ci+(t) ..2ml (t)] ilr i+m1 2m1 1 i+m J(t) which holds for each i = 2, ..., m. For each ci E qi (i = 1, ..., m2), let cpm: [0,1] > A be the function such that cPm (t) = (qp(t), ..., c ,2(t)) for each t E [0,1]. Since [em] = e, so that (e, ..., e) is also an madic identity, then pm_1(0) = e. Define (Cl' ",' cm2)l = $m,. Then for each c, [CpCP *mcp = Thus far, we have shown that (a, [...]) is an mgroup. Next, we show that the functions m) [ .~. ] (' ''2) M2 1' 2 are continuous. Let cpi E i (i = 1, ..., m) such that pi(l) = p.(i = 1, ..., m). Let V be any neighborhood of [19 2 ..~Pm] = [1p2 pm] so that every element of 9 is of the form [cp 2"..E.cm]*f for some f: [0,1] > A such that f(o) = p = [P]P2" pm = ['l(1)'2(1).."m(1)] = [ m](1) v. By the continuity of the mary operation on A, there exist neighborhoods U. containing pi (i = 1, ..., m) such that [U U ...U ] C V. Then [U1U2...Um] c V, for, if 'i E Ui (i = 1, ..., m), then every 'i E i is of the form .i = pi*fi (i = 1, ..., m) for some fi: [0,1] > U. such that f.(0) = p.. Thus, i(t) i(2t) for 0 < t < f.(2t1) for a t < 1. Then [1l(t)2(t) ..m(t) ] = [cp(2t)2(2t) ....(2t) ] = [cp](2t) for 0 < t < and [l(t)2(t)..*. m(t)] = [fl(2t)f(2t...f(2t = [2t1 (2t) [ t1) for < t < 1. Since fi(t) E Ui, then [fl]: [0,1] > V. Also, if [(f](0) = [fl(0)f2(0)...fm(0)] = [PIP2..'.m = p E V, then [5l = 2[2 = pm ]*[fm] E V. S ~ ~ 1 Next consider (1,' C2' ".''' 2) = m1 where pi(l) = pi (i = 1, ..., m2) so that (Pi, P2,' .'' Pm2)~ = Pml. Let V be a neighborhood of cPm with cp m(1) = Pm_i V. By the continuity of the inverse operation on A, there exist neighborhoods V. of pi (i = 1, ..., m2) such that 1 (V1 X V x ... x V 2) c V. Again, we claim (V X X ... _2) c V. For, if (i, 2' "'' m2) E V1 X V2 X ... Vm2 and iE 6 (i = 1, ..., m2), then i = cpi*fi(i = 1, ..., m2) for some fi: [0,1] > Vi. If fm: [0,1] > V is the continuous function such that fml(t) = (f(t), f 2(t)) then f i(1) = (fl(l), ... fm 2(1)) = Pm E V. Whence m (' 2' **' m2) = l 2 'm2 = (1p P2, "'I m2) *(f1' f2' "*' fm2 = ml m1 This completes the proof that (A, [...], ) is a topological mgroup. (4) (A, D) is a second countable space. If IS1 to, then also Iji .O (5) The covering function 8: > A such that 80() = gc(l) = p is an open continuous map which is locally a homeomorphism. If U is an arbitrary neighborhood of p, then e(U) c U, obviously, so that 9 is continuous. Let c E A and U be a neighborhood of c defined by the neighborhood U of (p() = p. By local arcwise connected ness there exists a neighborhood V of p contained in U such that for any x E V there is a continuous function f: [0,1] > U such that f(0) = p and f(l) = x. This implies that cp*f E U and 9(cp*f) = (cp*f)(l) = f(2*11) = f(l) = x. Whence V c 8(U), and 8 is open. Next, let c E A and (cp) = p(1) = p E A. Since A is locally simply connected, there is a neighborhood U of p such that every continuous f: [0,11 > U such that f(0) = f(l) = p is homotopic to zero in A. 8 is onetoone on U, for, suppose 6e(,) = 2(2) so that for some fi: [0,1] > U we have fi(0) = p(i = 1,2), and cp*fl E _l and cp*f2 E 2' If f{(t) = fl(l t) so that f2*f{: [0,1] > U is the con tinuous function such that (f2*f{)(t) = f (2t) for 0 < t < _ (f2*fY)(t) = lfi(2t1) for t < 1, then (f2*fi)(0) = f2(0) = e and (f2*f')(1) = f(1) = f1(0) = e. This means then that f2*fi is homotopic to zero and therefore c*fl is homotopic to p*f2 or 11 = 12' Since 9 is continuous, onetoone, and open on U it follows then that it is a homeomorphism on U. By virtue of this, then (6) (A, 7) is also locally arcwise connected, locally simply connected, and regular. (7) (A, <) is moreover arcwise connected. Consider any $ E A and 1 E A where w contains the null path. By homogenity of a topological mgroup, it suf fices to show that W and p are connected by a continuous path. Let cp E cp and cPs: [0,1] > A be the continuous function such that cps(t) = q(st). For any fixed s E [0,1], note that ps(0) = cp(0) = e and p s(1) = cp(s) = p E A; and hence cs E A. Define now 4: [0,1] > A such that 0(s) = s. s s Then I is continuous [14, p. 223]. Moreover, (0) = 0 = and 1(1) = 1 = p. Whence A is arcwise connected. (8) (A, ;) is simply connected. As in the previous, let E A be the homotopy class containing the null path. Consider any continuous function $: [0,1] > A such that (0) = w = (1). Define cp such that cp(t) = 9((t)) where 9 is the covering map. Define also for each s E [0,1], s: [0,1] > A such that p (t) = cp(st). Note that ps also depends continuously on t and Csp(O) = c(0) = 8(I(0)) = 8(;) = e so that Vs EA. Firstly, observe that 8((s)) = cp(s) = e(cs) for each s E [0,1]. We wish to show that $s = i(s) for all s E [0,1]. For s = 0, the equality obviously holds: P = (0) = = = (0). Let U be a neighborhood of = (0) for which 9 is a homeo morphism. Since both c and (s) are continuous functions S of s by an argument used in (7), then for some k sufficient ly small we have ([Ok)) c 0 and ([O,k)) c U. For each x E [O,k) so that ;x, 1(x) E 0, then since (qx) = 08((x)) and 9 is onetoone on U, then cx = (x). This shows that Ps = (s) for all s less than k and hence by the continuity of the function ps and I(s) we obtain k = lim cp = lim i (s) = 4(k). s>k s>k Repeating the process now for k instead of 0, we should eventually show that the above relation holds for all s E [0,11. For s = 1, in particular, we have qP = C'l 4(1) = (o) = W so that q is homotopic to zero. Suppose that this homo topy is effected by the continuous function F: [0,1]x [0,1] > A so that F(0,t) = c(t), F(l,t) = cp(0) = e, F(s,0) = p(0) = e, F(s,l) = p(1) = C(0) = e. For each fixed s and t, define G(s,t): [0,1] > A to be the function such that G(s,t)(x) = F(s, tx). Since F(s, tx) is continuous in s, t, and x, then G(s,t) is also continuous in s and t. For each fixed s and t, note also that G(s,t)(0) = F(s,0) = V(0) = e G(s,t)(l) = F(s,t) = p. Thus for each s and t, G(s,t) E A and G(s,t) also depends continuously on s and t. The following relations now show that G(s,t) effects the homotopy of I and 0: G(0,t)(x) = F(0,tx) = pt(x) or G(0,t) = Ct = (t) G(l,t)(x) = F(l,tx) = p(0) = or G(l,t) = G(s,0)(x) = F(s,0) = cp(0) = or G(s,0) = 0 = 1(0) G(s,l)(x) = F(s,x) = p = S = 1(1). (9) 8 is a homomorphism of A onto A. It is clearly onto. Let ci E i E A such that cpi(1) = p(i = 1, 2, ..., m). Then 9([P1cp2...' ]) = e( 2lP2 m] = [c tp.P2.n 1 = [cp(1)C2(L)...cm(1)] = [plp2"pm] = = [e(cp )o( tc )...(Cm) ]. The following theorem proves that A is the universal covering mgroup, i.e., A is unique up to iseomorphism. Theorem 4.5: Let h:A'> A be a continuous, open. local homeomorphism, homomorphism of an arcwise connected, locally arcwise connected and simply connected topological mgroup (A', [], U) with idempotent e' such that h(e') = e onto the given topological mgroup (A, [], t). Then A' and A are iseomorphic. Proof. Let 8:A > A be as previously defined and define a map k:A' A as follows: Let p' E A' and let p:([0,1] > A' such that cp(0) = e, c(l) = p' with p a con tinuous function. Then h o cp defines a curve in A with (h o p)(0) = e and (h o p)(1) = h(cp'). Since A is simply connected,the choice of curve p(t) connecting e' and ep' is unique up to homotopy, and since h is continuous, the image under h of any two such choices will be homotopic. The function h o cp[0,1] > A and hence defines a unique point h o p in A. Define k(cp') = h o c. It is clear that k is well defined. Now let U' be an open subset of A' and let cp E U'. Let U = (cplp:[0,1] > U', cp() = cp' and each cp is continu ous) and let # be continuous with #:[0,1] > A such that #(0) = e', *(1) = cp9. Since U' is open in A', h(U') is open in A and A being locally simply connected implies that there exists an open set V in A about the point h(cp'), V C h(U') such that any closed curve in V is null homo topic, or that any two curves contained in V beginning at qp and ending at the same point are homotopic. Letting W = (h o q1p E U, h o ~:[O,1]  V such that (h o p)(0) = h(cp)), we see that (1) W c h o U = (h o Ucpp e U), (2) **(h o cp)h o cp E W) defines an open set in A, precisely the open set V, and (3) Vc k(U'). Now, (1) is clear and (2) is clear if we note that the choice of V gives only one function (up to homotopy) con necting h(cp) and any point in V and this function will be homotopic to the himage of the corresponding function in A'. V c k(U') by the simple connectedness of A' and the definition of k. Hence, k is an open function. Next we must show that k is continuous. Let cp E A' and let V be o a basic open set in A about k(pO). Since e is an open map, 9(V) is open in A and A is locally simply connected so there exists a simply connected open set W in A such that W c e(V) and 9(k(co)) E W. Then h (W) is an open set in 1  A' about cp' such that k(h (W)) c V. To see this, let *:[O,1] > A' such that *(0) = e', ((1) = cp and # is con tinuous. Let X = [cpcp:[O,l] > h1(W) c(0) = *(1) and each c is continuous). Since A' is simply connected, *,X = [(*cpCP:[O,l] h(W), cp(0) = *(1) and each cp is continuous) represents all continuous functions i:[0,1] > A' such that f(O) = e' and f(l) E H (W). Since W is simply connected, h o X = (h o cplp:[0,1] > l h (W) cp(o) = (1) and each cp is continuous) represents all continuous function g:[0,1] > W such that g(0) = (h o cp) 1  /  _ (0) = h(cp). Hence, k(h (W))= h o (*X) = (h o *)*(h o X) =e 9CV. Next, let E', c' E A' and suppose k(') = k(cp'). If k(*') = k(cp'), the functions *:[0,1] > A' and c:[0,] > A' must be homotopic and hence, #(l) = (p(l), i.e., = '. Hence, k is 11. Now let tp, ...cp E A' with associated functions pi:[0,1] > A' such that pi(l) = ci, i = 2, ..., m 1. i = 27 ... Y ll. Then k[cp{ ... ps] = h o [1 cp ] : [h o cp "... h o cp] (since h is a homomorphism) = [h o p, ... h o pm] [k(p') ... k(cp)].. Thus, k:A' > A is an iseomorphism. Recalling that the iseomorphism of A/8.1 to A was demonstrated in Theorem 2.29, we see that the truth of Theorem 4.4 has been demonstrated. BIBLIOGRAPHY [1] Arens, Richard. "Topologies for homeomorphism groups," Amer. J. Math. 68 (1946), 593610. [2] Boccioni, D. "Symmetrizazione d'operation naria," Rend. Sem. Mat. Univ. Padova, 35 (1965), 92106. [3] Bruck, R. H. A Survey of Binary Systems. Springer Verlag (Berlin), 1958. [4] Dornte, W. "Untersuchungen uber einen verallgemeiner ten Gruppenbegriff," Math. Z., 29 (1928), 119. [5] Eilenberg and Steenrod. 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"An outline for a first course in algebraic topology," duplicated at the University of Florida, 1963. BIOGRAPHICAL SKETCH Robert Lee Richardson was born February 8, 1937, in Burlington, Vermont. In 1954, he graduated from Northfield High School and went to Castleton State Col lege, graduating in 1958. From 1958 to 1960, he taught at Middlebury Union High School leaving in 1960 to attend the University of Notre Dame. In 1961, he received his Master of Science degree with major in mathematics from the University of Notre Dame and became an instructor at Norwich University. In 1962, he came to the Univer sity of Florida as a halftime interim instructor to do further graduate work, remaining until August, 1966 when he received his Doctor of Philosophy degree. Robert Lee Richardson is married to the former Eleanor Rita Dundon of Orwell, Vermont. He is the father of three children, Robert Lee, Jr., Mary Margaret, and Patrick Joseph. 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, 1966 Dean, Colleg 'ot r s and Sciences Dean, Graduate School SUPERVISORY COMMITTEE: Chairman rl " (>5L/t6 e' e &2,L7zt UNIVERSITY OF FLORIDA 1 1262 0 6 1111111111111 85II6111 11111 3 1262 08556 7377 