
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00004944/00001
Material Information
 Title:
 Topological mGroups
 Creator:
 Richardson, R. L ( Robert Lee ), 1937
 Place of Publication:
 Gainesville
 Publisher:
 University of Florida
 Publication Date:
 1966
 Language:
 English
 Physical Description:
 iv, 65 leaves : illus. ; 28 cm.
Subjects
 Subjects / Keywords:
 Algebra ( jstor )
Continuous functions ( jstor ) Homeomorphism ( jstor ) Homomorphisms ( jstor ) Mathematical congruence ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Permutations ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Dissertations, Academic  Mathematics  UF Group theory ( lcsh ) Mathematics thesis Ph. D Topology ( lcsh )
 Genre:
 nonfiction ( marcgt )
Notes
 Bibliography:
 Bibliography: leaves 6364.
 General Note:
 Manuscript copy.
 General Note:
 Thesis  University of FLorida.
 General Note:
 Vita.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 021599309 ( ALEPH )
13174724 ( OCLC ) ACX0071 ( NOTIS )

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Full Text 
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 the Internet Archive
T&^lg&Wgading from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation
http://www.archive.org/details/topologicalmgrouOOrich
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
Page
DEDICATION , ii
ACKNOWLEDGEMENTS iii
INTRODUCTION 1
Chapter
I.SOME PROPERTIES OF ALGEBRAIC mGROUPS ... 4
II.TOPOLOGICAL mGROUPS 26
III.AN EMBEDDING THEOREM 37
IV.THE UNIVERSAL COVERING mGROUP 51
BIBLIOGRAPHY 63
BIOGRAPHICAL SKETCH 65
IV
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.
1
In 1935, G. A. Miller [13] obtained a result for
finite polyadic groups stating that every finite mgroup
2
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
posed elements x^
x x .
1 1+1
i fk
x. will be denoted by x. and if
k + i 1 i
k+1
... =  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 a^,
a2ml A
f(f(ai, am), am+1, > a2ml) f(al f(a2' **' am+l)
am+2 **' a2ml) *** f*al a2
a f (a .. ,
m1 m
a2m_i))* Following customary usage, we shall write
[a^ ... am] = [a] for f(aL, ..., am) and (A, []) for (A,f)
4
5
Proposition 1.2: For any k = 1, 2, {A, []) is
also a k(ml)+l semigroup (A. ()), where (a^m =
r r r 1 fc(ml)+l i
[ [ . laiJ . Ja(k_i) (mi) +1J
For example, a 2semigroup A is a k{2l)+l = k+1
semigroup for any k=l, 2, ... If k = 2, then A becomes
a 3semigroup by defining (a^a2a^) = [[a^a^Ja^].
Definition 1.3: An msemigroup (A, []) is an in
group iff for each i and for all a^, a^_^, ai+l
a b 6 A there exists uniquely an x A such that
r i 1 m i ,
[al xai+l! b
The following are oome examples of mgroups.
Example 1.4: Let R be the set of negative real
numbers and define [ ] : (R ) > R by [xyz] = xy*z, the
ordinary product of x, y, and z. Then (R []) forms a
3group.
Example 1.5: Let S^, S2 ... sm_i t>e anY sets
of the same cardinality. Let F(S^,..., Sm_^) be the family
of all bijective and surjective functions f: >
m ]_
U n S. such that f(S.) = S where p is any fixed per
i = l 1 1 p(i)
mutation of 1, 2, m1. Let []: Fm> F be defined as
the composition of m functions, i.e. [f^ f^] (x) =
(f^ (fm(x))). (F,[]) is an mgroup since it is clear
that [] is massociative and if x S^, fm(x) Sp'jj>
f .(f (x)) Â£ S (f. ...(f (x))) which is an
mlv mv p(p(i)) 1 m'
element of S where pm(i) is the mth permutation of i.
Pm(i)
pm L(i) is i, so the m*"*1
permutation of i is p(i). Hence,
6
[ f ^ ... f m ] : S ^> S (i) so that [f^ ... fm] Â£ 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 [x^ ... x ]
= x. + . + x + h for any h Â£ I.
1 m J
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 []:G > is defined as [x^ ... x^] =
X, *x0 x *h .
12 m
Example 1.8: Let %, *2 = 1 bd any complex
(m1) roots of unity. Define [ ] : T by [a^ ... a^] =
al + a2^ + "
+ am?m ^ (CC, [ ]) 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^Ja2^ ] = [a] +
am+l^ + + a2ml
1J m+1
.m1 j ,il .
> = aL + a25 + ... + ai^ + ai+1S
+
_ml ,
+ a Â§ + a Ll Â§ +
m+1^5
i+mi 2m1
+ a
. i + 1
m+i + 1
+
i1
+ a
.m1
2ml
r_iri_2rn~ 1 i , il r i+m_i
a1 ai+1 ai+m+1 al + a2* + ** + ai? + [ai+l]^ +
c i + 1
ai+m+l^
+ a_ 1 = a. + a_Â§ + ... + a.?11 +
2ml 1 2^
pi+mi1 i+m+li1
i+(mi)'= ^^i + im+li)13
+ a
.m1
Equality is apparent by noting that aj, +(m+ii) 5
ml
2ml^
i+m+li1
a
m+1
m
? am+l?
m+1^
Example 1.9: Let Z
odd
be the odd integers under the
operation [x^ x2 x^] = x^+ x2 + x^. Then ZQdd is a 3group.
7
, V .be finite
m1
Example 1.10: Let V^, ,
dimensional vector spaces of the same dimension n. Let
L(Vlf
. be the set of all (m1)tuples of non
singular linear transformations (A^, .... Am_^) where
A^:\A> V (jj for the permutation p = (12...m1). L is
1 12
an mgroup under the operation [(A^, ... A ^)(A^, >
2
m
m1 ,m
) / nm AUt ) 1 = (a'L A
> j \ ^ 2 * ^ f *2
. m
1 ,2 _m ,1 _2
...A., A A.
. .A
m2
m1
1 2 3
, A A, hZ
' m1 1 2
,m
. .Ajjj Associativity is clear,
1 2
and unique solvability follows from the condition that the
linear transformations be nonsingular.
Definition 1.11: An (m1)tuple (e., e,
e )
m1
of elements Â£ A is a left (right) madic identity iff
"1 X] = x ([xe1
for all x A, [e1" 1 x] = x ([xe'" ] = x) When
(e^, em ]_) s both a left and right madic iden
tity it is simply called an madic identity.
Proposition 1.12: For any a, e^,
> eml ^
if [e ^a ] = a([ae ^ ] = a), then (e , .... e is a
left (right) madic identity.
Proof. Let x 6 A be arbitrary and [e
m1
] = a.
Choose a.
Â£ A such that [aa^] = x. Then x =
[aa2l] [ [e ^ a ]] = [e 1 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.
m1
ident
Proof. Let (e, e_, ..., e ) be a left madic
12' m1
ity. Note that [ae2^^ e1]] = [a[e1 e^e1'1]
8
m " 1
= [ae^ ]. From the definition of an mgroup it then fol
lows that [e e, ] = e which by Proposition 1.12 im
m1 1 mi u
plies that (e^, .., era_]^) is also a right madic iden
tity .
Proposition 1.14: If (e^ e2, e^^) is an m
adic identity, then for each i, (e^, .., em_^, e^,
e^_L) is also an madic identity in the mgroup.
Proof. If (e. e~, .... e .) is an madic identity,
1 2 m1 1
then [e^ e2'*eml el^ = el anc^ ^ence by Propositions 1.12
and 1.13 it is also true that (e, ..., e e.) 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 in
group A, then (x, x, ..., x) (m1) times is both a left
and right madic identity.
Proof. (x, x, ..., x) (m1) times is a left and
right identity on x; hence, by Proposition 1.12, for all
z Â£ 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
(xj^, x^, ... elements from an mgroup is the
unique element x also denoted by (x. xn, ..., x _) ^
m1 2 1 2 m2
such that (x., x~. .. x ,) is an madic identity. We
1 2 m1
note that such an element always exists by the definition
of an mgroup and Proposition 1.12.
9
Definition 1.18: Let S . S
be any (m1)
sets. An madic function on S,, .... S is a function
1 mi
m1
f: U S
i = l
m1
U S.
i1 '
such that f(S.) c S
i = a(i) where a = (12...m1).
Proposition 1.19: The family of all surjective and
bijective madic functions on sets S^, .
cardinality forms an mqroup.
Proof. See Example 1.5.
, S . of the same
m1
Definition 1.20: Two ktuples (k < m) of elements
li+k'
from an mgroup A are equivalent, i.e., (a^+^, > a..^) ~
bi+l 1 bi+k), iff for all x,, ..., x., x. .,
1 i' i+k+1
xra A (0 < i, i+k < m),
Note
r i i+k m i r i, i+k m i
[xlai+lXi+k+l] = [xlbi+lxi+k+l]
that by the above definition (a., ..., a ) s (b.
J 1 m l
b ) if and only if [a,] = [b.].
m J 1J 1
Proposition 1.21: (ai+i>
* ai+k) fbi+l
b. ) iff there exists c, c., c. ., c f A such
i+k 1 i i+k + 1 m
that
[<=1
i i+k m
ai+lCi+k+l
i r i, i+k m i
^ clbi+lci+k+l
Proof. Let d ,, ...,d d~ . d A such
1+1 m1 2 i+k
that (di+1, ..., dm_1, c1, ..., c.) and (ci+k+1> > cm>
d, ..., d .. ) are madic identities of A. Then for each
2 i+k
10
, x., xi+k+1, ..., xm A,
r i i+k m
Lx^, ,x.
i+l i+k+1
ir ,ml i
r J.r ,1111 X 1 i+klr m ,ltKi m 1
^xl^di + lclai + Jai+2 ^ ai+kCi+k+ld2 xi+k+l
= rrxidm_1rciai+kcm rrdi+kxm i
L Lxidi + ilc1ai+1ci+k+1l la2 xi+k+1J
rr i,ralr i, i+k m ii.i+k ra ,
= I I x. d ,,[c.b. c ,. x. ,. I
11 1 i+l i i+l i+k+lJJ 2 i+k+1
, i+ki m
r ir ,ml i, i+klr, IH i+k n HI i
xldi+lclbi+lbi+2 ^bi+kCi+k+ld2 ^xi+k+l^
= [xibi+kxm 1
L 1 i+l i+k+lJ*
Proposition 1.22: e is an equivalence relation.
Proof. That s is reflexive and symmetric is clear.
Suppose (ai+1, ..., ai+k) s (bi+1, ..., ^i+k) and (bi+1,
.... b. .. ) (c. , .... c. L1 ) By Definition 1.20, for
i+k i+l i+k J
anY X1 xi xi+k+l xm A [xai+xi+k+l] =
r i, i+k m i r i i+k m i TT .
[xlbi+lxi+k+l] = [xlci+lxi+k+l]* Hence (ai+l ai+k}
(ci+l' ci+k)*
Let S. = A1/a. Note S, = A.
i 1
Theorem 1.2 3: Any mgroup is isomorphic to an in
group of bijective and surjective madic functions on dis
joint sets S., .... S ..
Proof. Let F(S^, .., sm_^^ be the family of all
surjective and bijective madic functions on S = A1/s,
i
ml
ml
i = 1, ..., ml. For each a A, define L : U S.> U S.
cl t 1 i 1
i=l i=l
such that La((x1( ..., x^/s) = (a, x^ x^/s, i = 1,
, m2, and La((x1> ..., x^^)/) = Caxi
ml
11
This is well defined since if (x^, .., x^) a
(y ^ > Yi) then also (b ( x ^, > x ^) = (a, y i
Vi
Suppose La((x1, ..., xi)/s) = I^Uy^ ... Yi)/3)
so that
(a, xf ..., xi) a (a, y1, ..., yi)
and hence [axja+] = [ay*a+^] for soine ai+,
a^ ^ A. Thus, by Proposition 1.21, (x^, .., x^)/s
= (yni .., y.)/s; that is to say, L is bijective.
Let (y ^, > yj.+i^'/3 ^ i+1 Th^1 for ^j.+2 * *
am g A, there exists uniquely (by definition of an mgroup)
an x A such that
r i
[ay2xa
m
i+2
r i + 1 m i
1 = [yj. a+21 *
This means that L ((y, ..., y.
Thus is also surjective.
Define f: A 5> F(S^, ..
Note that
([a], xL, ... xi)
([a2Xl1' xi) B
x)/= = (yL, ... yi+1)/'3.
. S ) such that f(a) = L .
m1 a
= (a^, [a2x^], xi)
(a2, [ax^], ..., xi)
0*0
3*0* 0**00**0*3
Thus,
([axj 1], xi) a {ai, [a+Ixj]),
([ai+lx]) a (a+1, ..., am, ..., x.)
L [ am ] ( (x X xi)/s) = ([a], X^
Xi)/.
Li^ ([^2^1^* x2 f )
?
12
= L L . .L ((x., .., x)/)
ci d ~ d X X
12 m
Whence fita,]) = L = L L . .L = f(a.)f(a)...f(a ) .
i r mi a, a^ a i m
L a ^ J 12 m
f is also onetoone. For, if L = f(a) = f(b) = L. ,
a b
then La((xlf xm_1)/8) Lb((xL, x^^/s) and
therefore [ax = [bx 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 [11] 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
[a] = a]_'a2*an fr a a i a 2 aiu ^ A* In fact, G/N
= (A,...,Am ^, N =Am ^ 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 S^,...,
S L. Let G be the group generated by A.
Note that for a fixed a Â£ A and any element b =
m
al*aml ^ there exist uniquely x, y A such that
r m1 i
La. xJ=a....aTYi1x = a
1 1 ml
r m 1 1
[yax ] = ya1...am_1 = a.
Thus every element of Am ^ can be expressed as ax ^ or y ''"a
for any fixed a A and x, y 6 A. Thus, if bb = y^a,
13
~tÂ¡2 ~ Y2^a are anY two elements of A \ then =
(yL a)(y2 a) = aa Y2 = Y2 ls also an element of
Am *". This means A ^ is a subgroup of G Note also that
, nml lAml ml
for each a Â£ A, aA = A = A a. Thus a A a = A =
m 1. L
aA a for each a 6 A. Since A generates G, then any
g G may be written as g = a^...aln for a. Â£ A and i, =
in 1 k
1 or 1. Then g A g = A and A is a normal sub
,ml . 1 ,ml 1_
group of G. From aA = A we obtain a aA = a A =
m 1 1 m1 l.,m2 m2
A Thus a A = (a A)A = A Similarly, for
_ 1 l,ml 1 m2 m3 ,
a^, a^ f A, a^ a^ A =a^A = A and so on. Thus
,ml , ,2 3 ml
5 A = A 0 A U A u ... J A
Some of the Ai's may be equal. In any case, the order of
G/A ^ is a divisor of m1.
Definition 1.25; G will be called the containing
group of A and AIU ^ 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.
The following theorem due to Sioson [20] will prove
useful in the sequel.
Theorem 1.26: The following conditions for an m
semigroup are equivalent:
14
(1) A is an mgroup;
(2) For all i = 1, ..., m, for a^, ..., a^ ,
ai+l' ** am> b A, there exists an x Â£ A such that
r i 1 ra i r i 1 m i
[al xai+iJ = b, i.e., [aL Aai + ]J = A;
(3) For some i between 1 and m, for a^, ..,
i 1_ m
a b Â£ A, there exists an x Â£ A such that [a, xa.,,1 =
m 1 l+l
b9 i.e. [a^ Aaifl* =
m 2_
(4) For each a1? a2, .., am_l $ A, [a a] =
A = [Aa^_i];
(5) For each a Â£ A, [aA ^] = A = [a ^a];
(6) For all a,, .... a Â£ A. there exists
1 m2
a Â£ A such that (a., .... a .) is an madic identity,
ml 1 ml
Proof. (1) implies (2) implies (3) implies (4) im
plies (5) are obvious. (2) implies (1) by definition of in
group; for, if x, y Â£ A such that [a} xa+^] =
ya
i+1
],
then for some elements b., .... b ,, cn,
i ml 2'
(bi > bm_L ai* aii) (ai+l'
are madic identities and hence x = y.
..., c^ Â£ A,
m
, c.)
(5) implies (2). Let (5) hold and a^, a.
a^ Â£ A. Noting Am = A, then
r i 1, m i r il,2ml m i
[ a ^ Aa [a, A a ; _i_ t J
i + 1'
r i 2 r mli rm 1 1 m 1
= Ia! IaIa ai+)>i+2]
r i2,3 mi ^m
 [a^ A a j_+2 J ~ A 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,
I
a b Â£ A.
m
Since
15
[Am] = A, then we may write the above equation as
r r r r 1, 1 r i.^. i i, ni ,iu ,im. i ,
[[[[xbib2..bm_i]b1b2..bm_1]...]bL b2 bm_I] b*
By applying (6), then
c c m2 wm2 m2 1 r.2 ,2 ,2 ilr, 1 *1
[b[bL ,b2 bm_1l [oI,b2, .mlJ [b1,o2,...
2, 2
m2, m2
m2
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 x^, x2 ..., x^_2 Â£ S, (x^,
x2 * xm2 ^ S *
Proposition 1.28: If S is a submgroup of an
mgrou j 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 m 1
As before, we may assume A c G = A \J A u U A Let
ITl b
S be a submgroup and H = 3 c G. Since S is a subm
group of A, for each x^, x2 .., xm_2 Â£ S, there exists
(x,, x_, . x ~) 1 = x S such that (x., x0,
' 1* 2 m2' m1 1 2'
x ,) is an madic identity. Hence x.x ... x n as an
m1 12 m1
element of G is the identity and is also an element of H.
Next, let x., X, .... x be m1 elements of S. Then for
any y^, y2, ..., Ym_ S there is a ym_^ Â£ S such that
([x 1y1] y2 .., ym_1) is an madic identity which im
plies then that (x.x0 ... x ,) (y,y~ .. y ) is the
identity of H. Hence y,y0 ... y is the inverse of
2 2 l2 2 m1
x.x0 ... x an arbitrary element of H. Also note that
1 2 m1 2
= sm_1 Sm_I = Sm Sm~2 = Sm_I = H. Thus H is closed
HH
16
both under the binary operation in G and inversion and
hence is a subgroup of G and hence of A \ the associated
group of A.
Definition 1.29: A submgroup S of A is called in
variant iff [at S am 11  S for each madic identity (a, ,
1 l + l. x 1
a. .... a ,) of A and each i = 1, 2, .... m2 and also
2 m i
[a"1 ^ [ S a2m 2 ] ] = S for each a., a~, . a such that
L 1 m 1 2' 2m2
([a1!1], a a J 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 L31P Every invariant submgroup is
semiinvariant.
Proof. Let S be an invariant submgroup of A
with the associated subgroup H Sm ^ in the containing
group G = A U A2 U .... u A ^ of A. Let x 6 G so that
x = a,a,..a. for some i < m1 and a,, a, .... a, f
12 i = 1 2 i v
A ,
If i < m1, then there are a.,, ..., a A such that
i+l m1
(a, a ...
) is an madic identity. Thus x
1
a.,, .... a is the inverse of x in G. S is invariant,
l+l m1
so a^ S a+^ = S. Since the group operation gives the m
. TJ ml i m1, m1
group operation, then H = S (a^ S ay+^)
, m1 i. m2 ,
(ai a2 .... ai) (S ai + I aL) S^ai+1
m1
x sm ^ x ^ =xHx ^. If i = m1, then there exist
m'
, a2m~2 A such that a]> am+l
' a2m2) 1S
an madic identity and hence also
17
, r ro
a2ra2 a
m+1, .., a2m2^* Note aiso that then
, a / and (a . a ,
2m3 m' 2m3
, r mli
a2m2a1 ^ am
[a a ^ ]) are also madic identities. Then
2m2 1
xa, a0 ... a ,
12 m1
1
, x = a a
m m+1
TT ml ml/o 2m2..ml ,
H = S = (ax (S am )) = (aL a2
a_ and
2m2
a )
m1
. _ml 1
a2m2 = x S x
/r* 2m 3. m I, .m 2 ,
(S a (a_ 0 a. ) ) S (a
m 2m2 1 m
x H x 1. Thus in both cases H is invariant in G.
Proposition 1.32: A submgroup S of an mgroup
A is semiinvariant iff [a Sm_I] = [sm~L a] for all a A.
Proof. Suppose [a Sm ^] = [sm ^ a] for all a 6 A.
Let H = Sm *" be the associated subgroup of S and G the
containing group of A. Thus in G we have a H a ^ = H.
Since a A, then a = (a a ..., am_i^ such that
(a, a2,
.a ,) is an madic identity. Since G is
m1 1
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{yyxH}.
Consider the set {xHxÂ£A}. Since H is normal in G, if
xl x2 either x^H = x2H or they are disjoint. An mary
operation may now be defined on the set {xHxA} by defin
ing (x^H x2H ... x^H) = [x]H. This operation is associ
ative for ((^H ... ... x^H) =
18
.. ,,H ... x H)
i+ra+1 2ml '
it is necessary to show that for any xÂ£A there exists aÂ£A
such that for x^, x^ xm^A* (x^H X^H a H x^+^H .
x H) = xH or [x a x ]H = xH. Since A is an mgroup,
this can clearly be done. The set {xHjxÂ£A} 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 subra
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 re la tion R on A is a subset of
A x A.
Definition 1.35: The domain n^(R) of a relation R
on A is the set tt^(R) = (x: (x,y) Â£ R for some y 6 A} and
its range is the set rr0(R) = (y: (x,y) 6 R for some x A].
Notation 1.36: If U is a subset of A, denote
RU = Tt1(r n (A X U))
UR = tt9 (R n (U X A) ) .
Definition 1.37: A congruence relation R on an in
group A is a relation which is reflexive, symmetric, and
transitive, and such that if (x^,y^) Â£ R
then ([x],[y]) R.
9
(x ,y ) Â£ R
m m
19
Theorem 1.38: Let R be a congruence relation on an
mgroup A. Then R* = (A x A) u {(x,y)  x^^A1 for some i = 2,
... m such that x = e ^ . .e^ _^x y = ej_ *ey_j_Y implies
(x',y') Â£ R] is a congruence in the covering group of A.
Proof. Let (e^, eÂ£, .., em_^) ^ an m~adic identity
of A. Note that each x 6 G is either an element of A or can
be uniquely written in the form
x = e ^2 *ex '
for some x' Â£ A and i = 1, 2, m2 For each x, y G
define (x,y) Â£ R* if and only if either x, y Â£ A or
x, y 6 A1 for some i = 2, m1 such that x = e ^2 *ei]_x
and y = e^e^ . .e ^ _^y with (x y') Â£ R Reflexivity, sym
metry, and transitivity of the relation R* are clear. Sup
pose (x^, y^) $ R* (t = 1, 2) so that either x^, y. 6 A or
x. = e,e0...e. ,x,', y, = e,en...e. ,y' with (x y') 6 R
1 12 ll 1' J 1 12 ilJl 1'
and either y'2 A or XÂ£ = eie2 * *e 'ix2 ^2 =
e ^2 .e ^ _^y^ with (xj yj,) Â£ R. Obviously, x^x,, y ^y2 6 A
for some k = 1, 2, ..., m1. Thus it suffices to consider
the case when k is greater than 1. Let xx2 = efe2'eklX'
and y1y2 = e ^e2 . .e^ _^y We shall show (x y') R. Ob
serv
, r m1 k1 ,1 r m1 1 r m1 i 1 j1 i
x' = Lev e, x'J = Lek XLX2J = Le,, e, xjef x^ J
ek el Xlel X2
and y' = [ek e1
m1 k1 ,i r m1 ) r m1 i1 j1 ,
y J = Le. YnYo] = [e>. e
1 *ie
Since (x^, y^) Â£ R, (x^, y^) R and R contains the diagonal
of R, then ([ej^e^x'ej'^], [ ~ Le j ~ Ly e j1y ]) Â£ R.
Whence (x, y') 6 R; and (x^, YLY2) R*. Lastly,
also show that (x, y) Â£ R* implies (x \ y R*.
we
20
Suppose (x, y) Â£ R* so that either x, y A or
x. = e1...ei_Ix' and y = e1...ei_1y' with (x' y') 6 R. We
shall consider two cases. In case i = m2, so that
x = e ^e2 . .em_3x and y = e ^e2 .' with (x y') R,
then x ^ = x^ Â£ A and y 1 A. Thus
x^ = [eL x] = L Lex xoJeL y yQJ
, r rn1 i r mlr rn3 i i i
Yo = *el Vq J = LeL Lx^e L xyoJJ.
Since (x', y) Â£ R and R contains the diagonal of A, then
([eLx^e~3x *y] [e1xe_3y 'y]) R. Whence (x 1, y L)
= (x^, y^) R Otherwise, in case i / m2, let x ^ =
_ 1
e,e_...e ~x and y = e.e...e ~y'. As in the pre
1 2 mi2 o 1 12 mi2ro
vious case, from the fact that
ml [e;;J_1e12ye j'Ly 'e'12*;])
111 J.
e ,e,
mi1 1
Â£ R and
y e. x e.
1 o 1 1
'mi1 1
r m1 mi2r i1 mi2 l1n
Lemiiei [yoei y ei xo]], It follows that
(x, y) (z R and henee (x 1, y~L) R*.
Since e1em_i is the identity in G and R* is a
congruence in the group G, then e,e0...e .R* is a normal
12 m1
subgroup of G. Observe also that e.e0...e ,R* =
12 m1
(ele2"em2) *(emlR) IÂ£ x (ele2 * *em2 ) (emlR) 30
that x = e ]^e2 em_2x where (x em_^) R, then
(x, e^e2 ern_i) R* Thus, the right side of the above
equality is contained in the left. If (x, e^e2...e Â£ R*,
21
then x = e,e9...e _~x' for some x1 Â£ A with (x', e ) Â£ R
i. u m m i.
and hence also x Â£ (e,e~...e n)*(e ,R). The above equal
12 m2' m1 ^
ity is thus demonstrated.
Theorem 1.39: Let R be a congruence relation on an
mgroup A. The congruence class em_^R of_ R is a subm
group of A if and only if there are elements e^, ...,
e G e ,R such that (e,, e e ,) is an madic
m2 m1 1 2 m1
identity .
Proof. If em_^R is a submgroup of A, then obvious
ly it contains elements e^, en, .., em_2 suc^ that (e^, e2
. .., em_^) is an madic identity.
Conversely, suppose em_]_R contains elements e ^, e2,
..., e such that (e1f e~, ..., e .) is an madic identi
ty. By the preceding theorem, R and (e^, e2, .., emi^
determine a congruence relation R* on the covering group
G of A whose congruence class of the identity e,e....e R*
= (e.e . .e ~) (e .R) = N is a normal subgroup of G. We
L fa iii fa m
shall show that e ,R = (e ,e,...e ~) (e .R) = e ,N is
m1 m1 1 m2' m1 m1
a submgroup of A. It is clearly a subset of A. The
closure of e .R = e .N is also clear, for, if x. Â£ e .R
m1 m1 i ^ m1
for i
 1, 2,
. . m, so that
(xi
e .) R, then
m1
x],
[ (e .
m1
)m]) 6 R. Since
a Iso
R U = 1,
2
* }
, m1) ,
then (e .,[ (e
m1 m
_L)m])
, r m~i i
= ([el eml
[(em_L)m]) Â£ R. Hence ([x], e R and therefore
^Xl^ emlR Finally let x! x2 xm_L em_LR =
e .N. Then x. = e .n. (i = 1, ..., m1) and
mi i m1 i
22
[ (e n ) (e n)...(e n ) (e ,N)] = e ,ne ,nv.
m1 1 m1 2 mI m1 m1 mi 1 m1 2
e ,n ,e .N = [(e ,)m]n'n'...n' ,N for some n* n ....
m1 m1 m1 m1 1 2 m1 1' 2
n N. From the steps above recall (e ., [(e )ml) 6 R
m1 ^ m1 L m17 J/ ^
or [ (e ,)m] e .R = e ,N so that [ (e .)m] = e n for
L m1 J m1 m1 L x m1 J m1
some n C N. Hence [x ^x2 . ^ (em ) ] = [ (em_ini) (emln2 )
...(e .n . ) (e .N)] = [{e )m ] n n .n .N =
m1 m1 m1 J v m1 J 1 2 m1
e 1(nn'nl...n' .N) = e .N. In an analogous manner
m1 1 2 m1 m1 ^
[(e .N)x.x...x ,] = e .N. This completes the proof
m1 1 2 m1 m1 c r
that e .N = e ,R is a submgroup of A.
m1 m1 ^
Corollary 1.40: If R is a congruence relation on an
mgroup A with idempotent e, then eR is a submgroup 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 n1 tuple (a^, a^, .., an_]^
will be said to be commutative with an element x if and
only if
(x, aL, a2, ..., an_1)s(a1 a2 ***' anl x)*
Proposition 1.42: An mgroup is reducible to an n
group. m k(nl) + 1, k a positive integer, ir and only
if there exists an n1 tuple (a, a^ a ,) which com
1 c 12 n1
mutes with every element of the mgroup and such that
(a!* a2 anl al a2 anl' al a2' **' anl)7
23
with the n1 tuple (a^, a2, an_1) repeated k times, is
an madic identity.
The nadic operation may be defined as (x....x ) =
j. 'In
r n1
Lx1...a^
n 1
a^ ] where the righthand side of the
n1
above equation is the ordinary mgroup operation, and a^
occurs k1 times. (k1) (nl)+n = knn+lk+n = knk+1 =
k(nl)+l so the operation is properly defined.
Proof. If there exists an n1 tuple (a^, a9, ...,
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
tive for ((xi) xn+1 ) = L(xL) xn+1 aL ...aL J =
rr n n1 ni.T nr nr nii r r
[[x1 aL ...a1 ] xn + L aL ...aL J 
n1
2nl n1
n1
n+1 n1
a,
n11
a, J
2nl n1 nli , n + 1 2nlx r n1
Xn+2 al J (x1(x2 ^ Xn+2 ) ~ *** ^X1
r 2nl n1 nli n1 nln n1, 2nl^
[xn ax *a ]_ Ia! ai 1 = (*1 <*n ))
Next, if the mgroup is reducible to an ngroup, the
ngroup has an nadic identity. (a^, a2, ..., a a^, ,
*' an1' al a2 anl'
a1' a2
, a ) k times
n1
, a )
n1
will then be an madic identity and that (a^, a^, 
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
25
i 1 m
[sn a s ,, 1 = x, and all these a must be equal by the
1 T 1+1 t
unique solvability in a^ = a Â£ for all t. Hence
a Â£ S and S is a submgroup of A^.
(iii) Associativity is clear. Let y, y^, . .y^
f(A ). Hence there exists x, x., .... x Â£ A with f(x)
= y and f(x^) = y^ i = 1, ..., m. There exists a Â£ A_ such
that [x* ^ a x ] = x and for this a, f([xj^ ^ a x+^]) =
f(x) or [f(xL)...f(xi_I)f(a)f(xi+1)...f(x^)] = f(x). So
i 1. m
there exists b = f(a) Â£f(A ) such that [y^ b y^+^] = y
and we have unique solvability in f(A^). Note that if y^,
. .., ym 6 f(A^), [y] f(A^) for fn the notation above,
[y] = [f(xL)...f(xm)] = f([x]) f(a ).
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, [], t)
is an mgroup (A, [ ] ) together with a topology t on_ A under
which the functions f and q defined by f(x^, x2, xm)
= [x] and g(x1# x2, xm_2) =xm_i, where (x^ x2, ...
x L) is an madic identity, are continuous.
Proposition 2.2: If (A, [], t) is a topological in
group, then for any k = 1, 2, ..., (a, [], t) is also a
topological k(ml)+l group (A, (), t) when we define
, kimU+l* r r r mi .k(ml)+l l
U1 ] LL LalJ Ja(k1) (ml)+lJ *
Proof. The proof is clear.
Example 2.3: Let R be the set of negative real
numbers with []:(R > 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.
j
Example 2.4: Let S' = {zz=e 0 < x < 2tt ) .
Define []:(S)m> S' as follows: If z^ .., z S' and
26
27
IX.
= e m, then [z] = e;(xi+'
*m>
= a z = e tnen lz, j = e L1 '"m* if
T is the usual topology on S', then (S', [], t) is a topo
logical mgroup.
Example 2.5: If (G, , t) is an ordinary topologi
cal group, and if h is in the center of G, (G, [], t) will
be a topological mgroup if [ ] : G > G is defined as [x]
= x, *x~ *x *h.
12 m
Theorem 2.6: The function h: A > A defined by
d. 2. m
h(x) = [a^ xa+ ^] is a homeomorphism for each choice of
i = 1*
. m and elements a a A (where bv con
m
vention [ad ^xa+^] iÂ§_ when i = 1 and [a ^x] when
i = m) .
Proof. The function h is the restriction of the map
f to the subset {ax...x{a^}x A x (a^+^)x...xfa^} of
A x A X...X A (m times) and hence is continuous. Let b^,
b~ ..., b A such that (an, a_, .... a. b., ...,
2 m1 1 2 11 i
bml^ and ^bl b2 bi_i> ai+i am) are madic
identities and define k(x) = [b^ ^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 a^, ..., am in the in
group such that [a^ ^aai + ]J = b the maP h(x) = [a^ ^xai+l^
is a homeomorphism that takes a to b.
28
Proposition 2.8: Let A be a topological mqroup
and let A, A A be anv m subsets of A. If A. is
12' m i
open for some i, then [A^, A^, A ] is open. If A^,
.... A are compact, then [A,...A 1 is compact.
> m c 1 m c
Proof. If A^ is open, then by Theorem 2.6
[alai_l Ai+1aml is open, and [A^A^ =
U{ta1"ai_1 Ai ai+i*'am^lai ^ Am) is Pen* since
f:(x., x ) r> [xTl is continuous, if each A. is com
pact, A^ x A2 x***x A is compact, so [A^A2...Am] 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 [V ] c U.
Proof. Since f:(e,...,e) >e is continuous if U
is any neighborhood of e, there exists U^...U neighbor
hoods of e with [U....U ] c U.
m
Let V = 0 U. and we see that [v] c U.
i=l 1
Proposition 2.10: Let A be a topological mqroup
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
A be subsets of A. Then
m
and let A^, A2,
#
29
(l) [Ax..Am] C [A1...Am]
(ii) (A1, . y A^_2 ) C ((A^y y A^_2 ) )
(ill) [XL .A_. xi+1J = Lxl A xi+1J .
Proof. (i) It is known that for any continuous
function f, f(A) c f(A).
(ii) Same as (i).
(iii) By Theorem 2.6 f:x
r 1 m 
Lxx x xi+1J is a
home orno rphism.
Proposition 2.12: If H is a submsemigroup. subm
qroup. or semiinvariant submqroup of a topological in
group A, then H is. respectively, each of these.
Proof. Let H be a submsemigroup of A. Then
[H111] c H and by Proposition 2.11 (i) [ (H )m] c [H111] C H .
Next, let H be a submgroup of A. Then, as in the
first part of this proof, [H10] c H Since (H, H, . H) ^
c H, by Proposition 2.11 (ii) (H H ... H ) ^ c
((H, H, ... H)_1) c H~.
H is defined to be an invariant submgroup iff for
each madic identity (e^, eml^'^el H ei + l^ = H
and by Proposition 2.11(iii) [e^ H e+^] = H
= 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 submqroup is closed.
Proof. Suppose the interior is not empty and let
em_^ be an interior point of H. Then there exists an open
30
neighborhood V of em_^ with V c H. Let e^, em_2 b
a collection of m2 elements from H such that (e^, ..,
e ,) is an raadic identity. (Such a collection exists
m1
since H is a submgroup.) Hence for any h 6 H,
[h e *"] = h, so [h V] = H and by Proposition 2.8, H is
open. By Theorem 2.6, [x 1 H] is open for any choice of
x^, ..., A. Hence, U {[x 1 H]h ^ [x 1 H] for
any h $ H] is open and is a\h. Hence, H is closed.
Proposition 2.14: A submgroup H of a topological
mgroup 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 f H = (x}. Let
y 6 H. Since H is a submgroup of A, there exists x^, x2,
..., xm_^ H such that [x1^1 x] = y. Since U is open
m"" 1.
about x, [x^ U] is open about y by Theorem 2.6. Since
U 0 H = {x}, {y} [x_1 U] n H. If y + yQ [x1 u] n H,
m 1
yQ [x U] which implies that for some xq Â£ U, yQ =
m * 1
[x, x ]. Now x., x0, ..., x 6 H, y Â£ H and H being
1 o 1J 2 m1 Jo ^
a submgroup implies that xq 6 H, and therefore, that
XQ e u n H. But xq ^ x by uniqueness and the choice of
m 1
yo ^ y ,a contradiction. Hence, [x 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.
31
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 topology.
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 A.
The following theorem has been proved for general
algebraic systems in which the congruences commute by
Mal'cev [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 Â£ UR such
that for any neighborhood V of x, we have V fi (A\UR) ^
Since x Â£ UR, then there is a y U such that (x, y) R.
Let x~, x_, ..., x be elements in A such that (x, x~, .
2 3 m1 2
..., x ) is an madic identity so that [xx^ "'"y] U. By
the continuity of the mary operation on A, then there
exists an open set V containing x such that
[Vx2_1y] c U.
By hypothesis, since UR is not open, then there is a v V
such that v Â£ UR. Since (x,y) Â£ R, (v,v) 6 R, and (x^, x^)
R for all i = 2, ..., m1, then ([vx ], [vx ^x]) R.
However, (X2, .., xm_]_> x)> being a cyclic permutation of
32
an raadic identity, is also an madic identity so that
([vx ^y], v) 6 R. Note [vx Xy ] U and hence
([vx ^y], v) R PI (U x A) This implies that
tt2([vX2 y], v) = v UR, which is a contradiction. There
fore, UR must be open.
Definition 2.19: Let (A, [], t) be a topological in
group and let R be an equivalence relation on A. Define
n: A > A/R by_ n(a) = aR for each a A. n will be called
the natural map. Let 11 be the family of subsets of A/R
defined by U 6 31 iff n ^(U) is open in A.
Remark 2.20: If R is a congruence and U 21 then
U may be expressed as {xRx Â£ T Â£ t} for if U Â£ 21 set
T ~ n ^(u) Conversely, if T is open in A, n(T) 31 for
n ^(n(T)) = TR which is open in A since R is lower semi
continuous .
Theorem 2.21: The family of sets 31 in Definition
219 is a topology for A/R. The mapping n is continuous
and 21 is the strongest topology on A/R under which n is
continuous.
Proof. Let (uRu T } T 6 T be an arbitrary
\ \ K
collection of sets in 21. Then (J (uRu T ] =
X A X
{uRu 6 u T } Â£ 21 since u T is open in A. If
XA X X A X
fuRlu T.l^ n T. T is a finite collection of members of
L i i1 i=l i
n n
21, n{uR(uÂ£T.'} = {uRu n T} which is in 21 since
i=l 1 e=l
n
0 is open in A. Hence 21 is a topology on A/R. It is
i=l
33
clear that n is continuous. Next, let 3 be another topolo
gy on A/R such that 21 c a. Let F 6 a be an 3open set
which is not 21 open and suppose n is continuous under the
topology 5. Then n 1(F) is open in A, so suppose n ^(F) =
T t. Then n(T) is an element of 21 by the remark, so n(T)
is 21 open and n(T) = n(n 1 (F) ) = F, i.e., F is 21open, a
contradiction. So n is not continuous under the topology 0
and 21 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 A, xR is compact for all x A.
Proof. Let x A and choose a,, .... a 6 A such
1 ml
that [a1^ ^ a] = x. Observing that [a (aR) ] =
{z(([a i q], z) R and (q,a) R} = {z([ai a], z) R}
= [a 1 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 mqroup
with a congruence relation R on. A. Then A/R is discrete
if and only if aR is open in A for some a 6 A.
Proof. Suppose aR is open for some a 6 A. For any
m __ "l
x A, let a,, .... a be chosen so that [a a] = x.
1 ml 1
m
Then [a^ (aR)] = xR which is open by Theorem 2.6. Hence,
if aR is open for some a 6 A, xR is open for any x 6 A.
34
By Proposition 2.22, {yRy xR] = xR g 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
open in A/R, so n ^{aR} = aR is open in A.
Proposition 2.25: If an equivalence relation R gn.
a_ To topological space A is closed, then A/R is a topo
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 tt: A x Y > Y is closed. If R is closed, then
R 0 (A x {y}) is also closed for any y 6 A. Hence,
n^R H (A x [y})) is closed and A/R is .
Theorem 2.26: If h: A > A is a topological (con
tinuous) homomorphism between two topological mqroups and
A is. Tq, then the congruence relation R = h h ^ 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 h*h 1 = (h x h) 1 (a). A being Tq implies it is
in fact T2 and hence a is closed in A X A. Hence, hh ^ is
also closed. By the same reasoning as in the proof of Pro
position 2.25, Ry = rr^ ((h x h)1 (a) fl (A x {y})) and yR =
TT2 ((h X h) 1 (a) f ({y} x A)) are closed for each y 6 A and
hence R is both lower and upper semiclosed.
Theorem 2.27; If h: A > a is an open continuous
epimorphism of Tq topological mgroups, then A/R where R =
h*h is iseomorphic to A under the natural mapping.
35
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 mqroup. 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} Â£ 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 6 X such that h(x) = x.
Theorem 2.3Q: Let A be a topological mgroup. Then
A does not have the fixed point property.
Proof. Define a function h: A > A by choosing
Xp ... xm_^ Â£ A such that (x^, ..., xm_^) is not an madic
identity and letting h(a) = [x 1 a ] By Corollary 2.7,
h is continuous and suppose h(a) = a for some a Â£ A. Then
[x ^ a] = a, so by Propositions 1.12 and 1.13, (x^, ...,
xm_^) is an madic identity contradicting the choice of
(X]_ xml*
Remark 2.31: If A is a topological mgroup, then A
is not homeomorphic to [0,1] X...X [0,1] = [0,l]n for any
n, nor is A homeomorphic to the Tychonoff cube since both
36
have the fixed point property [5, p. 301].
Theorem 2.32: Let A be a topological mgroup, and
for some x Â£ A, let C be the component of x. Then, if
[x ^ x] = y, [x 1 C] is the component of y.
Proof. Let x and y be given as in the statement of
, x i A be chosen so that
m1
the theorem and let x^,
[x ^ x] = y. Then y [x ^ C] and [x 1 C] is connected
and closed by Corollary 2.7. [x ^ c] is the component of
m 1
y, for if not, let K be the component of y. Then [x^ C]
c K. Let x.
1,1
x
x. n A be chosen such that (x. .,
i,m2 1,1
Then
^ m_2 is an madic identity, i = 1, ... m1
[x ....x 0...x. ....x. 0 x. x0. .x C] = Cc
m1,1 ml,m2 1,1 l,m2 1 2 m1 ^
[x ....x ...x, ....x, 0 K] which is closed and
m1,1 m1,m2 1,1 l,m2
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
as its associated group according to the Post Coset
Theorem. If (e,, e0, ..., e is any madic identity of
, m1
A so that e,. e, .... e A. then for each x A
1' 2 m1
and any fixed index i = 1, ... m1, there exists an ele
ment a A such that
e . . e . ae e x.
1 ll l + l m1
Proof. Since x 6 Am \ then x = aj^^.a ^ for
some a., a, .... a .. Â£ A. For any a A, there exists,
by definition of an mgroup, a unique a $ A such that
[e^ ^aei+ianJ = ta] Considered in the containing group
of A, the equality above reduces to e....e. ,ae......e .a
^ J 1 ll l+l m1
== ci cl ~ .3
.a = xa and hence e. . .e ae ..... .e .
1 2 m1 mm 1 ll l + l m1
m
= x.
37
38
, e ,) be an
ra1
Proposition 3.2; Let (e^, e2>
madic identity of a topological mqroup (A, [], t) with
21 (e ) a local open basis at e Then 21 (e, e . .e ,) =
m1 m1 1 2 m1
fe,e^...e I U f 51 (e ,)1 is a local open basis of the
L 1 2 m2 1 m1 J 1
identity e = e^e0...em_^ of the containing group of A under
some topology.
Proof. It is sufficient to show closure under inter
section. If U., U_ 31 (e then clearly we have
1 2 m1 1
e.e0...e U, (1 e.,e~...e = e.e0...e 0(U, D U~).
12 m2 1 12 m2 2 12 m2'1 2'
Remarks. Preparatory to the demonstration that
21 (e^e2 e ^) 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 (e,, e^ e. , ) be an madic
1 2 m1
identity of a topological mqroup (A, [], t). For any
i = 1, ..., m2 and each open set V containing e^, there
exists a basis element U f $I(e ,) such that e,e...e c
^ m1 1 2 m2
1 m1 l+l m1
Proof. For each v $ V and fixed element Â£ A,
there exists, by definition of an mgroup, an element w $ A
such that
r m2 r i 1 m1 i
Le ^ wx 1J = Lex vei + 1x1 J .
Define two functions h and k by h(v) = [e^ ^ve + x^] and
m2
k(w) = [e^ wx^]. Let (x^, x2
identity.
, x i
m1
) be an madic
39
Then
W = [w [e 2wxJ = [e^ for some v Â£ V] =
r i r r i1 m1 i mli
{w vei+ixi
[eml e"lvei + xl x2~1] = [emlh(V)xr1] = klh(V)
is an open subset of A containing em_^ since both h and k
are homeomorphisms by Theorem 2.6.
Since 51 (e ,) is a local basis at e then there
m1 m1
exists a U 6 such that U c W. Whence [e* ^Ve + x^]
[e ^Wx^] 3 [e 2UxJ Considered in the containing group
this gives e... .e .Ve. ,. ...e x. 3 e. ...e 0Ux, and hence
^ 1 x1 i+l m1 1 1 m2 1
e. ...e .Ve..... .e 3 e. . .e ~U .
1 ll i + l m1 1 m2
Notation. In Lemmas 3.4 and 3.5 let f and g be the
functions of Definition 2.1.
V
Lemma 3.4: For each U Â£ Site ,), there exists a
^ m1
6 5l(em_^) such that (e ^e2 . e^ _2V) (e ^e2 . .e^ _2V) c
e.e~ . .e .
12 m2
Proof. Let U 6 51 (e^ ]_) and define a function
m_2
h: A x A > A by h(x,y) = [xe y]. Being a restriction
of the continuous function fonA x fe.l x...x fe ~ 1 xA,
h is therefore also continuous. h ''"(U) is thus an open
subset of A x A. Since the first and second projection
maps rr^ and rr2 are open maps, then V = rr^h (U) 0 rr2h ^(U)
is also open. It is also nonempty, since h(e ^ ,e =
m ]_
[e e. 1 = e 6 U and therefore e Â£ V. We now claim
m1 1 m1 m1
2
(e.e~...e ~V) c e.e~...e 0U; for, if v. v~ 6 V, then
12 m2' 12 m2 1 2 ^
40
h(vi,v2) U and hence (e j^e2 *em_2vi ie2 * *em2v2 ^ =
e,e 6
12 m
m2
2[vlel V21 = ele2"em2h(vl. v2) ele2em2
u,
Lemma
3.5: For each U 6 ll(e .), there exists a
m1
V 8l(e ) such that (e,e^...e V)
m1 12 m2
1
c e,e~...e ~U,
12 m2
1,
Proof. By definition of a topological mgroup,
g L(U) is open in A X A x...x A (m2 times). Since
U 6 2l(e ) and therefore e U. then (e., e~, .... e )
ml ml 1 2 m2
g 'L(U). Thus, there exist Â£ aUe^), W2 6 81 (e2), ...,
W Â¡Â¡j(e ) such that W. X W x . x W c g ^(U). Let
m c. m *4 x m 4
W = W^W2 . .W 2 Observe that W ^ c U in the containing
group. Thus
(W.e...e ) 1 c (We .) ^ = e ^ = e,e...e U.
12 m1 v m1 m1 12 m2
By Lemma 3.3, there exists a V 6 2J(em_^) suc'"1 that
ele2*
,e V c Wne . .e .
m2 1 2 m1
and therefore
(ele2
m 2
1
c e. e0 . .e ^U .
12 m2
Lemma 3.6: For each U Â£ 21 (e ) and x e.e~ . .e _9U,
m ~~ L x 4 m "fc
there exists a V 6 $I(e ,) such that x(e,e...e ,V) c
^ m1 1 2 m2
e e~ . .e .
12 m2
Proof. Let x = e.e~...e ~u for u U. Define the
12 m2
2
function h such that h(v) = [ue^ v]
Since this is a
1,
homeomorphism by Theorem 2.6, then h (U) = W is an open
and nonempty subset of A. Inasmuch as em_^ W, then
there is an element V 6 8i(em_^) contained in W. Hence
x (e ie2
m2
V ) Cl X ( 0 ^6 9^^
m2
12
m2
41
m2.
(e,e0...em ,W) = e,e0...em 0 ue W = e e9...e 0h(w)
'12 m2 12 m2 1 12 m2
e e~ . .e 0U.
12 m2
1
Lemma 3.7: For each U 2J (em_^) and any element x
in the containing group of A, there exists a V Â£ 81 (em )
such that x(e,e^...e ^V)x ^ c e,e...e U.
1 2 m2 1 2 m1
Proof. Recall that the containing group of A is
given by A U A2 U...U Am ^. We consider two cases:
m
Case I. Suppose x Â£ A so that x = a^2. .a^.
Let a ... ...,a i A such that a,a....a.a. n ...a is
l+l' m1 1 2 i l+l m1
the identity element in the containing group. Denote x
a. .....a n For any fixed element a A. let
l+l m1 2
tt f i r ir m2 i m 1 i r in2 i .
W = [w Laj^Lej^ wai + 1Jai+2aJ L e L Ua J } .
m 2
Since [e^ Ua] is an open subset in A (in fact a basis ele
ment at the point a Â£ A), then, being the inverse under a
homeomorphism, W is also open. Since e^ ^ Â£ W, there exists
a V Â£ ^(emi^ such that W contains V and hence
c ir m2TI ) m1 i r ir m2TT i m1 r m2T7
[a1[e1 Vai + 1Jai+2a J c La L Le x Wai+]_ Jai + 1a ] = le1 Ua J .
Thus, after simplication in the containing group of A, we
obtain x(e.e...e 0V)x ^ = (a. ...a.)(e.e~ . .e nV)
12 m2 '1 i 1 2 m2
(a. ....a d e e0 . .e ^U*
l+l m1 1 2 m2
m1
Case II. Suppose x Â£ A so that x = a.a...a
12 m1
for a^,
, a .
' m1
Â£ A. Let a ..... an n Â£ A such that
m 2m2
([ a1^1 ] am+L .., a2m2^ an mad^c identity and denote
42
x'L=aa11...a . For any fixed element a 6 A, let
m m+1 2m2
W = {w [ [a^_1[e_2wam] ] a^~2a] [e_2Ua]}. 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 8 (e ^2 *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 G, U Â£ 91(eie2* *"eml^ or
{Ux: x G, U ^ (e]_e2 .em_^) } .
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 91 (e1e2 enj_^) 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_^ A.
If U is any basis element of A containing em_^, then e^
e.e...e U is a basis element in G of e = e.en...e e ..
1 2 m2 1 2 m2 m1
Hence, e 1(e.e~...e 0U) = U is also a basis element of G
m1 1 2 m2
at e .. Thus, every basis element of A at e (and hence
m1 J m1
43
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 Tq topological mgroup 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 Tq, then
N is T and hence completely regular. Thus A is also
o
completely regular.
Theorem 3.10: Anv compact Tq topological mgroup 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 in
groups, the given mgroup A is isomorphic to a submgroup
of the mgroup of madic functions F(S^,S2 .., ^) on
the sets = A x...x A/s (i times). Since A is compact
Hausdorff, then A x...x A (i times) is compact Hausdorff
and hence also S^ under its natural quotient topology.
m1
Thus u S. is also compact Hausdorff under its sum topology
i=l 1
m1
Any bijective continuous function on U S. is also
i=l 1
a homeomorphism and thus the collection of all such func
m1
tions H( U S.) is a group (of homeomorphisms). By a result
i=l 1
44
ra1 m1
of Arens [l, p. 597], since U S. is compact, then H( U S.)
i1 i1 1
is a topological group under the compact open topology and
the operation of composition. Since F(S^,S2,..,sm_^) c
m1
H( S), then for any fL> f2, ..., fm 6 F(SL, S2, ..., Sm_1),
the functions
and
(fi* f2 fm2}
> (f..f9...f 0)~l = fSi"1,.. .fT1
1 m z mZ m3 1
are continuous. Thus F(S^, ... S is a topological in
group under the compact open topology (relativized).
Consider now the regular representation
m1
h: A > F(,S2, _^) d H( U S ^)
i=l
defined by h(a) = The following Lemma will be needed.
m1 m1 .
Lemma 3.11: The function k: U S.> U A1 = G such
i=l 1 i=l
that k((a^. a2, ..., a^)/s) =a^a2...a^ is a continuous
open map onto the containing group G of the topological in
group A.
Proof. Let U be any open subset in G such that
ala2*..ai U. By the continuity of the multiplication in
G, then there exist open subsets in G, a^ V^, a2 V2,
..., a^ such that
k(vL x v2 x...x v^s) = v1v2...vi C U.
Thus, h is continuous. That h is open is obvious.
45
Let L (C, U) or
d
{ (a,ap a^)/s for some i = 1, ..., m1 and
(a^, ..., a^)/s Â£ C} c U. Applying k on both sides, we
have
{aa^a2...a^ for some i = 1, m1 and
(a ai)/a Â£ 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
(a^, ... a^)/s Â£ C, by the continuity of the multiplica
tion in G, there exist open subsets
a Â£ V_ ^ an 6 V= a2 Va ..., a V
a.a . .a 1 ~ a.
1 2 i 1
a.
i
such that
aa,...a. V V c k(U).
x x d 2* 1 ^* ^ * d ^
Thus,
U a ^ C
f c ^
v ...v o k(c).
1 l
By compactness of k(C), then there exists a finite number
of ituples, (a^, ..., a^), ... (a^, ..., a) such that
n
U v kv kv k => Me)
k=1 ai a2 ai
n
Let U = n V
k=l
tains a. Then
k k*** k wh:*ch is nonempty since it con
k = l a.a. .a.
12 i
n
n
U uv kV k...V k = D( u V kV V ) C k(U)
k=l a. a a. k = l a. a~ a.
1 2 i 1 2 i
46
and hence
n
U UXVkX...XV /acU.
k=l a* a*
This means,
Vc)
x V k /
a .
l
) c u,
in other words,
h(U) = LyC (C, U).
Whence h is continuous and the final result follows.
Corollary 3.12: Let (A, [], t) be a topological in
group F a compact subset of A, and U an open subset of A
containing F. Then for each madic identity (e^, ..., e^
of A there exists open subsets U U with e. Â£ U..
ml i
i = 1, ..., ml such that [FU^1] U [U?1F] c U. If A is
locally compact, then U^, ..., um_^ may be chosen so that
([FU^ 1] U [^F]) 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 = e,...e ,
1 ml
Â£ V and by the continuity of the operation in G there exist
open subsets U. ..., U with e. Â£ U.f i = 1, .... ml
r 1' ml l x
such that U....U c V. Thus [FU1?1] U [UJ?1f] =
1 ml 1 1
(FU....U .) U (U....D .F) c FV U VF c U.
i ml l ml
If A is locally compact so that its containing group
is locally compact, the second part of [8, 4.10] states
47
that V may be chosen so that (FV U VF) is compact. Hence,
([FU1] U [u "^F ]) as a closed subset of a compact space
is compact.
Proposition 3.13: Let A^ and A2 be two mgroups
such that A^ is iseomorphic to A2 Then their covering
groups G^ and G2 are iseomorphic.
Proof. Let f: A^> be an iseomorphism and G^ =
Ax U...U A1, G2 = A2 U...U A2_1.
Define g: G^> G2 as follows. If x G^, x = x^...
x^ with x^, ., x^ A^ let g(x) = f(x^)...f(x^). Then if
y G1 and y = yLyk with yL, ..., yk A^ we have g(xy)
= g(xx...xiy1...yk) = f(x^...f(xi)f(yL)...f(yk) = g(x)g(y)
so g is a homomorphism. Next, suppose g(x) = g(y). Then
f(xL)..,f(xi) = f(y1)...f(yk) so i = k and if z+1
zm G A1 we have (XL)*f(xi)f(zi+1)*f(zm)] = [f(yL)...
f(yi)f(zi+1)fUm)] and f[x^z+1] = f[y^z+1]. f is bi
jective, so [x^z+^] = [y^z+^] which implies that x^...x^
= yx...yi, i.e., x = y and g is bijective.
Now let U be an open neighborhood of the identi
ty of G.. Then e. = x....x with (x., ..., x .) an m
2 1 11 m1 1 m1
adic identity, xj_ A^ i = 1, ..., m1. U can be expressed
as U = x....x o U' with U' open in A, and x Â£ U'.
1 mz 1 m1
g(U) = gixj^ xm_2 u') = f (xj_) .f (xm2)f (U ) and f (U ) is
open in A2 so f(x^)...f(xm_2)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.
48
Theorem 3.14: If a topological mqroup A is compact,
locally compact, rrcompact, 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 cosets of G, F compactness is clear.
Proposition 3.15: Let A be a topological mgroup
and let A, A^ A be subsets of A. If A is T and
1 2' m o
= [x] for any permutation p and any choice of
x. f A., then [x^1?! 1 = fx1?} for any choice of x. 6 A.,
i p(l) 1 1 i i
Proof. Let e be the identity of the containing
group G of the mgroup A. Let
1
1
H {(aL, ..., am) G X...X GlaLamap(l)*ap(m)
= ie}}.
Since A is T G is T and fe} 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 A. X... X A c H. Hence (A. X...X A ) c A. X...X A c H.
1 m 1 m 1 m
Proposition 3.16: If A is a Tq topological mqroup
and H is an abelian submsemigroup or submgroup 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.
49
Lemma 3.17: Let A be a topological mgroup with
congruence relation R on it. If aR is compact for some
a $ R, xR is compact for all x Â£ A.
Proof: Let x A and choose a,, .... a Â£ A
1 m1
in X m X
such that [a^ a] = x. Then [a^ (aR)] = xR is compact
since by Theorem 2.6 it is the continuous image of a com
pact set.
Lemma 3.18: Let A^, ...A^ be a collection of sets
such that A^, are compact and A^ is closed. Then
A....A ] is closed.
1 m
Proof. Since in a group the product of compact sets
is compact, we have compact in the containing
group so [A^...Am] is closed in A [8, 4.4].
Theorem 3.19: Let A be a topological mgroup with
a congruence relation R on A. I_f A/R and aR are compact
for some a 6 A, then A is compact.
Proof. By Proposition 2.23, since aR is compact,
xR is compact for any x 6 A. Let e^, ..., em_^ A such
that (e^, ..., e is an madic identity. As shown in
Theorem 1.37, N = e. .e (e _.R) is a normal subgroup of
j. mz rn *x
the covering group G. Since G is a topological group,and
em_1R is compact, N is compact. We next show that G/N is
compact. Since A/R is compact and A/R = (xNx 6 A],
(xNx 6 A] is compact. Hence, in G, y*{xNx A] is com
pact for any y 6 A and y*{xNx A] = {yxNx A} =
2
(zNz 6 A }. Continuing this process, we see that for any
i = 1, ..., m1, (xNx 6 A1} is compact so
50
G/N = U [xNx A1} 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, gcompact
topological mgroup. Let f be a continuous homomorphism of
A onto a locally countably compact Tq 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 T ,
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, Tq, and second countable
space A is arcwise connected (locally arcwise connected) if
and only if for each pair a, b A there exists a continu
ous function cos [ 0,1 ] > A such that cp(0) = a and cp(l) = b
(for each a $ A and every neighborhood U of a there exists
a neighborhood V of a contained in U such that for all x V
there is a continuous function cp: [ 0,1 ] > U such that eo(0)
= a and (i) = x).
Definition 4.2: A space A is simply connected (lo
cally simply connected) if and only if for each a Â£ 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 (cp: [0,l] > V) such that cp (0) = cp (1) ,
then cp is homotopic to 0 in A (in U).
Lemma 4.3: If f: [0,1] > A is homotopic to zero
and cp: [0,l] > A is an arbitrary continuous function.
51
52
then co*f is homotopic to cp, where cp*f is defined as fol
lows:
(cp*f) (t) =
cp(2t) for 0 < t < Â£
f(2tl) for Â£ ^ t ^ 1.
Proof: Note f (0) = cp(l). Then the horaotopy is
effected by the following function:
^ 1 + s
F(s,t) = <
# 21 *
*(1 + 3
for
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 0: A > A is the covering homomorphism, then A/00 1 is
iseomorphic to A.
Proof: Let A be the family of homotopy classes of
continuous functions cp: [0,1] > A such that cp(0) = e.
The homotopy class containing cp will be denoted by qj.
Consider any cp 5 A with cp(1) = p Â£ A. If 21 is
an open basis of the topology t on A, then for each U con
taining p in 31, let
U = [tp*f f: [0,1] > U such that f(0) = p}
and 2 = {u[ U 31}. It is easy to see that if cp is re
placed by any ii Â£ iÂ¡? U, then U and \)i will determine ex
actly the same 0 [14, p. 221],
53
on A.
(1) 5 is an open basis for some topology t
If , V 2J, so that U, V ai, then U n V 6 5. It
. r ^ ~ ~
is not difficult to show that U n V = U n V.
(2) (A, t) is a Tq topological space.
Consider any pair qj^, cp2 A such that / cp2. bet
cp (i = 1, 2) such that cpi(l) = p^
If p^ ft p2, then since A is Tq there exists a neigh
borhood U of p^ such that p2 Â£ U. In this case, q?]_ 0 but
cp2 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(1) = p^ is homotopic
to zero. If U = (cp^*f f: [ 0, 1 ] > U with f(0) = p^} ,
then p2 i U; for, if cp2 U, then there is an f: [ 0,1 ] > U
with f (0) = p^ such that cp^*f cp2 where
f cpn (2t) for 0 < t < ^
(cpi *f) (t) = I
[f(2tl) for ^ ^ t < 1.
Note: cp ^ (1) = f(0) = = (cp^*f) (1) = f(l). Thus
f is homotopic to zero and cp^ = cp^*f = cp2, which is a con
tradiction .
(3) If for
f ine s [ cp 2^2 * ^ ^ [ cp ^ (t) cp2 (t) ... cp^ ( f) 1 s ^d [ cp ^cp2 cp^ ^
then (A, [...], t) is a topological mgroup.
Since [cp]Lcp2 . .cpm] (0) = [cp^(0)cp2 ( 0) ...cpm(0) ] = [em] = e
and [cp^cp2 .cp ] is a continuous function on [0,l] to A, the
above operation on A is clearly well defined. Associativity
54
follows from the following relations
= [[tp1(t)cp2(t) .=Pm(t)]=PTO+1(t) . cP2m_L(t) ]
= [cpL(t) . cpi1(t)[cpi(t) .. .cpi+in_1(t)]
^i+m^ cp2ml^t^ ] = _CPi(fc)
[^(t) * cPi+ml(t)]q5i+m(t) cP2ml(t).
= [cp1(t) . cpi_1(t) [cPi+m_1](t)cpi+in(t) . cp2m_1(t)
r ilr i+mll 2ml'
K K >
i+m ](t)
which holds for each i = 2, .m. For each cp^ cp^
(i = 1, m2) let [0,1] > A be the function
such that cp (t) = (cp (t), cp _9(t)) ^ for each
t Â£ [0,1]. Since [em] = e, so that (e, e) is also an
madic identity, then cpm ^(0) = e. Define (cp^, cpm_2)
= Then for each cp,
[cpfCP2 cpm_Lcp] cp
Thus far, we have shown that (&, [...]) is an mgroup.
Next, we show that the functions
(cp 2_ > cpm) "> [ V 4^ ]
(cpj_, > cPjQ 2 ^ [*Â£]_> > cPm_2 ^
are continuous. Let cp^ cp^ A {i = 1, . m) such that
cp^(l) = p^(i = 1, ..., m) Let V be any neighborhood of
[cpj_cp2 . cpj^ ] = [^r2 . cp^ ] so that every element of is of
the form [cp^cp2 .cpm] *f for some f: [0,l] A such that
55
f(0) = p = [p1P2...pm] = [cp1(l)cp2(l) cpra(l) ] = [^H1) V>
By the continuity of the raary operation on A, there exist
neighborhoods containing p. (i = 1, ..., m) such that
[U.U...U ] C V.
12 m
Then [,U0... ] c V, for, if if. U. (i = 1, . m) then
12 in y i l
every i/^ if^ is of the form \i ^ = cp^*f^ (i = 1, . m) for
some f^: [0,1] > Ih such that f^(0) = p^. Thus,
(t)
cp. (2t) for 0 < t < $
<
f.(2tl) for 4 < t < 1.
1 =
Then
[ijiL(t)l)/2(t) .. .v^m(t) ] = [cp1(2t)rp2(2t) cpm (21) ]
= [cp](2t) for 0 < t ^ Â£ and [ i) l (t) 2 (t) . i/m(t) ]
= [f1(2tl)f2(2tl)...fm(2tl)] = [f^](2tl) for
i  g 1.
Since f^(t) th then [f]: [ 0,1 ] > V. Also, if [f](0)
= [f1(0)f2(0)...fm(0)] = [pLp2...pm] = p V, then
[*l*2***^m^ = = V.
Next consider (cp^, cp2 cpm_2) ^ = ^m 1 w^ere
cPi(i) = (i = 1, .... m2) so that (pi, p2, ..., Pm_2)_1
= Pm_j_* Let V be a neighborhood of with cpm ^(1)
= Pm_^ V. By the continuity of the inverse operation on
A, there exist neighborhoods of p. (i = 1, .., m2)
such that
(V1 X V2 x
X V ~) 1 C V.
m2
Again, we claim
(VL X V2 X
xv 0)
m2
1
c V.
56
For, if
1
continuous function such that
p V. Whence
Mnl
1
= (cpL
1
. .X
V 0
and
m2
i (1
= 1,
f rn
.] 
~> V
is the
fl<
t) .
>
m2
(1))"
1
J
2'
>
V2>
f 2
)
fm2) ~
cp T t V,
Tm1 m1
This completes the proof that (A, [...], t) is a topological
mgroup.
(4) (A, t) is a second countable space.
If 1311 ^ then also S ^ .
(5) The covering function 0: A > A such that
9(cp) = cp(1) = p is an open continuous map which is locally
a homeomorphism.
If U is an arbitrary neighborhood of p, then
0(U) c U, obviously, so that 0 is continuous.
Let cp 6 A and U be a neighborhood of cp defined by
the neighborhood U of cp(l) = p. By local arcwise connected
ness there exists a neighborhood V of p contained in U such
that for any x 6 V there is a continuous function f: [0,1_
> U such that f(0) = p and f(l) = x. This implies that
cp*f 6 U and @{cp*f) = (cp*f)(l) = f (211) = f (1) = x. Whence
V c 0(U), and 0 is open.
57
Next, let cp A and Q (cp) = cp(l) = p A. Since A is
locally simply connected, there is a neighborhood U of p
such that every continuous f: [ 0,1 ] > U such that f(0) =
f(l) = p is homotopic to zero in A. 0 is onetoone on ,
for, suppose 0(^) = 0(j2) so that for some f^: [0,1] > U
we have f^(0) = p(i = 1,2), and cp*f^ ^ and cp*f2 ^2'
If f(t) = f^(l ~ t) so that f 2 f : [ 0,1 ] > U is the con
tinuous function such that
(f2*fp(t)
(f2*f')(t)
f2(2t)
for 0 < t ^ Â£
^f{(2tl) for ^  t $ 1,
then (f2*fp (0) = f2() = e and (f2*fp(i) = f[(l) = fL(0)
= e. This means then that f2*f^ horootopic to zero and
therefore cp*f^ is homotopic to cp*f2 or Ip ^ = 1/ 2 .
Since 0 is continuous, onetoone, and open on U it
follows then that it is a homeomorphism on U.
By virtue of this, then
(6) (, t) is also locally arcwise connected,
locally simply connected, and regular.
(7) (A, t) is moreover arcwise connected.
Consider any cp A and Â£ A where uj contains the
null path. By homogenity of a topological mgroup, it suf
fices to show that uu and cp are connected by a continuous
path. Let cp 6 cp and cpg: [0,1] > A be the continuous
function such that cps(t) = cp(st) For any fixed s Â£ [ 0,1 ] ,
note that cp (0) = cp(0) = e and cp (1) = cp(s) = p 6 A; and
s s
hence cp A. Define now $: [ 0, 1 ] > A such that $(s) =
s
58
Then Â§ is continuous [ 14, p. 223]. Moreover, $(0) = cpQ
= uu and $(1) = cp^ = cp. Whence A is arcwise connected.
(8) (A, t) is simply connected.
As in the previous, let w A be the homotopy class
containing the null path. Consider any continuous function
$: [ 0, 1 ] > A such that $(0) = uj = $(1). Define cp such
that cp(t) = 0 ($ (t) ) where 9 is the covering map. Define
also for each s Â£ [ 0,1 ] cp : [ 0, 1 ] > A such that cp (t) =
s s
cp(st) Note that cps also depends continuously on t and
cp (0) = cp c 0) = e(Â§(0) ) = e (uj) = e so that cp 6 A.
Firstly, observe that 0($(s)) = cp(s) = 9 (cps) for
each s Â£ [0,1].
We wish to show that cpg = Â§ (s) for all s 6 [ 0,1 ] .
For s = 0, the equality obviously holds:
cp0 = cp(0) = jj = $(0) .
Let U be a neighborhood of c?q = $(0) for which 0 is a horneo
morphism. Since both cfT and $(s) are continuous functions
of s by an argument used in (7), then for some k sufficient
ly small we have
cp ([ 0 k) ) c 0 and $([*k)) <= u.
For each x Â£ [0,k) so that J $(x) ^ 0, then since 9 (cp ) =
0($(x)) and 0 is onetoone on U, then cp = Â§(x). This
shows that cp = $(s) for all s less than k and hence by the
s
continuity of the function cps and $(s) we obtain
cp, = lim cp = lim $(s) = $(k).
s>k S si>k
59
Repeating the process now for k instead of 0, we should
eventually show that the above relation holds for all
s Â£ [0,1]. For s = 1, in particular, we have
cp = cp^ = Â§(1) = $(0) = U)
so that cp 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) = cp (t) F (1, t) = cp(0) = e, F(s,0) =
cp(0) = e, F (s 1) = cp (1) = cp(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) = cp (0) = e
G(s,t)(1) = F(s,t) = p.
Thus for each s and t, G(s,t) Â£ 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 $ and 0:
G~(0, t) (x) = F (0 tx) = 'pt (x) or G(0,t) = cpfc = $(t)
G( 1, t) (x) = F (1, tx) = 'cp(O) = or gTT, t) = i
G( s, 0) (x) = F (s 0) = qT(0) = a> or G(s,0) = uj = $(0)
G(s,1)(x) = F(s,x) = p = 5 = $(1).
(9) 0 is a homomorphism of A onto A.
It is clearly onto. Let cp^ A such that
cpi(l) = pi(i = 1, 2, ..., m) Then
I
60
e([cp1cp2.. Cpm] ) = 0 ([cpxcp2 *cpm] ) = [cpLCp2 cpnl] (1)
= [cp1(l)cp2(L) . .cpm(l) ] = [p1P2...pm] =
= [ 9(cp1) 9(cp2) . 9(cpm) ].
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 1> 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 A1 and
M#
A are iseomorphic.
Proof. Let 9:A > A be as previously defined and
define a map k:A'> A as follows: Let cp1 A* and let
cp:[0,l] > A' such that cp(0) = e', cp(l) cp' with cp a con
tinuous function. Then h o cp defines a curve in A with
(h o cp) (0) = e and (h o cp)(l) = h(cp'). Since A is simply
connected, the choice of curve cp(t) connecting e' and cp' 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,l] > A and hence defines a unique point
li~'o~~cp in A. Define k(cp) = /h~~o~'cp;. It is clear that k is
well defined.
Now let U1 be an open subset of A* and let cp^ U'.
Let U = {cp  cp: [ 0,1 ] "> U', cp (0) = cp^ and each cp is continu
ous] and let ij/ be continuous with ijf:[0,l] > A such that
^ (0) = e 1 y (1) = cp.
Since U' is open in A', h(U') is
61
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
cp^ and ending at the same point are homotopic.
Letting W = {h o cp  cp U, h o cp: [ 0,1 ] > V such
that (h o cp) (0) = h(cp^)}, we see that
(1) W c h o U = {h o cp  cp 6 U},
(2) ii*(h o cp) h o cp 6 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' A' and let V be
o
a basic open set in A about k(cp^) Since 0 is an open map,
0(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 Q(V) and 9(k(cp^)) 6 W. Then h ^(w) is an open set in
A* about cp^ such that k(h ^ (W) ) c V. To see this, let
if:[0,l] > A' such that \Ji (0) = e', \r (1) = cp^ and tjf is con
tinuous. Let X = {cpcp:[0,lj > h ^(W) cp (0) = (1) and
each cp is continuous}. Since A' is simply connected,
62
\!*X = [ iji *cp  cp: [ 0,1 ] > h ^(W), cp(0) = ijf(L) and each cp is
continuous} represents all continuous functions
i/:[0,l] > A' such that f(0) = e' and f(l) 6 H ^(W).
Since W is simply connected, h o X = {h o cpcp:[0,l] >
h L(W) cp(o) = ii(1) and each cp is continuous] represents all
continuous function g:[0,l] > W such that g(0) = (h o cp)
(0) = h(cp^) Hence, k(h ^(W)) = 'h o = (h~ o ii ) (h o X j
= fi c v.
Next, let ^i', cp1 A' and suppose k(iji ) = k(cp'). If
k(i)i') = k(cp'), the functions ^ : [ 0,1 ] > A' and cp: [ 0,1 ] *>
A' must be homotopic and hence, vj (1) = cp(l), i.e., f = cp .
Hence, k is 11.
Now let cp[, .., cp^ 6 A' with associated functions
cp : [ 0,1 ] > A' such that cp, (1) = cp , ? _
1 > 1 1 ^ . ill,
/ ) ^ )
Then k[cp ... cp^] = ho [cp1 ... cpml = [hocp^ ... hocpm]
(since h is a homomorphism) = [h o cp^ ... ho cpm] =
[k(cp) ... k(cp^)]. Thus, k:A' > A is an iseomorphism.
^ ~ / 1
Recalling that the iseomorphism of A/Q*8 to A was
demonstrated in Theorem 2.29, we see that the truth of
Theorem 4.4 has been demonstrated.
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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.
65
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
s and Sciences
Dean, Graduate School
SUPERVISORY COMMITTEE:
fy ft, tftulm/ub
5*
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UNIVERSITY OF FLORIDA
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Full Text 
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 the Internet Archive
T62t31g&Mbading from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation
http://www.archive.org/details/topologicalmgrouOOrich
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 disserÂ¬
tation .
I am grateful to Dr. John E. Maxfield and the
University of Florida for providing the necessary finanÂ¬
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, withÂ¬
out whose constant encouragement this work would not have
been possible.
TABLE OF CONTENTS
Page
DEDICATION Â«, . ii
ACKNOWLEDGEMENTS iii
INTRODUCTION 1
Chapter
I.SOME PROPERTIES OF ALGEBRAIC mGROUPS ... 4
II.TOPOLOGICAL mGROUPS 26
III.AN EMBEDDING THEOREM 37
IV.THE UNIVERSAL COVERING mGROUP 51
BIBLIOGRAPHY 63
BIOGRAPHICAL SKETCH 65
IV
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 extenÂ¬
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 opÂ¬
eration of which the polyadic operation was merely an exÂ¬
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 terÂ¬
minology, apparently without knowledge of Dornte's work
and proceeded to develop a theory of these triplexes.
1
2
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 invariÂ¬
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 paÂ¬
pers by him [17], [18 ] , [19], [2 0], [21], [22], [23], varÂ¬
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 semiÂ¬
group. A representation theorem for msemigroups then reÂ¬
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 sufÂ¬
ficient conditions will be given.
Notation 1.0: In the sequel, a sequence of juxtaÂ¬
posed elements x^
x .  x . , .
1 1+1
i â– fk
x. , . will be denoted by x. , and if
k + i 1 i
k+l
... = â€¢ x simply as x , k + l denoting the
number of times the element x occurs.
Definition 1.1: An msemigroup is a set A and a
function f: Amâ€”> A such that for all a^,
â€™ a2ml â‚¬ Aâ€™
f(f(ai, am), am+1, â€¢â€¢â€¢> a2ml) f(alâ€™ f
am+2â€™ â€¢**' a2ml) *** f*alÂ» a2â€™
â– a , â– f (a , .. . ,
â€™ m1 m' â€™
a2m_i))* Following customary usage, we shall write
[aL ... am] = [aâ„¢] for f(aL, ..., am) and (A, []) for (A,f)
4
5
Proposition 1.2: For any k = 1, 2, {A, []) is
also a k(ml)+l semigroup (A. ()), where (a^m =
r r r 1 fc(ml)+l i
[ [ . . â€¢laiJ . . . Ja(k_i) (mi) +iJ â€™
For example, a 2semigroup A is a k{2l)+l = k+1
semigroup for any k = 1, 2, ... . If k = 2, then A becomes
a 3semigroup by defining (a^a2a^) = [[a^a2Ãa3^*
Definition 1.3: An msemigroup (A, []) is an inÂ¬
group iff for each i and for all a^, a^_^, ai+lâ€™ â€¢â€¢â€¢Â»
a , b 6 A there exists uniquely an x â‚¬ A such that
r i â€” 1 m i ,
[al xai+l! b
The following are come examples of mgroups.
Example 1.4: Let R be the set of negative real
numbers and define [ ] : (R ) â€”â€¢> R by [xyz] = xy*z, the
ordinary product of x, y, and z. Then (R , []) forms a
3group.
Example 1.5: Let S^, S2 , ...Â» sm_i t>e any m1 sets
of the same cardinality. Let F(S^,..., Sm_^) be the family
of all bijective and surjective functions f: â€”>
m ]_
U n S. such that f(S.) = S where p is any fixed per
i = l 1 1 p(i)
mutation of 1, 2, m1. Let []: Fmâ€”> F be defined as
the composition of m functions, i.e. [f^ f^] (x) =
(f^ (fm(x))). (F,[]) is an mgroup since it is clear
that [] is massociative and if x â‚¬ S^, fm(x) â‚¬ Sp'jj>
f .(f (x)) Â£ S , (f. ...(f (x))) which is an
mlv mv p(p(i))â€™ â€™ 1 ' m'
element of S where pm(i) is the mth permutation of i.
Pm(i)
pm 1(i) is i, so the m*"*1
permutation of i is p(i). Hence,
6
[ f ^ ... f m ] : S ^â€”> S (ij so that [f^ ... fm] Â£ 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 []: Zmâ€”â– '> Z is defined by [x^ ... x ]
= x. + . . . + x + h for any h Â£ I.
1 m J
Example 1.7: More generally, if (G,0 is an ordiÂ¬
nary group and if h is in the center of G, (G,[]) will be
an mgroup if []:Gâ„¢ > is defined as [x^ ... x^] =
X, *x0 â€¢ â€¢ x *h .
12 m
Example 1.8: Let %, *2 = 1 bd any complex
t ll
(m1) roots of unity. Define [ ] : T by [a^ ... a^] =
al + a2^ + "
+ am5m ^ â€¢ (
clear that all conditions except massociativity are satisÂ¬
fied, and that condition being satisfied is apparent if the
following two expansions are studied: [ [aâ„¢]a2â„¢7^] = [aâ„¢] +
am+l? + â€¢â€¢â€¢ + a2ml
1J m+1
.m1 , _ . , ril ,
> = aL + a25 + ... + ai^ + ai+1S
_ml ,
+ a Â§ + a Â§ +
m3 m+13
i+mi 2m1
+ a
. i + 1
m+i+1
+
iâ€”1
+ a
.m1
2ml
r ir i+mi 2ml "i , _ , , ,.11 , r i+uiih
a1 ai+1 ai+m+1 " al + a23 + + ai? + [ai+l]^ +
c i + 1
ai+m+l^
+ a_ 1 = a. + a_Â§ + ... + a,?11 +
2ml3 1 23 i3
pi+mi1 i+m+li1
i+(mi)3 ai+(m+li)3
+ a
.m1
Equality is apparent by noting that aj, +(m+ii) 5
â€žml
2ml3
i+m+li1
a
m+13
m
3 m+13
m+13
Example 1.9: Let Z
odd
be the odd integers under the
operation [x^ x2 x^] = x^+ x2 + x^. Then ZQdd is a 3group,
7
, V . be finite
m1
Example 1.10: Let V^,
dimensional vector spaces of the same dimension n. Let
l(vl,
. , Vm_^) be the set of all (m1)tuples of nonÂ¬
singular linear transformations (A^, A^, .... Am_^) where
A^:\Aâ€”> V (jj for the permutation p = (12...m1). L is
1 12
an mgroup under the operation [(A^, ..., Ara_^)â€¢â€¢â€¢>
2
m
m1 ,m
) / Am AUt ) 1 = fA'L A
' â€¢ â€¢ â€¢ \**^j â€¢ â€¢ â€¢ j / j \ ^ 2 * * â€¢ ^ f Ã*2
. m
1 ,2 m A1 *2
...A., Aâ€ž A.
. .A
m2
m1
1 2 3
,A . A. hZ
' m1 1 2
,m
. .Ajjj . Associativity is clear,
1 2 â€™
and unique solvability follows from the condition that the
linear transformations be nonsingular.
Definition 1.11: An (m1)tuple (e., eâ€ž,
e , )
m1
of elements e^ Â£ A is a left (right) madic identity iff
"1 X] = x ([xeâ€1
for all x â‚¬ A, [e1" 1 x] = x ([xe'" x ] = x) . When
(e^, em j_) â€¢â€˜s both a left and right madic idenÂ¬
tity it is simply called an madic identity.
Proposition 1.12: For any a, e^,
> eml ^
if [eâ„¢ ^a ] = a ([ ae"1 L] = a), then (e^, , .... e is a
left (right) madic identity.
Proof. Let x 6 A be arbitrary and [e
m1
] = a.
Choose a.
Â£ A such that [aa^] = x. Then x =
[aa2l]  [ [eâ„¢ ^ a ]] = [eâ„¢ 1 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.
m1
ident
Proof. Let (e., e_, ..., e , ) be a left madic
12' â€™ m1
ity. Note that [aeâ„¢2^^ eâ„¢1]] = [a[eâ„¢1 e^eâ€1'1]
8
m â– " 1
= [ae^ ]. From the definition of an mgroup it then fol
lows that [e , e, ] = e , which by Proposition 1.12 im
m1 1 mi 2
plies that (e^, , . .., era_]^) is also a right madic idenÂ¬
tity .
Proposition 1.14: If (e^, e2> . .., e ^ is an triÂ¬
adic identity, then for each i, (e^, . .., em_^, e^,
e^_^) is also an madic identity in the mgroup.
Proof. If (e. , e~, .... e .) is an madic identity,
1 2 â€™ m1 1
then [e^ e2**â€™eml el^ = el anc^ ^ence by Propositions 1.12
and 1.13 it is also true that (eâ€ž, ..., e e.) is an m
adic identity. By a repetition of this argument, the reÂ¬
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 inÂ¬
group A, then (x, x, ..., x) (m1) times is both a left
and right madic identity.
Proof. (x, x, ..., x) (m1) times is a left and
right identity on x; hence, by Proposition 1.12, for all
z Â£ 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
(x^, x^, ...Â» Â°f elements from an mgroup is the
unique element x also denoted by (x. , xn, ..., x _) ^
m1 2 1 2 m2
such that (x., x~, . .. , x ,) is an madic identity. We
1' 2â€™ m1
note that such an element always exists by the definition
of an mgroup and Proposition 1.12.
9
Definition 1.18: Let S , ..., S
be any (m1)
sets. An madic function on S,, .... S , is a function
1 mi
m1
f: U S
i=l
m1
U S.
iâ€”1 '
such that f(S.) c S ..
v i = a(i) where a = (12...m1).
Proposition 1.19: The family of all surjective and
bijective madic functions on sets S^, .
cardinality forms an mqroup.
Proof. See Example 1.5.
, S . of the same
m1
Definition 1.20: Two ktuples (k < m) of elements
Li+k]
from an mgroup A are equivalent, i.e., (a^+^, â€¢â€¢â€¢> a..^) ~
bi+lâ€™ â€˜* * * bi+k) , iff for all x, , .. ., x., x. . .,
â€™ 1 â€™ â€™ i' i+k+1
xm â‚¬ A (0 < i, i+k < m),
Note
r i i+k m i r i, i+k m t
[xlai+lXi+k+l] = [xlbi+lxi+k+l]
that by the above definition (a., ..., a ) s (b.
J 1 m l
b ) if and only if [a ] = [b., ] .
m J 1J L 1
Proposition 1.21: (a^+1,
b^+^) iff there exists c^, ..., 
a .
. . ) s (b.
i+k * i+
1â€™
c , . . . , c â‚¬ A such
'i+k + 1:
m
that
[<=1
i i+k m
ai+lCi+k+l
i _ r i, i+k m i
^ clbi+lci+k+l
Proof. Let d . ,, , ...,d . , d~ , . . . , d . Ã‰ A such
1+1â€™ â€™ m1â€™ 2â€™ â€™ i+k
that (di+1, ..., dm_1, cL, ..., c.) and (ci+k+1> â€¢â€¢â€¢> cm>
dâ€ž, ..., d . ,, ) are madic identities of A. Then for each
2 i +k
10
, xi, xi+k+1, ..., xm â‚¬ A,
r i i+k m
Lx^, , 1xJ
i+l i+k+1â–
ir ,ml i
r Xr_,IUJ. X 1 i+klr m ,ltKi m 1
txJdi + lclai + Jai+2 ^ ai+kCi+k+ld2 K+k+l^
= frxidm_1rciai+kcm rrdi+kxm i
L Lxidi + ilc1ai+1ci+k+1l la2 xi+k+1J
rr i,ralr i, i+k m â– M,i+k ra i
= l[x.1d.l1ic.b.,.1c.,. , t J J d x.,. , . I
11 1 i+l i i+l i+k+lJJ 2 i+k+1
, i+ki m
r ir .ml i, 1, i+klr. IH , i+k n IH i
xldi+lclbi+lbi+2 ^bi+kCi+k+ld2 W+k+J
= [xV+V* 1
L 1 i+l i+k+lJ*
Proposition 1.22: e is an equivalence relation.
Proof. That s is reflexive and symmetric is clear.
Suppose (ai+1, ..., ai+k) B (bi+1> â€¢â€¢â€¢Â» bi+k) and (bi+1>
.... b. ,. ) * (c. , 1 , .... c. L1 ) . By Definition 1.20, for
â€™ i+k i+lâ€™ â€™ i+k J
any x^ ..., xÂ±, x.+k+1, ..., xm â‚¬ A, [X^a^kxâ„¢+k+1] =
r i, i+k m i r i i+k m i TT , .
[xlbi+lxi+k+l] = [xlci+lxi+k+l] Henceâ€™ (ai+lâ€™ ai+k}
(ci+l' ci+k)*
Let S. = A1/a. Note S. = A.
i 1
Theorem 1.2 3: Any mgroup is isomorphic to an inÂ¬
group of bijective and surjective madic functions on dis
joint sets S., .... S ..
Proof. Let F(S^, . .., sm_i^ ^ tbe ^am^Y of all
surjective and bijective madic functions on S â€¢ = A1/s,
i
ml
ml
i = 1, ..., ml. For each a â‚¬ A, define L : U S.â€”â– > U S.
cl â€¢ t 1 â€¢ i 1
i=l i=l
such that La((x1( ..., x^/s) = (a, x^ x^)/s, i = 1,
, m2, and La((x1> ..., x^^)/Â») = CaXj^
ml
11
This is well defined since if (x^, x^) s
(y ^ > â€¢â€¢â€¢Â» Y i) , then siso (b ( x ^, â€¢â€¢â€¢> x ^) = (a, y ^Â» â€¢ â€¢ â€¢Â»
ViÂ»
Suppose La((x1, ..., xi)/s) = La((y1# yi)/a)
so that
(a, xL, xÂ±) â– (a, yL, ..., yj_)
and hence [axjaâ„¢+] = [ay*aâ„¢+^] for some a^ +  Â» ...,
^ â‚¬ A. Thus, by Proposition 1.21, (x^, . .., x^)/s
= (yni . .., y.)/s; that is to say, L is bijective.
Let (y ^, â€¢ â€¢ â€¢ > Yi+i^^3 ^ Â®i+1 * ^ for ^j.+2 â€™ * * * â€™
am g A, there exists uniquely (by definition of an mgroup)
an x â‚¬ A such that
r i
[ay2xa
m
i+2
r i + 1 m i
1 = [yL ai+2 ^*
This means that L ((yâ€ž, ..., y.
cl Z X
Thus La is also surjective.
Define f: A â€”5> FÃS^, ..
Note that
([aâ„¢], xL, ..., xi)
([a2Xl1' xi) B
x)/Â« = (yL, ...Â» yi+1)/s.
, S . ) such that f(a) = L .
â€™ m1 a
= (a ^ ^a 2X1^â– i Xi)
(a2, [aâ„¢x^], ..., xi)
â€¢ 0*0
3*3* â€¢â€¢0**00Â«**0Â«*3
Thus,
(UK li> xi> " (ai> [ai+ixi^â€™
([aâ€+1x^]) . (ai+1 am, x^
L [ am ] ((x X , â€¢â€¢â€¢Â» xi)/s) = ([aâ„¢], xL,
Xi)/.
La (ta2x^], x2, ..., x^)/â€”) ..
â€¢ â€¢ â€¢ f
12
= L L . . .L ((x., . .., xâ€¢)/â– )
ci d ~ d X X
12 m
Whence fita,]) = L = L L ...L = f(a.)f(a) ...f(a ) .
i r mi a, a^ a i Â¿ m
L a ^ J 12 m
f is also onetoone. For, if L = f(a) = f(b) = L. ,
â€™ a bâ€™
then La((xlf xm_L)/a) = Lb((x1, x^^/s) and
therefore [axâ„¢ = [bxâ„¢ 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 anaÂ¬
logue of this theorem for msemigroups has been proved by
Los [11] 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
[aâ„¢] = a]_'a2â€™â€™*an fÂ°r aÃ¼ a i â€™ a 2 T â€˜ â€™ aiu ^ A* In fact, G/N
= (A,...,Am ^, N =Am ^ 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 S^,...,
Sm_1. Let G be the group generated by A.
Note that for a fixed a Â£ A and any element b =
m
al**'aml ^ A Â» there exist uniquely x, y â‚¬ A such that
r m1 i
La. xJ=a....aTri1x = a
1 1 ml
r m â€” 1 1
[yax ] = ya1...am_1 = a.
Thus every element of Am ^ can be expressed as ax ^ or y ''"a
for any fixed a â‚¬ A and x, y e A. Thus, if b1 = y^a,
13
b2 = y2^a are anY two elements of Am \ then =
(yL a)(y2 a) = aa y2 = y2 is also an element of
Am â– *". This means Aâ„¢ ^ is a subgroup of G Note also that
for each a Â£ A, aA = A = A a. Thus a A a = A =
m 1. â– â€œ L
aA a for each a 6 A. Since A generates G, then any
g 6 G may be written as g = a^...aln for a. Â£ A and i, =
ill 1 rv
1 > l,ml 7>ml J 7vm_l â– 1 T
1 or 1. Then g A g = A and A is a normal sub
â€ž ,ml â€ž , . . 1 â€žml 1_
group of G. From aA = A we obtain a aA = a A =
m â€” 1 1 m1 , 1, > m2 ,m2
A . Thus a A = (a A)A = A . Similarly, for
_ _ 1 l,ml 1 m2 â€žm3 ,
a^, 82 f A, a^ 82 A =a^A = A , and so on. Thus
â€ž â€žml â€ž â€ž 2 3 m1
G, A == A 0 A UA u ... U A
Some of the Ai's may be equal. In any case, the order of
G/Am ^ is a divisor of m1.
Definition 1.25; G will be called the containing
group of A and AIU ^ 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.
The following theorem due to Sioson [20] will prove
useful in the sequel.
Theorem 1.26: The following conditions for an m
semigroup are equivalent:
14
(1) A is an mgroup;
(2) For all i = 1, ..., m, for a^, ..., a^ ,
ai+l' â€¢**Â» ara> b â‚¬ A, there exists an x Â£ A such that
r i â€”1 ra i , r i1 m t
[al xai+iJ = b, l.e., [aL Aai + ]J = A;
(3) For some i between 1 and m, for a^, . ..,
i â– â€œ m
a , b Â£ A, there exists an x t A such that [a, xa.,,1 =
m 1 l+l
b9 i.e. La^ Aa^+^J = A.
m 2_
(4) For each a1? a2, . .., am_1 $ A, [aâ„¢ a] =
A = [Aa^_i];
(5) For each a Â£ A, [aAâ„¢ ^] = A = [aâ„¢ ^a];
(6) For all a,, .... a . Â£ A. there exists
1 m2
a t â‚¬ A such that (a., .... a .) is an madic identity,
ml 1 ml
Proof. (1) implies (2) implies (3) implies (4) imÂ¬
plies (5) are obvious. (2) implies (1) by definition of inÂ¬
group; for, if x, y Â£ A such that [a} xaâ„¢+^] =
ya
i+1
],
then for some elements b., .... b ., câ€ž.
(biâ€™ bmlâ€™ al* **â€¢' ail)> (ai+l' â€™â–
are madic identities and hence x = y.
..., c^ â‚¬ A,
m â–
, c.)
(5) implies (2). Let (5) hold and a^, a.
a^ 6 A. Noting Am = A, then
r i1, m r il,2ml m i
[ a ^ Aa â€” [a i A a â– ; _i. i J
i+r
r i â€”2 r mâ€”li rm â€” 1 n m 1
= Ia! [a^A 1a[a ai+1]ai+2]
r i2,3 mi ,m
 [a^ A a j_+2 J ~ â€¢â€¢â€¢ â€œA  A.
(2) implies (6) as we have already seen.
(6) implies (4). We shall show the existence of an
x such that [xaâ„¢] = b for any a2,
â€¢ â€¢ â€¢ t
a , b â‚¬ A.
m
Since
15
Xm2 â‚¬ Sâ€™ (xlâ€™
[Am] = A, then we may write the above equation as
rr rr ,1,1 ,1 1,2,2 ,2 i m2 m2 ,m2. _ v,
[[...[[xbxb2..â€¢bm_i]b1b2..â€¢bm_1l...]bL b2 b*
By applying (6), then
c c,m2 ,m2 ,m21 r,2 ,2 ,2 nir, 1 *1
X  ,b2 â€™ " * ' bml â€¢ â€¢ â€¢ l Ã¼iâ€™b2 â€™ â€¢ â€¢bml^ 'â– blâ€™b2â€™"
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 x^, x2,
Proposition 1.28: If S is a submgroup of an
rn i_
mgrou j 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 m â€”1
As before, we may assume A c G = A \J A u â€¢ â€¢ â€¢ â€¢ U A . Let
ITl
S be a submgroup and H = S c G. Since S is a subm
group of A, for each x^, x2, . .., xm_2 Â£ S, there exists
Â£ S such that (x^, x2,
, x n) Â¿z S .
â€™ m2'
(x, , x~ , . . . , x â€ž) ^ = x
' 1 â€™ 2 â€™ â€™m2 m
x ,) is an madic identity. Hence x.xâ€ž ... x n as an
"il 12 m1
m
element of G is the identity and is also an element of H.
Next, let x^, x2,
, x .be m1 elements of S. Then for
â€™ m1
any y^, y2, ..., Ym_2 â‚¬ S there is a ym_L Â£ S such that
([xâ„¢ Ly L], y2 , ..., ym_1) is an madic identity which imÂ¬
plies then that (x,x0 ... x ,)(y,yâ€ž ... y ,) is the
c 12 m1 1J 2 â€¢'m1
identity of H. Hence y.y ... y . is the inverse of
l1 2 â€¢'m1
xm_^ an arbitrary element of H. Also note that
= sm_1 Sm_I = Sm Sm~2 = Sm_I = H. Thus H is closed
xlx2 *
HH
16
both under the binary operation in G and inversion and
hence is a subgroup of G and hence of Aâ„¢ 1, the associated
group of A.
Definition 1.29: A submgroup S of A is called inÂ¬
variant iff [at S am,t1 = S for each madic identity (a,.
1 l + l. 7 ' 1'
aâ€ž, .... a ,) of A and each i  1, 2, .... m2 and also
l m â€” i
m â€” 1 , 9m â€”9
such that
[a1,11 x[s aÂ¿Ui ^]] = S for each a , a~, . .., aâ€ž
L 1 L ra J J 1' 2' â€™ 2mi.
([aâ„¢], am+^> . .., a2m2^ ^s an mac3ic 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 Every invariant submgroup is
semiinvariant.
Proof. Let S be an invariant submgroup of A
with the associated subgroup H  Sm ^ in the containing
group G = A U A^ U .... u Aâ„¢ ^ of A. Let x 6 G so that
x = a,aâ€ž,..a. for some i < m1 and a,, aâ€ž, .... a, f
12 i = 1 â€™ 2 * ' i v
A ,
If i < m1. then there are a.,,. .... a , â‚¬ A such that
â€™ i+lâ€™ â€™ m1
(aii a j i â€¢â€¢â€¢
) is an madic identity. Thus x
1
a.,, .... a . is the inverse of x in G. S is invariant,
l+l m1
so a^ S ajL + j_ = S. Since the group operation gives the m
. . â€ž _ml , i â€ž m1. m1
group operation, then H = S  (a^ S ay+^)
, . m1 i. m2 ,
(ai a.2 .... ay) (s ai + i ai^ S^ai + 1
m1
x sm ^ x ^ =xHx ^. If i = m1, then there exist
m'
, a2m_2 â‚¬ A such that Ãœaâ„¢]> am+lâ€™
' a2m2) 1S
an madic identity and hence also
17
(a2m2â€™ [aT]' a
. .., a2m2^* Note siso that then
. a â€ž / and (a , . . . , a _ ,
â€™ 2m3 m' â€™ 2m3â€™
, r m 11
a2m2a1 amâ€™
[aâ€ž ~ aâ„¢ â– *â– ] ) are also madic identities. Then
2m2 1
xâ€”a, a0 ... a ,
12 m1
1
, x = a a
m m+1
TT â€žml , ml/o 2m2..ml ,
H = S = (ax (S am )) = (a1 a2
a_ â€ž and
2m2
a , )
m1
. _ml 1
a2m2 = x S x
/r* 2m 3. m 1Â» .m 2 _ ,
(S a (aâ€ž â€ž a. ) ) S (a
m 2m2 1 m
x H x L. Thus in both cases H is invariant in G.
Proposition 1.32: A submgroup S of an mgroup
A is semiinvariant iff [a Sm_I] = [sm~L a] for all a â‚¬ A.
Proof. Suppose [a Sm ^] = [sm ^ a] for all a 6 A.
Let H = Sm â– *" be the associated subgroup of S and G the
containing group of A. Thus in G we have a H a ^ = H.
Since a Â£ A, then a = (a , a , ..., am_i^ such that
(a, a2,
.a ,) is an madic identity. Since G is
m1
generated by A, then H is normal in G and the result folÂ¬
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{yyâ‚¬xH}.
Consider the set {xH)xÂ£A}. Since H is normal in G, if
xl Â¿ x2â€¢ either x^H = x2H or they are disjoint. An mary
operation may now be defined on the set {xHxâ‚¬A} by definÂ¬
ing (xx2H ... x^H) = [xâ„¢]H. This operation is associ
ative for Â«x^ ... ... x^H) *
18
. , ,,H ... xâ€ž ,H)
i+ra+1 2ml '
it is necessary to show that for any xÂ£A there exists aÂ£A
such that for x^, x^ , â€¢â€¢â€¢Â» xm^A* (x^H â€¢â€¢â€¢ X^H a H x^+^H .
x H) = xH or [x a xâ„¢ , ]H = xH. Since A is an mgroup,
i+L
this can clearly be done. The set {xHjxÂ£A} is A/S.
As opposed to a 2group, a congruence relation on an
mgroup need not determine a submgroup. Consider the conÂ¬
gruence relation of equality. This will determine a subra
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 re la tion 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 tt^(R) = (x: (x,y) Â£ R for some y 6 A] and
its range is the set rr0(R) = (y: (x,y) 6 R for some x â‚¬ A].
Notation 1.36: If U is a subset of A, denote
RU = Tt1(r n (A X U))
UR = tt9 (R n (U X A) ) .
Definition 1.37: A congruence relation R on an inÂ¬
group A is a relation which is reflexive, symmetric, and
transitive, and such that if (x^,y^) Â£ R
then ([xâ„¢],[yâ„¢]) â‚¬ R.
â€¢ â€¢ â€¢ 9
(x ,y ) 6 R
m 2 m
19
Theorem 1.38: Let R be a congruence relation on an
mgroup A. Then R* = (A x A) u {(x,y)XjygA1 for some i = 2,
...Â» m such that x = e^...e^_^xâ€˜, y = ej_ * â€¢ *e ' implies
(x',y') Â£ R] is a congruence in the covering group of A.
Proof. Let (e^, e2 > â€¢â€¢â€¢Â» em_]^ ^ an m~adic identity
of A. Note that each x 6 G is either an element of A or can
be uniquely written in the form
x = e^2 â€¢ â€¢ .e^x â€™
for some x' Â£ A and i = 1, 2, m2. For each x, y 6 G
define (x,y) Â£ R* if and only if either x, y 6 A or
x, y 6 A1 for some i = 2, m1 such that x = e ^e2 â€¢ â€¢ *ei]_x'
and y = e^e^. . *e j__^y ' with (x ' , y') Â£ R. Reflexivity, symÂ¬
metry, and transitivity of the relation R* are clear. SupÂ¬
pose (x^, y ) â‚¬ R* (t = 1, 2) so that either x^, y. â‚¬ A or
x. = e,e0...e. ,x,', y, = e,en...e. ,y' with (x ' , y') 6 R
and either x2 , y2 6 A or x^ = e, e2 â€¢ * *e _^x2, y2 ~
e ^2 â€¢ . .e _^2 with (xj , yj,) Â£ R. Obviously, x^x,, y ^y2 â‚¬ A
for some k = 1, 2, ..., m1. Thus it suffices to consider
the case when k is greater than 1. Let x^x2 = eie2'â€™â€˜ekLX'
and y1y2 = e^e2...e^_^y'. We shall show (x', yâ€™) â‚¬ R. ObÂ¬
servÃ©
, r m1 k1 ,1 r m1 1 r m1 i1 . j1 â€¢ \
x' = Lev e, x'J = Lek x^J = Le,, e, xjef x^ J
ek el Xlel X2
and y' = [ek e1
m1 k1 ,i r m1 â– ) r m1 i1 , j1 ,
Y J = Le, y^y,] = [e,_ e Â«'
1 yieÃ
Since (x^, y^) Â£ R, (x2, y2) 6 R and R contains the diagonal
of R, then ([ej^e^x'ej^x,;], [ej^e^yâ€™ef Ly â€¢ ]) â‚¬ R.
Whence (xâ€™, y') 6 R; and (x^, y^) â‚¬ R*. Lastly,
also show that (x, y) Â£ R* implies (x \ y â‚¬ R*.
we
20
Suppose (x, y) Â£ R* so that either x, y â‚¬ A or
x. = e1...ei_Ix' and y = e1...ei_1y' with (x' , y') â‚¬ R. We
shall consider two cases. In case i = m2, so that
x
= e^ . . .em_3x ' and y = e ^e2 . â€¢ em_3Y ' with (x ' , y') â‚¬ R,
then x 1 = x^ Â£ A and y Â£ â‚¬ A. Thus
x^ = [e1 xÂ¿] = L Lex xoJeL y yQJ
, r IB1 , i r ITl â€”lr , ITlâ€”3 , ill
Yo = ^S1 Yq J = LeL Lx^e L x'yoJJ.
Since (x', y') Â£ R and R contains the diagonal of A, then
([eâ„¢_Lx^eâ„¢ 3x'yÂ¿], [eâ„¢_Lx^eâ„¢~3y'yÂ¿]) â‚¬ R. Whence (x 1, y L)
= (x^, y^) â‚¬ R. Otherwise, in case i / m2, let x Â£ =
__ 1
e,en...e . ~x ' and y = e.e0...e . ~y ' . As in the pre
1 2 mi2 o 1 12 mi2jro
vious case, from the fact that
Ãœ 111 J.
e . ,e,
mi1 1
â‚¬ R and
y e. x e.
J o 1 1
'mi1 1
*Â¿ â–  tÂ«s:Ãl[ri"2yÂ¿'Ã'V.1].ri_2*Â¿]
r m1 mi2r . i â€”1 , mi2 l1n
 Lemiiei fyQei y ei xo]]â€™ lt: folIows that
(xÂ¿, YÂ¿) â‚¬ R and hence (x Â£, y~l) â‚¬ R*.
Since e1em_i is the identity in G and R* is a
congruence in the group G, then e,e0...e ,R* is a normal
12 m1
subgroup of G. Observe also that e.e0...e ,R* =
12 m1
(ele2em2} *(emlR) â€¢ IÂ£ x â‚¬ (ele2 * * *em2 ) * (emlR) 30
that x = e ]^e2 â€¢ â– â€¢em_2x ' where (x ' , em_^) â‚¬ R, then
(x, e^e2 â€¢ â€¢ ,eni_i) â‚¬ R* â€¢ Thus, the right side of the above
equality is contained in the left. If (x, e ^e2 . . .e^^) â‚¬ R*,
21
then x = e,e9...e _~x' for some x1 Â£ A with (x', e _,) Â£ R
j. u mÂ¿ m â€ i
and hence also x Â£ (e,e~...e n)*(e ,R). The above equal
12 m2' m1 ^
ity is thus demonstrated.
Theorem 1.39: Let R be a congruence relation on an
mgroup A. The congruence class em_]_R of_ R is a subm
group of A if and only if there are elements e^, ...,
e â€ž G e ,R such that (e,, eâ€ž e ,) is an madic
m2 m1 1 â€™ 2 â€™ â€™ m1
identity .
Proof. If em_^R is a submgroup of A, then obvious
ly it contains elements e^, en, . .., em_2 suc^ that (e^, e2
. .., em_^) is an madic identity.
Conversely, suppose em_]_R contains elements e ^, e2,
..., e â€ž such that (e., eÂ». .... e .) is an madic identi
ty. By the preceding theorem, R and (e^, e2, . .., e
determine a congruence relation R* on the covering group
G of A whose congruence class of the identity e,e....e , R*
= (e.eâ€ž . . .e ~) â€¢(e .R) = N is a normal subgroup of G. We
L fa iii fa m
shall show that e ,R = (e .e, . . .e ~) â€¢ (e ,R) = e ,N is
m1 m1 1 m2' ' m1 â€™ m1
a submgroup of A. It is clearly a subset of A. The
closure of e .R = e .N is also clear, for, if x. 6 e .R
m1 m1 ' â€™ i ^ m1
for i
 1, 2,
. . . , m, so that
(xiâ€™
e .) â‚¬ R, then
m1 â€™
([xâ€œ].
[ (e .
m1
)m]) 6 R Since
a Iso
â‚¬ R U = 1,
2
^ j â€¢ â€¢ â€¢
, m1) ,
then (e .,[ (e
m1â€™ m
_L)m])
, r â„¢l i
= ([el emL â€™
[(em_L)m]) â‚¬ R. Hence ([xâ„¢], e â‚¬ R and therefore
[x1!1] Â£ e .R. Finally, let x. , x~ , ..., x , â‚¬ e ,R =
1 m1 21 1â€™2â€™ â€™ m1 m1
e .N. Then x. = e .n. (i = 1, ..., m1) and
mi i m1 i â€™ â€™
22
[ (e n ) (e nâ€ž)...(e n . ) (e ,N)] = e ,ne nâ€ž. .
m1 1 m1 2 mI m1 m1 mi 1 m1 2
e ,n ,e ,N = [(e ,)m]n'n'...n' ,N for some n ' , n â€™ , ....
m1 m1 m1 m1 1 2 m1 1' 2â€™ â€™
nâ€˜ . Â£ N. From the steps above recall (e ., [(e , )ml) 6 R
m1 ^ ' m1â€™ L m17 J/ ^
or [(e ,)m] â‚¬ e .R = e ,N so that [(e .)m] = e .n for
L m1 J m1 m1 L x m1 J m1
some n C N. Hence [x ^x2 . . . ^ (em _ ) ] = [ (em_ini) (emln2 )
...(e .n . ) (e .N)] = [{e . )m ] n * n ' . .n ' ,N =
m1 m1 m1 J v m1 J 1 2 m1
e 1(nn'nl...n' .N) = e .N. In an analogous manner
m1 1 2 m1 m1 ^
[(e 1N)x1xâ€ž...x ,] = e .N. This completes the proof
m1 12 m1 m1
that e .N  e ,R is a submgroup of A.
m1 m1 ^
Corollary 1.40: If R is a congruence relation on an
mgroup A with idempotent e, then eR is a submgroup 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 reÂ¬
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 n1 tuple (a^, a^, . .., an_]^
will be said to be commutative with an element x if and
only if
(x, aL, a2, ..., an_1)s(a1Â» a2â€™ ***' anlâ€™ x)*
Proposition 1.42: An mgroup is reducible to an n
group. m  k(nl) + 1, k a positive integer, ir and only
if there exists an n1 tuple (a, , a^ a ,) which com
1 câ€” 12 n1
mutes with every element of the mgroup and such that
(ax, a2, ..., an_1, a2 anl' â€¢â€¢â€¢, aL> a2, ..., an1)Â»
23
with the n1 tuple (a^, a2, an_1) repeated k times, is
an madic identity.
The nadic operation may be defined as (x....x ) =
j 'In
r n1
[x^...x^ a ^
n â– â€œ 1
a^ ] where the righthand side of the
n1
above equation is the ordinary mgroup operation, and a^
occurs k1 times. (k1) (nl)+n = knn+lk+n = knk+1 =
k(nl)+l so the operation is properly defined.
Proof. If there exists an n1 tuple (a^, a9, . ..,
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
tive for ((xi) xn+1 ) = L(xL) xn+1 aL ..J =
rr n n1 nii Â¿nr nr nii r r
[[x1 aL ...a1 ] xn + L aL ...aL ] 
n1
2nl n1
n1
n+1 n1
a,
n11
â€¢ a, J
2nl n1 nli , , n + 1Â» 2nl, r n1
Xn+2 al J  (x^(x2 ) xn+2 )â€¢â€¢â€¢ lxi
r 2nl n1 nli n1 nln , n1, 2nl^
[xn ax ...aj ] at . . .ax ] = C^L (*n ))â–
Next, if the mgroup is reducible to an ngroup, the
ngroup has an nadic identity. (a^, a2, ..., a a^, a2,
â€¢**' an1' alÂ» a2â€™ anl' â€¢â€¢â€¢â€™ al' a2â€™ anl* k times
will then be an madic identity and that (a^, a^, . .., anl^
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 condiÂ¬
tions of the preceding theorem are satisfied.
Example 1.6 is an example of an mgroup, m > 2, that
Pages
Missing
or
Unavailable
25
i â€œ 1 m
[sn a s.,n] = x, and all these a must be equal by the
1 T 1+1 T
unique solvability in = a ^ for all r. Hence
a Â£ S and S is a submgroup of A^ .
(iii) Associativity is clear. Let y, y^, . . .y^
â‚¬ f(A ). Hence there exists x, x., .... x Â£ A with f(x)
a 1 m a
= y and f(x^) = y^ i = 1, ..., m. There exists a Â£ A_ such
that [x* â– *" a xâ„¢+,] = x and for this a, f([x^ ^ a xâ„¢+^]) =
f(x) or [f(xL)...f(xi_I)f(a)f(xi+1)...f(xm)] = f(x). So
i 1. m
there exists b = f(a) Â£â€˜f(A ) such that [y^ b y^+^] = y
and we have unique solvability in f(A^). Note that if y^,
. .., ym 6 f (A^) , [yâ„¢] â‚¬ f(A^) for in the notation above,
[yâ„¢] = [f(xL)...f(xm)] = f([Xâ„¢]) â‚¬ f(a ).
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, [], t)
is an mgroup (A, [ ] ) together with a topology t on_ A under
which the functions f and q defined by f(x^, x2, xm)
= [xâ„¢] and g(x1# x2, xm_2) =xm_i, where (x^ x2, ...Â»
x L) is an madic identity, are continuous.
Proposition 2.2: If (A, [], t) is a topological inÂ¬
group, then for any k = 1, 2, ..., (a, [], t) is also a
topological k(ml)+l group (A, (), t ) when we define
( kimU+lj rr . r m, k(ml)+l ]
U1 ] LL LalJ Ja(k1) (ml)+lJ *
Proof. The proof is clear.
Example 2.3: Let R be the set of negative real
numbers with []:(R â€”> R defined by [xyz] = x*y*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.
â€¢j
Example 2.4: Let S' = (zz=e , 0 < x < 2tt ) .
Define []:(Sâ€™)mâ€”â– > S' as follows: If z^ . .., z â‚¬ S' and
26
27
IX.
= e m, then [zâ„¢] = e*(xi+'
â– **ra>
= a z = e , tnen lz, j = e ' L1 ' 'hn' . If
T is the usual topology on S', then (S', [], t) is a topoÂ¬
logical mgroup.
Example 2.5: If (G, â™¦, t) is an ordinary topologiÂ¬
cal group, and if h is in the center of G, (G, [], t) will
be a topological mgroup if [ ] : Gâ„¢ â€”â€¢> G is defined as [xâ„¢]
= x, XÂ» â€¢ â€¢ *x *h.
12 m
Theorem 2.6: The function h: A â€”> A defined by
x 1. m
h(x) = [a^ xaâ„¢+ ^] is a homeomorphism for each choice of
i = 1*
. m and elements a a â‚¬ A (where bv con
m
vention [ad ^xaâ„¢+^] iÂ§_ when i = 1 and [aâ„¢ ^x] when
i = m) .
Proof. The function h is the restriction of the map
f to the subset {ax...x{a^}x A x (a^+^)x...xfa^} of
A x A X...X A (m times) and hence is continuous. Let b^,
b~ , .... b . â‚¬ A such that (an, a_, .... a. b., ...,
2â€™ â€™ m1 1â€™ 2â€™ â€™ ll' i'
bml^ and ^blâ€™ b2â€™ bi_i> ai+i' â€¢â€¢â€¢Â» am) are madic
identities and define k(x) = [b^ "'"xbâ„¢ 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 homoÂ¬
genous .
Proof. if a and b are any two elements of a topoÂ¬
logical mgroup, then for elements a^, ..., am in the inÂ¬
group such that [a^ ^aai + ]J = â– bÂ» the maP h(x) = [a^ ^xai+l^
is a homeomorphism that takes a to b.
28
Proposition 2.8: Let A be a topological mqroup
and let A, , Aâ€ž A be anv m subsets of A. If A. is
12' m â€” i â€”
open for some i, then [A^, A2, . A ] is open. If A^,
.... A are compact, then [A,...A 1 is compact.
â€™ m c â€™ 1 m c
Proof. If A^ is open, then by Theorem 2.6
[a.**a. , A. A.,,*a ] is open, and [A â€¢A ] =
L 1 ll i i+l mJ L 1 m
Ai ai+i*'am^lai ^ Am} is open. Since
f:(x., x ) â€”> [xTl is continuous, if each A. is com
pact, A^ x A2 x***x Am is compact, so [A^A2...Am] 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 [Vâ„¢ ] c U.
Proof. Since f:(e,...,e) â€”>e is continuous if U
is any neighborhood of e, there exists U^.. .neighborÂ¬
hoods of e with [U....U ] c u*
m
Let V = D U. and we see that [vâ„¢] c U.
i=l 1
Proposition 2.10: Let A be a topological mqroup
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
A be subsets of A. Then
m
and let A^, A2,
â€¢ â€¢ â€¢ I
29
(i)[Ax...Am] C [A1...Am]
(ii)(A1, â€¢ i â€¢ i Am_2 ) ((Aj^, â€¢ â€¢ â€¢ i Am_2 ) )
(m) [xL â€¢ .A_. xi+1J = Lxl A xi+1J .
Proof. (i) It is known that for any continuous
function f, f(A) c f(A).
(ii) Same as (i).
(iii) By Theorem 2.6 f:x
r Ã1 m 
Lxx x xi+1J is a
home orno rphism.
Proposition 2.12: If H is a submsemigroup. subm
qroup. or semiinvariant submqroup of a topological inÂ¬
group A, then H is. respectively, each of these.
Proof. Let H be a submsemigroup of A. Then
[H111] c H and by Proposition 2.11 (i) [ (H )m] c [H111] C H .
Next, let H be a submgroup of A. Then, as in the
first part of this proof, [Hâ„¢] c H . Since (H, H, ... H) ^
c H, by Proposition 2.11 (ii) (H , H , ... H ) ^ c
((H, H, ... H)_1)" c H~.
H is defined to be an invariant submgroup iff for
each madic identity (e^, ' â€¢â€¢â€¢Â» H ei + l^ = H
and by Proposition 2.11(iii) [e^ H eâ„¢+^] = H
= 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 submqroup is closed.
Proof. Suppose the interior is not empty and let
e ^ be an interior point of H. Then there exists an open
30
neighborhood V of em_^ with V c H. Let e^, em_2 be
a collection of m2 elements from H such that (e^, . ..,
e ,) is an raadic identity. (Such a collection exists
m1
since H is a submgroup.) Hence for any h 6 H,
[h eâ„¢ â– *"] = h, so [Hâ„¢ V] = H and by Proposition 2.8, H is
open. By Theorem 2.6, [xâ„¢ 1 H] is open for any choice of
x^, ..., Â£ A. Hence, U {[xâ„¢ 1 H]h ^ [xâ„¢ 1 H] for
any h $ H) is open and is a\h. Hence, H is closed.
Proposition 2.14: A submgroup H of a topological
mgroup 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 fl H = (x}. Let
y 6 H. Since H is a submgroup of A, there exists x^, x2,
..., xm_^ â‚¬ H such that [x1^1 x] = y. Since U is open
about x, [x^ U] is open about y by Theorem 2.6. Since
U 0 H = {x}, {y} â‚¬ [xâ„¢_1 U] n H. If y + yQ 6 [xâ„¢1 u] n H,
m â– â– 1
yQ â‚¬ [xâ„¢ U] which implies that for some xq â‚¬ U, yQ =
[x, x ]. Now x., x0, ..., x , 6 H, y Â£ H and H being
1 o 1J 2â€™ â€™ m1 â€™ Jo ^
a submgroup implies that xq 6 H, and therefore, that
xq â‚¬ U n H. But xq ^ x by uniqueness and the choice of
yo ^ y ,a contradiction. Hence, [x1^ 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.
31
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 topology.
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 â‚¬ A.
The following theorem has been proved for general
algebraic systems in which the congruences commute by
Mal'cev [12, p. 136]. It can be shown that the congruÂ¬
ences in an mgroup commute but we will prove the theoÂ¬
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 $ UR such
that for any neighborhood V of x, we have V fi (A\UR) ^
Since x Â£ UR, then there is a y Â£ U such that (x, y) â‚¬ R.
Let x~, x_, ..., x , be elements in A such that (x, x~, .
..., x , ) is an madic identity so that [xx^ "'"y] â‚¬ U. By
the continuity of the mary operation on A, then there
exists an open set V containing x such that
[Vx2_1y] c U.
By hypothesis, since UR is not open, then there is a v â‚¬ V
such that v Â£ UR. Since (x,y) Â£ R, (v,v) 6 R, and (x^, x^)
â‚¬ R for all i = 2, ..., m1, then ([vxâ„¢ ], [vxâ„¢ ^x]) â‚¬ R.
However, (X2, . .., xm_]_> x)> being a cyclic permutation of
32
an madic identity, is also an madic identity so that
([vxâ„¢ ^y], v) 6 R. Note [vxâ„¢ Xy ] 6 U and hence
([vxâ„¢ ^y], v) â‚¬ R H (U x A). This implies that
tt2([vx2 y], v) = v â‚¬ UR, which is a contradiction. ThereÂ¬
fore, UR must be open.
Definition 2.19: Let (A, [], Â«r) be a topological inÂ¬
group and let R be an equivalence relation on A. Define
n: A â€”> A/R by_ n(a) = aR for each a â‚¬ A. n will be called
the natural map. Let 11 be the family of subsets of A/R
defined by U 6 31 iff n ^(U) is open in A.
Remark 2.20: If R is a congruence and U â‚¬ 21 , then
U may be expressed as {xRx Â£ T Â£ t} for if U Â£ 21 , set
T ~ n ^(u) . Conversely, if T is open in A, n(T) â‚¬ 31 for
n ^(n(T)) = TR which is open in A since R is lower semi
continuous .
Theorem 2.21: The family of sets 31 in Definition
219 is a topology for A/R. The mapping n is continuous
and 21 is the strongest topology on A/R under which n is
continuous.
Proof. Let (uRu â‚¬ T } T 6 T be an arbitrary
\ \ l\ K
collection of sets in 21. Then (J (uRu â‚¬ T } =
X â‚¬ A X
{uRu 6 u T } 6 91 since u T is open in A. If
Xâ‚¬A X X â‚¬A X
fuRlu $ T.!1? , T. 6 t is a finite collection of members of
L i i1 i=l i
n n
21, n{uR(uÂ£T.'} = {uRuâ‚¬ n T} which is in 21 since
i=l 1 e=l
n
0 is open in A. Hence 21 is a topology on A/R. It is
i=l
33
clear that n is continuous. Next, let 3 be another topoloÂ¬
gy on A/R such that 21 c a. Let F $ a be an 3open set
which is not 21 open and suppose n is continuous under the
topology 5. Then n 1(F) is open in A, so suppose n ^(F) =
T â‚¬ t. Then n(T) is an element of 21 by the remark, so n(T)
is 21open and n(T) = n(n ^(F)) = F, i.e., F is 21open, a
contradiction. So n is not continuous under the topology 3
and 21 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 â‚¬ A, xR is compact for all x â‚¬ A.
Proof. Let x â‚¬ A and choose a,, .... a . 6 A such
1 ml
that [aâ„¢ ^ a] = x. Observing that [aâ„¢ (aR) ] =
{z(([aâ„¢ i q], z) â‚¬ R and (q,a) â‚¬ R} = {z([aâ„¢i a], z) â‚¬ R}
= [aâ„¢ â– *" 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 mqroup
with a congruence relation R on. A. Then A/R is discrete
if and only if aR is open in A for some a 6 A.
Proof. Suppose aR is open for some a 6 A. For any
m __ "l
x â‚¬ A, let a,, .... a . be chosen so that [aâ„¢ a] = x.
1â€™ ml 1
rn â€œâ€œX
Then [a^ (aR)] = xR which is open by Theorem 2.6. Hence,
if aR is open for some a 6 A, xR is open for any x 6 A.
34
By Proposition 2.22, {yRy â‚¬ xR] = xR g 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
open in A/R, so n ^{aR} = aR is open in A.
Proposition 2.25: If an equivalence relation R gn.
a_ To topological space A is closed, then A/R is a topoÂ¬
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 proÂ¬
jection map tt: A x Y â€”> Y is closed. If R is closed, then
R 0 (A x {y}) is also closed for any y 6 A. Hence,
â– n^ÃR H (A x [y})) is closed and A/R is .
Theorem 2.26: If h: A â€”> A is a topological (conÂ¬
tinuous) homomorphism between two topological mgroups and
A is. Tq , then the congruence relation R = h â€¢ h ^ 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 h*h 1 = (h x h) 1 (a). A being Tq implies it is
in fact T2 and hence a is closed in A X A. Hence, hh ^ is
also closed. By the same reasoning as in the proof of ProÂ¬
position 2.25, Ry = tt ^ ((h X h)1 (a) ft (A x {y})) and yR =
TT2 ((h x h) 1 (a) f ({y} x A)) are closed for each y 6 A and
hence R is both lower and upper semiclosed.
Theorem 2.27: If h: A â€”> a is an open continuous
epimorphism of Tq topological mgroups, then A/R where R =
h*h is iseomorphic to A under the natural mapping.
35
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 mqroup. 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} Â£ 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 â€”> X there exists x 6 X such that h(x) = x.
Theorem 2.3Q: Let A be a topological mgroup. Then
A does not have the fixed point property.
Proof. Define a function h: A â€”> A by choosing
Xp ...Â» x ^ â‚¬ A such that (x^, ..., xm_^) is not an madic
identity and letting h(a) = [xâ„¢ 1 a]. By Corollary 2.7,
h is continuous and suppose h(a) = a for some a Â£ A. Then
[xâ„¢ ^ a] = a, so by Propositions 1.12 and 1.13, (x^, ...,
xm_^) is an madic identity contradicting the choice of
(X]_Â» * * * Â» xml* '
Remark 2.31; If A is a topological mgroup, then A
is not homeomorphic to [0,1] X...X [0,1] = [0,l]n for any
n, nor is A homeomorphic to the Tychonoff cube since both
36
have the fixed point property [5, p. 301].
Theorem 2.32: Let A be a topological mgroup, and
for some x Â£ A, let C be the component of x. Then, if
[xâ„¢ ^ x] = y, [ xâ„¢ C ] is the component of y.
Proof. Let x and y be given as in the statement of
, x . i A be chosen so that
â€™ m1
the theorem and let x^,
[xâ„¢ ^ x] = y. Then y â‚¬ [xâ„¢ ^ C] and [xâ„¢ 1 C] is connected
and closed by Corollary 2.7. [xâ„¢ ^ c] is the component of
m 1
y, for if not, let K be the component of y. Then [x^ C]
c K. Let x.
t 1,1
x
x. n â‚¬ A be chosen such that (x. .,
i,m2 1,1â€™
Then
^ m_2 > xj_) is an madic identity, i = 1, ...Â» m1
[x ....x . 0...x. ....x. ~ x,x0.. .x . C] = Cc
m1,1 ml,m2 1,1 l,m2 1 2 m1
[x . ....x , â€ž...x, ....x, 0 K] which is closed and
m1,1 m1,m2 1,1 l,m2
connected, contradicting the assumption that C is the comÂ¬
ponent of x.
CHAPTER III
AN EMBEDDING THEOREM
In this chapter we will prove the topological verÂ¬
sion of the Post Coset Theorem by showing that each topoÂ¬
logical mgroup can be considered as the coset of a normal
(in the group sense), open, and closed subgroup of an ordiÂ¬
nary topological group. Several consequences of this theoÂ¬
rem will also be exhibited.
Proposition 3.1: Let (A, []) be an mgroup with
m1
as its associated group according to the Post Coset
Theorem. If (e,, e0, ..., e is any madic identity of
,ml
A so that e,. eâ€ž, .... e , PA. then for each x P A
1' 2â€™ â€™ m1 â€”
and any fixed index i = 1, ...Â» m1, there exists an eleÂ¬
ment a p A such that
e..... e . . ae e . â€” x.
1 ll l + l m1
Proof. Since x P Am \ then x = aj^.a ^ for
some a1f aâ€ž, .... a .. p A. For any a P A, there exists,
by definition of an mgroup, a unique a p A such that
[e^ J'aei + iam^ ~ [aâ„¢]. Considered in the containing group
of A, the equality above reduces to e,...e. . ae ...... .e .a
^ J 1 ll l+l m1
== ci , 3. ~ . â€¢ .3
.a = xa , and hence , e. ...e . , ae ..... .e .
1 2 m1 mmâ€™ â€˜1 ll l + l m1
m
= x.
37
38
, e ,) be an
m1
Proposition 3.2; Let (e^, e,,Â»
madic identity of a topological mqroup (A, [], t) with
21 (e , ) a local open basis at e Then 21 (e, eâ€ž . . .e ,) =
m1 m1 1 2 m1
fe,e^...e I U Â£ 21 (e , ) 1 is a local open basis of the
L 1 2 m2 1 m1 J 1
identity e = e^e0...em_^ of the containing group of A under
some topology.
Proof. It is sufficient to show closure under interÂ¬
section. If U. , U_ â‚¬ 31 (e then clearly we have
1â€™ 2 m1 1
e.e0...e 0U. (1 e.,e~...e = e.e0...e 0(U, H U~).
12 m2 1 12 m2 2 12 m2'1 2'
Remarks. Preparatory to the demonstration that
21 (e^2 . . *em_^) defines a topology that converts the conÂ¬
taining group of A into a topological group, we shall prove
a series of Lemmata.
Lemma 3.3: Let (e,, e^ e. , ) be an madic
â€”â€” 1 â€™ 2 â€™ â€™ m1
identity of a topological mqroup (A, [], t). For any
i = 1, ..., m2 and each open set V containing e^, there
exists a basis element U f $I(e ,) such that e,eâ€ž...e c
^ m1 1 2 m2
1 m1 l+l m1
Proof. For each v $ V and fixed element Â£ A,
there exists, by definition of an mgroup, an element w $ A
such that
r m2 r i â€” 1 m1 i
Le ^ wx 1J = Lex vei + 1x1 J .
Define two functions h and k by h(v) = [e^ ^veâ„¢ + x^] anc^
m2
k(w) = [e^ wx^]. Let (x^, x2
identity.
, x i
â€™ m1
) be an madic
39
Then
W = [w [eâ„¢ 2wxJ = [e^ for some v Â£ V] =
r i r r iâ€”1 m1 i mâ€”li _ Â»
{w vei+ixi
[eml eÃ"lvei + Ãxl x2~1] = [emlh(V)xr1] = kâ€˜lh(V)
is an open subset of A containing em_^ since both h and k
are homeomorphisms by Theorem 2.6.
Since 51 (e ,) is a local basis at e . , then there
m1 m1â€™
exists a U 6 such that U c W. Whence [e^ ^Vei + ix]_^
[eâ„¢ ^Wx^] 3 [eâ„¢ 2UxJ . Considered in the containing group
this gives e... .e . .Ve. ,. ...e . x. 3 e. ...e 0Ux, and hence
^ 1 ll i+l m1 1 1 m2 1
e. ...e . .Ve..... .e . 3 e. . . .e ~U .
1 ll i + l m1 1 m2
Notation. In Lemmas 3.4 and 3.5 let f and g be the
functions of Definition 2.1.
V
Lemma 3.4: For each U 6 51 (e ,), there exists a
^ m1 â€™
6 5l(em_^) such that (e ^e2 . . . e^ _2V) (e ^e2 . â€¢ .e^ _2V) c
e.en.. .e .
12 m2
Proof. Let U 6 51 (e^ ]_) and define a function
m_2
h: A x A â€”> A by h(x,y) = [xeâ„¢ y]. Being a restriction
of the continuous function fonA x fe.l x...x fe ~ 1 xA,
h is therefore also continuous. h ''"(U) is thus an open
subset of A x A. Since the first and second projection
maps rr^ and rr2 are open maps, then V = rr^h (U) 0 rr2h ^(U)
is also open. It is also nonempty, since h(e ^ ,e =
m 1_
[e . e. 1 = e . 6 U and therefore e . Â£ V. We now claim
m1 1 m1 m1
2
(e.e~...e ~V) c e.e~...e 0U; for, if v. , v~ 6 V, then
12 m2' 12 m2 â€™ 1â€™ 2 ^ â€™
40
h(vi,v2) â‚¬ U and hence (e j^e2 â€¢ â€¢ *em_2vi ie2 * * *em2v2 ^ =
e,e~ â€¢ Â» â€¢ s
12 m
m2
2[vlel v2] = ele2"em2h(vl.â€™ v2) Â« ele2â€˜em2
u,
Lemma
3.5: For each U 6 ll(e .), there exists a
m1
V â‚¬ 2l(e . ) such that (e,e^...e â€žV)
m1 12 m2
1
c e,e0...e ~U,
12 m2
1,
Proof. By definition of a topological mgroup,
g â– L(U) is open in A X A x...x A (m2 times). Since
U 6 2l(e , ) and therefore e â‚¬ U. then (e., e~, .... e â€ž)
ml ml 1 2 m2
â‚¬ g 'L(U). Thus, there exist Â£ Sl(e^), W2 6 91 (e 2) , ...,
W â‚¬ Â¡Â¡j(e _) such that W. X Wâ€ž x. . . x W c g ^(U). Let
W = W^W2 . . .W 2 â€¢ Observe that W ^ c U in the containing
group. Thus
(W,e0...e ,) 1 c (We .) ^ = e Stf ^ = e,e....e â€žU.
12 m1 v m1 m1 12 m2
By Lemma 3.3, there exists a V 6 ^emi^ such that
ele2*
,e â€žV c Wneâ€ž . . .e ,
m2 1 2 m1
and therefore
le le2
m ~~ A
i
c e. e0 . . .e ^U .
12 m2
Lemma 3.6: For each U Â£ 21 (e ) and x â‚¬ e.e~ . . .e _9U,
m L i. a. m â€œ"fa
there exists a V 6 $I(e , ) such that x(e,eâ€ž...e ,V) c
' m1 1 2 m2
e . e~ . . .e .
12 m2
Proof. Let x = e.eâ€ž...e ~u for u â‚¬ U. Define the
12 m2
2
function h such that h(v) = [ue^ v]
Since this is a
1,
homeomorphism by Theorem 2.6, then h (U) = W is an open
and nonempty subset of A. Inasmuch as em_^ â‚¬ W, then
there is an element V 6 il(em_^) contained in W. Hence
x (e ie2
m2
V ) Cl X ( 0 2 9^^
m2
12
â– Â» ^ v
m2
41
m2.
(e,e0...em ,W) = e,e0...em 0 ue W = e e9...e 0h(w)
'12 m2 12 m2 1 12 m2
e , e~ . . .e 0U.
12 m2
1
Lemma 3.7: For each U â‚¬ 2J (em_^) and any element x
in the containing group of A, there exists a V Â£ 81 (em )
such that x(e,e^...e ^V)x ^ c e,eâ€ž...e , U.
â€” 1 2 m2 1 2 m1
Proof. Recall that the containing group of A is
given by A U A2 U...U Am ^. We consider two cases:
m â€”
Case I. Suppose x Â£ A so that x = a^2. . .a^.
Let a ... , ...,a . Â£ A such that a,a....a.a, ,..a . is
1+1' â€™ m1 1 2 i l+l m1
the identity element in the containing group. Denote x
a. .....a 7 . For any fixed element a â‚¬ A. let
l+l m1 J â€™
, . i r ir mâ€”2 â– > m â€”1 i r rnâ€” 2 i .
W = [w La^ej^ wai + 1Jai+2aJ â‚¬ Le L Ua J}.
Since [e^ Ua] is an open subset in A (in fact a basis eleÂ¬
ment at the point a Â£ A), then, being the inverse under a
homeomorphism, W is also open. Since e^ ^ 6 W, there exists
a V Â£ ^(emi^ such that W contains V and hence
r ir m2TI â€¢) m1 _ r ir m2TT i m1 r m2T7
[a1[e1 Vai + 1Jai+2aJ c L a L1 e x Wai+]_ Jai + 1a ] = [el Ua J .
Thus, after simplication in the containing group of A, we
obtain x(e.eâ€ž...e 0V)x ^ = (a. . . . a . ) (e.e~ . . .e nV) â€˜
12 m2 '1 i ' 1 2 m2
(a. ....a d e e0 . . .e ^U*
l + l m1 1 2 m2
m1
Case II. Suppose x â‚¬ A so that x = a^a2***aml
for a^,
, a .
' m1
6 A. Let a ..... a., n Â£ A such that
m* â€™ 2m2
([ a1^1 ] , am+^, . .., a2m2^ *s an mad^c identity and denote
42
x'L=aa11...aâ€ž â€ž. For any fixed element a 6 A, let
m m+1 2m2
W = {w [ [a^_1[eâ„¢_2wam] ] a^~2a] â‚¬ [eâ„¢_2Ua]}. 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 topoloÂ¬
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 8Ã (e ^2 . â€¢ *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 â‚¬ G, U Â£ y(eie2 * * "eml^ or
{Ux: x â‚¬ G, U â‚¬ ^ (e]_e2 â€¢ â€¢ .em_^) } .
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 quoÂ¬
tient topology.
It thus remains to show that the topology defined in
G by 91 (e1e2 â€¢ â€¢ â€¢enj_^) 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_^ â‚¬ A.
If U is any basis element of A containing em_^, then e^
e.eâ€ž...e â€žU is a basis element in G of e = e.eâ€ž...e â€že ..
1 2 m2 1 2 m2 m1
Hence, e .(e.e..e 0U) = U is also a basis element of G
m1 1 2 m2
at e .. Thus, every basis element of A at e . (and hence
m1 â€™ J m1
43
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 Tq topological mgroup 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 subÂ¬
group N which is homeomorphic to A. Since A is Tq, then
N is T and hence completely regular. Thus A is also
o
completely regular.
Theorem 3.10: Anv compact Tq topological mgroup 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 inÂ¬
groups, the given mgroup A is isomorphic to a submgroup
of the mgroup of madic functions ..., Sm ^) on
the sets = A x...x A/s (i times). Since A is compact
Hausdorff, then A x...x A (i times) is compact Hausdorff
and hence also S^ under its natural quotient topology.
m1
Thus u S. is also compact Hausdorff under its sum topology
i=l 1
m1
Any bijective continuous function on U S. is also
i=l 1
a homeomorphism and thus the collection of all such func
m1
tions H( U S.) is a group (of homeomorphisms). By a result
i=l 1
44
ra1 m1
of Arens [l, p. 597], since US. is compact, then H( U S.)
iâ€”1 iâ€”1 1
is a topological group under the compact open topology and
the operation of composition. Since F(S^,S2,..â€¢,sm_^) c
m1
H< U SÂ±>, then for any fL> f2, ..., fm 6 F(SL, S2, ..., Sm_1),
the functions
and
(fi* f2â€™ fm2}
> (f..f9...f 0)~l = .f'1
1 Â¿ m z mm3 1
are continuous. Thus F(S^, ...Â» S is a topological m
group under the compact open topology (relativized).
Consider now the regular representation
m1
h: A â€”> F(S1,S2, ..., Sm_1) c H( U SL)
i=l
defined by h(a) = . The following Lemma will be needed.
m1 m1 .
Lemma 3.11: The function k: U S.â€”â€¢> U A1 = G such
i=l 1 i=l
that k((a^, a2, ..., a^)/s) =a^a2...a^ 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
ala2â€™*â€˜ai ^ u* BY the continuity of the multiplication in
G, then there exist open subsets in G, a^ â‚¬ V^, a2 â‚¬ V2,
..., a^ â‚¬ such that
k(vL x v2 x...x vi/s) = v1v2...vi C U.
Thus, h is continuous. That h is open is obvious.
45
Let L â‚¬ (C, U) or
d
{ (a,a1( a^)/s for some i = 1, m1 and
(a^, ..., a^)/ss Â£ C} c U. Applying k on both sides, we
have
{aa^a2...a^ for some i = 1, . .., m1 and
(a , ai)/a Â£ C} c k(U).
By Lemma 3.11 k(C) is compact and k(U) is open.
Thus, for any fixed index i = 1, ...Â» m1 and
(a^, ...Â» a^)/s Â£ C, by the continuity of the multiplicaÂ¬
tion in G, there exist open subsets
a Â£ V_ â€ž ^ , an 6 V= , a2 â‚¬ Va , ..., aÂ± â‚¬ V
a.aâ€ž . . .a.' 1 ~ a.
1 2 i 1
a.
i
such that
aa....a. â‚¬ V V c k(U).
X x d 2* * ' 1 ^* ^ * * * d ^
Thus,
U a ^ C
â€¢ â€¢ â€¢ j C v
v ...v o k(c).
Q > d â€¢
1 l
By compactness of k(C), then there exists a finite number
of ituples, (a^, ..., a^), ...Â» (a^, ..., aâ€) such that
n
U v kv k*â€¢v k => Me)
k=1 ai a2 ai
n
Let U = n V
k=l
tains a. Then
k k'** k wh:*ch is nonempty since it con
k=l a,a~...a .
12 i
uv kv k
k=l aj_ a2
n
.V
k
a .
i
u( U v kV k. . .V ) C k(U)
k=1 ai a2 ai
46
and hence
n
U UXVkX...XV /acU.
k=l a* a*
This means,
Vc>
â€¢Xv k7
a .
l
) c u,
in other words,
h(U) = c (C, U).
Whence h is continuous and the final result follows.
Corollary 3.12: Let (A, [], t) be a topological inÂ¬
group , F a compact subset of A, and U an open subset of A
containing F. Then for each madic identity (e^, ..., e^
of A there exists open subsets U U . with e. Â£ U..
â€” _LÂ» ml â€” i
i = 1, ..., ml such that [FU^1] U [U^1F] c U. If A is
locally compact, then U^, ..., um_^ may be chosen so that
([FU^ 1] U [^F]) 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 y VF c U. Then e = e,...e ,
1 ml
Â£ V and by the continuity of the operation in G there exist
open subsets U., ..., U , with e. Â£ U.f i = 1, .... ml
r 1' â€™ ml i â€™ â€™
such that U....U . c V. Thus [FU1?1] U [UI?1F] =
1 ml 1 1
(FU....U .) U (U....D nF) c FV U VF c U.
l ml 1 ml
If A is locally compact so that its containing group
is locally compact, the second part of [8, 4.10] states
47
that V may be chosen so that (FV U VF) is compact. Hence,
([FUâ„¢1] U [uâ„¢ "^F ]) as a closed subset of a compact space
is compact.
Proposition 3.13: Let A^ and A2 be two mgroups
such that A^ is iseomorphic to . Then their covering
groups G^ and G2 are iseomorphic.
Proof. Let f: A^â€”> be an iseomorphism and G^ =
Ax U...U Aâ„¢1, G2 = A2 U...U A2_1.
Define g: G^â€”> G2 as follows. If x â‚¬ G^, x = x^...
x^ with x^, x^ â‚¬ A^ let g(x) = f(x^)..,f(x^). Then if
y â‚¬ G1 and y = yLÂ«yk with y^, . .., yk e A^ we have g(xy)
= g(xL...xiy1...yk) = f(x^...f(xi)f(yL)...f(yk) = g(x)g(y)
so g is a homomorphism. Next, suppose g(x) = g(y). Then
f(xL)...f(xi) = f(yL)...f(yk) so i = k and if zÂ¿+1
zm G Ai we have (XL)â€¢â€¢*f(xi)f(zi+1)â€¢â€¢*f(zm)] = [f(yL)...
f(yi)f(zi+1)â€¢.fUm)] and f[x^zâ„¢+1] = f[y^zâ„¢+1]. f is bi
jective, so [x^zâ„¢+^] = [y^z*j*+^] which implies that x^...x^
= y^..,yi, i.e., x = y and g is bijective.
Now let U be an open neighborhood of e^ the identiÂ¬
ty of G.. Then e. = x....x , with (x., ..., x .) an m
2 1 11 m1 1â€™ â€™ m1
adic identity, xj_ â‚¬ A^ i = 1, ..., m1. U can be expressed
as U = x....x ~ U' with U' open in A, and x . g U'.
1 mÂ¿ 1 m1
g(U) = gixj^ xm_2 u') = f (Xj,) . . . f (xm_2 ) f (U ' ) and f (U ' ) is
open in A2 so f(x^)...f(xm_2)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.
48
Theorem 3.14: If a topological mqroup A is compact,
locally compact, rrcompact, or locally countably compact.
then its containing group G is, respectively, each of these.
Proof. Since G is a topological group, multipliÂ¬
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 cosets of G, F compactness is clear.
Proposition 3.15: Let A be a topological mgroup
and let A,. Aâ€ž A be subsets of A. If A is T and
1Â» 2' â€™ m â€” â€” o
= [xâ„¢] for any permutation p and any choice of
x. f A., then [x^1?! 1 = fx1?} for any choice of x. e A.,
i p(l) 1 1 i i
Proof. Let e be the identity of the containing
group G of the mgroup A. Let
1
1
H {(aL, ..., am) â‚¬ G X...X GlaLâ€¢*araap(l)â€¢*ap(m)
= ie}}.
Since A is T , G is T and fe} 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 A. X... X A c H. Hence (A. X...X A ) c A. X...X A c H.
1 m 1 m 1 m
Proposition 3.16: If A is a Tq topological mqroup
and H is an abelian submsemigroup or submgroup 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.
49
Lemma 3.17: Let A be a topological mgroup with
congruence relation R on it. If aR is compact for some
a $ R, xR is compact for all x Â£ A.
Proof: Let x â‚¬ A and choose a,, .... a . Â£ A
1â€™ m1
in X m X
such that [a^ a] = x. Then [a^ (aR)] = xR is compact
since by Theorem 2.6 it is the continuous image of a comÂ¬
pact set.
Lemma 3.18: Let A^, ...A^ be a collection of sets
such that A^, are compact and A^ is closed. Then
ÃA....A 1 is closed.
1 m
Proof. Since in a group the product of compact sets
is compact, we have compact in the containing
group so [A^...Am] is closed in A [8, 4.4].
Theorem 3.19: Let A be a topological mgroup with
a congruence relation R on A. I_f A/R and aR are compact
for some a 6 A, then A is compact.
Proof. By Proposition 2.23, since aR is compact,
xR is compact for any x 6 A. Let e^, ..., em_^ â‚¬ A such
that (e^, ..., e is an madic identity. As shown in
Theorem 1.37, N = e, . .e (e _.R) is a normal subgroup of
J. mz rn *â„¢x
the covering group G. Since G is a topological group,and
em_1R is compact, N is compact. We next show that G/N is
compact. Since A/R is compact and A/R = (xNx 6 A],
(xNx 6 A] is compact. Hence, in G, y*{xNx â‚¬ A] is comÂ¬
pact for any y 6 A and y*{xNx 6 A] = {yxNx â‚¬ A} =
2
(zNz 6 A }. Continuing this process, we see that for any
i = 1, ..., m1, (xNx 6 A1} is compact so
50
G/N = U [xNx â‚¬ A1} being a finite union of compact sets
iâ€”1
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, gcompact
topological mgroup. Let f be a continuous homomorphism of
A onto a locally countably compact Tq 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' beÂ¬
tween G and G'. Since A is locally compact and acompact,
G also is. Since A* is locally countably compact and T ,
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 conÂ¬
nected universal covering mgroup [14, p. 232].
Definition 4.1; A regular, Tq, and second countable
space A is arcwise connected (locally arcwise connected) if
and only if for each pair a, b â‚¬ A there exists a continuÂ¬
ous function cos [ 0,1 ] â€”> A such that cp(0) = a and cp(l) = b
(for each a $ A and every neighborhood U of a there exists
a neighborhood V of a contained in U such that for all x â‚¬ V
there is a continuous function cp: [ 0,1 ] â€”> U such that eo(0)
= a and co(i) = x) .
Definition 4.2: A space A is simply connected (loÂ¬
cally simply connected) if and only if for each a Â£ 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] â€”> A (cp : [0,l] â€”> V) such that cp (0) = cp (1) ,
then cp is homotopic to 0 in A (in U).
Lemma 4.3: If f: [0,1] â€”> A is homotopic to zero
and cp: [0,l] â€”> A is an arbitrary continuous function.
51
52
then cp*f is homotopic to cp, where cp*f is defined as fol
lows:
(cp*f) (t) =
cp(2t) for 0 < t < Â£
f(2tl) for Â£ ^ t ^ 1.
Proof: Note f (0) = cp(l). Then the horaotopy is
effected by the following function:
^ _ 1 + s
F(s,t) = <
# 21 *
'P'l + s
for
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 0: A â€”> A is the covering homomorphism, then A/0 â€¢ 0 1 is
iseomorphic to A.
Proof: Let A be the family of homotopy classes of
continuous functions cp: [0,1] â€”> A such that cp(0) = e.
The homotopy class containing cp will be denoted by qj.
Consider any cp â‚¬ 5 â‚¬ A with cp(l) = p 6 A. If 21 is
an open basis of the topology t on A, then for each U conÂ¬
taining p in 31, let
U = [tp*f f: [0,1] â€”> U such that f(0) = p}
and 2 = [u[ U â‚¬ 31}. It is easy to see that if cp is reÂ¬
placed by any ii Â£ iÂ¡? â‚¬ U, then U and \)i will determine exÂ¬
actly the same 0 [14, p. 221],
53
(1)2J is an open basis for some topology t
on A.
If Ãœ, V â‚¬ 5, so that U, V â‚¬ ai, then U n V 6 5. It
is not difficult to show that U n V = U n V.
(2) (A, t) is a Tq topological space.
Consider any pair , cp2 â‚¬ A such that / cp2. Let
â‚¬ cp^ (i = 1, 2) such that cpi(l) = p.Â¡_.
If p^ fi p2, then since A is Tq there exists a neighÂ¬
borhood U of p^ such that p2 Â£ U. In this case, q?]_ â‚¬ 0 but
cp2 Â¿ Ãœ 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 conÂ¬
tinuous f: [0,1] â€”> U with f(0) = f(1) = p^ is homotopic
to zero. If U = {cp^*f f: [ 0, 1 ] â€”> U with f(0) = p^} ,
then 5>2 Â¿ U; for, if cp2 â‚¬ U, then there is an f: [ 0,1 ] â€”â– > U
with f (0) = p^ such that cp^*f â‚¬ cp2 , where
f cpn (2t) for 0 < t < ^
(cpi *f) (t) = I
l^f(2tl) for ^ ^ t < 1.
Note: cp ^ (1) = f(0) = = (cp^f) (1) = f(l). Thus
f is homotopic to zero and cp^ = cp^*f = cp2, which is a conÂ¬
tradiction .
(3) If for cp^ 6 cf>i â‚¬ A (i = i, . . . , m) one de
f ine s [ cp 2^2 * * â€¢ ^ ^ [ cp ^ (t) cp2 (t) ... cp^ ( f) 1 s ^d [ cp ^cp2 â€¢ â€¢ â€¢ cp^ ^
then (A, [...], t) is a topological mgroup.
Since [cpIcp2 cpm](0) = [ cp1 (0) cp2 ( 0) cpm(0)] = [eâ„¢] = e
and [cp^cp2 â€¢ â€¢ .cp ] is a continuous function on [0,l] to A, the
above operation on A is clearly well defined. Associativity
54
follows from the following relations
rf m") 2ml"
1 KK+i
(t) =
rÂ«p1j(t)cpm+1(t)..
= [[cPi(t)cp2(t) ...cpm(t)]cpni+1(t)   cp2m_1(t)]
= [cpL(t) .. .cpi_1(t) cpÂ±(t) .. â€¢cpi+m_1(t)]
n (t) â€¢ â€¢ cp2m_i () J â€” j_ _]_ (t)
[^(t) â€¢ â€¢ â€¢cPi+ml(t)]q5i+m(t) â€¢ â€¢ â€¢cf>2ml(t).
= [cpL(t) . . .cpi_i(t)[cp^+m'1](t)cpi+m(t) . . cp2m_1 (t)
Cp
r ilr i+mln 2ml'
L'pi K
i+m ](t)
which holds for each i = 2, . . m. For each cp^ â‚¬ cp^
(i = 1, m2) , let [0,1] â€”â– > A be the function
such that cp (t) = (cp (t), cp _9(t)) ^ for each
t Â£ [0,1]. Since [em] = e, so that (e, e) is also an
madic identity, then cpm ^(0) = e. Define (cp^, . .., cpm_2)
= Then for each cp,
[cpfCp2 â€¢ â€¢  1.^ ~~ *
Thus far, we have shown that (X, [...]) is an mgroup.
Next, we show that the functions
(cp 2_ > â€¢ â€¢ â€¢ Â» cpm) "â– > [ cp ^ â€¢ â€¢ â€¢ cPj^ ]
(cp , â€¢ â€¢ â€¢ > cPjq 2 ^ > â€¢ â€¢ â€¢ > cPm_2 ^
are continuous. Let cp^ â‚¬ cp^ â‚¬ A {i = 1, . . . , m) such that
cp^(l) = p^(i = 1, . . . , m) . Let V be any neighborhood of
[cpj_cp2 . . .cp,^] = [^T2 . . . cp^ ] so that every element of is of
the form [cp^cp2 . . .cpm] *f for some f: [0,l] â€”A such that
55
f(0) = p = [p1p2...pm] = [cp1(l)cp2(l) . â€¢  cpm (1) ] = [*â€](!) â‚¬ V.
By the continuity of the raary operation on A, there exist
neighborhoods containing p^ (i = 1, . . ., m) such that
[U.U...U ] C V.
12 m
Then [Ãœ.U...Ãœ ] c V, for, if if. â‚¬ U. (i = 1, . . . , m) , then
12 in â€™ yi i â€™ â€™ â€™
every i/^ â‚¬ if^ is of the form \i ^ = cp^*f^ (i = 1, . . . , m) for
some f^: [0,1] â€”â– > Ih such that f^(0) = p^. Thus,
^(t)
cp. (2t) for 0 < t < $
<
f.(2tl) for 4 < t < 1.
1 = =
Then
[ iji ^ (t) iÂ¡2 (t) . . .^m( t) ] = [cp1(2t).4>2 (2t) q>m ( 21) ]
= [cpâ„¢](2t) for 0 < t ^ $ and [ l (t) f2 (t) . . . i/m(t) ]
= [f1(2tl)f2(2tl) . . .fm(2tl) ] = [f^](2tl) for
t ^ t ^ 1.
Since f^(t) â‚¬ U^, then [fâ„¢]: [ 0,1 ] â€”â– > V. Also, if [fâ„¢](0)
= [f1(0)f2(0)...fm(0)] = [p1p2...pm] = P â‚¬ v, then
[ â™¦ ]_â™¦ 2 â€¢ * *i = = [cpâ„¢]* [ fâ„¢] â‚¬ V.
Next consider (Ã©p^, cp2, ..., cpm_2) ^ = cpm i w^ere
cpÂ± (i) = pi (i = 1, .... m2) so that (pi, p2, ..., Pm_2)1
= Let V be a neighborhood of rPm_^ with cpm_^(l)
= Pm_^ â‚¬ V. By the continuity of the inverse operation on
A, there exist neighborhoods V. of p. (i = 1, . .., m2)
such that
(V, X Vâ€ž x . . . x V â€ž)_1 c V.
1 2 m2
Again, we claim
(Vi x v2 x ... x vm_2)
c V.
56
For, if
1Â»
continuous function such that
p , â‚¬ V. Whence
Mnl
1
= (cpLÂ» CP2â€™
1
. .X
V 0
and
m2
i (1
= 1,
â€¢ â€¢ â€¢ f m
.] 
~> V
is the
fl<
t) , .
â€¢ â€¢ )
m2
(1))"
1
J
â™¦ 2'
â€¢ â€¢ â€¢ >
V2>
f 2 â€™
â€¢ â€¢ * )
fm2) ~
cpâ€ž T * t ! â‚¬ v.
Tm1 m1
This completes the proof that (A, [...], t) is a topological
mgroup.
(4) (A, t) is a second countable space.
If 9l ^ , then also  911 ^ .
(5) The covering function 0: A â€”â€¢> A such that
0(cp) = cp(1) = p is an open continuous map which is locally
a homeomorphism.
If U is an arbitrary neighborhood of p, then
0(U) c U, obviously, so that 0 is continuous.
Let cp 6 A and U be a neighborhood of cp defined by
the neighborhood U of cp(l) = p. By local arcwise connectedÂ¬
ness there exists a neighborhood V of p contained in U such
that for any x 6 V there is a continuous function f: [0,1_
â€”â– > U such that f(0) = p and f(l) = x. This implies that
cp*f 6 U and @{cp*f) = (cp*f)(l) = f (211) = f (1) = x. Whence
V c 0(U), and 0 is open.
57
Next, let cp â‚¬ A and Q (cp) = cp(l) = p â‚¬ A. Since A is
locally simply connected, there is a neighborhood U of p
such that every continuous f: [ 0,1 ] â€”â– > U such that f(0) =
f(l) = p is homotopic to zero in A. 0 is onetoone on Ãœ,
for, suppose 0(^) = 8(^2) so that for some f^: [0,1] â€”> U
we have f^(0) = p(i = 1,2), and cp*f^ â‚¬ ^ and cp*f2 â‚¬ ^2'
If f(t) = f^(l ~ t) so that f 2 * f : [0,1] â€”> U is the conÂ¬
tinuous function such that
(f2*fpit)
(f2*f{)(t)
f2(2t)
for 0 < t ^ Â£
^f{(2tl) for ^ $ t $ 1,
then (f2*f) (0) = f2(0) = e and (1) = f[(l) = f]_(0)
= e. This means then that f2*f{ homotopic to zero and
therefore cp*f^ is homotopic to cp*f2 or Ip ^ = 1/ 2 .
Since 0 is continuous, onetoone, and open on U it
follows then that it is a homeomorphism on U.
By virtue of this, then
(6) (Ã, t) is also locally arcwise connected,
locally simply connected, and regular.
(7) (A, t) is moreover arcwise connected.
Consider any cp â‚¬ A and S Â£ A where uj contains the
null path. By homogenity of a topological mgroup, it sufÂ¬
fices to show that uu and cp are connected by a continuous
path. Let cp 6 cp and cpg : [ 0,1 ] â€”â– > A be the continuous
function such that cp (t) = cp(st). For any fixed s Â£ [ 0,1 ] ,
note that cp (0) = cp(0) = e and cp (1) = cp(s) = p â‚¬ A; and
s s
hence cp â‚¬ A. Define now $: [ 0, 1 ] â€”â– > A such that $(s) =
s
58
Then Â§ is continuous [ 14, p. 223]. Moreover, $(0) = cpQ
= uj and $(1) = cp^ = cp. Whence A is arcwise connected.
(8) (A, t) is simply connected.
As in the previous, let w â‚¬ A be the homotopy class
containing the null path. Consider any continuous function
$: [ 0, 1 ] â€”â€¢> A such that $(0) = uj = $(1). Define cp such
that cp(t) = Q ($ (t) ) where 9 is the covering map. Define
also for each s Â£ [0,l], cp : [ 0, 1 ] â€”> A such that cp (t) =
s s
cp(st) . Note that cps also depends continuously on t and
cp (0) = cp(0) = e(Â§(0) ) = e(ui) = e so that cp â‚¬ A.
Firstly, observe that 0($(s)) = cp(s) = 9 (cps) for
each s Â£ [0,1].
We wish to show that cpg = Â§ (s) for all s 6 [ 0,1 ] .
For s = 0, the equality obviously holds:
cp0 = cp(0) = jj = $(0) .
Let U be a neighborhood of c?q = $(0) for which 0 is a horneo
morphism. Since both cp^ and $(s) are continuous functions
of s by an argument used in (7), then for some k sufficient
ly small we have
cp ([ 0 x)) c E and $([o,k)) c u.
For each x 9 [0,k) so that cp , $(x) Â£ 0, then since @(cp ) =
0($(x)) and 0 is onetoone on U, then cp = Â§(x). This
shows that cp = $(s) for all s less than k and hence by the
s
continuity of the function cps and $(s) we obtain
cp, = lim cp = lim $(s) = $(k).
sâ€”>k S sâ€”i>k
59
Repeating the process now for k instead of 0, we should
eventually show that the above relation holds for all
s Â£ [0,1]. For s = 1, in particular, we have
cp = cp^ = $(1) = $(0) = U)
so that cp is homotopic to zero. Suppose that this homo
topy is effected by the continuous function F: [0,1] x [0,1]
â€”> A so that F(0,t) = cp (t) , F (1, t) = cp(0) = e, F(s,0) =
cp(0) = e, F (s , 1) = cp (1) = cp(0) = e.
For each fixed s and t, define
G(s,t): [0,1] â€”> 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) = cp (0) = e
G(s,t)(1) = F(s,t) = p.
Thus for each s and t, G(s,t) Â£ 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 $ and 0:
G~(0, t) (x) = F (0 , tx) = 'pt (x) or G(0,t) = cpfc = $(t)
G( 1, t) (x) = F (1, tx) = 'cp(O) = ai or G(1, t) = o>
G(s,0)(x) = F(s,0) = cpTo) = i or G(s,0) = J = $(0)
G(s,1)(x) = F(s,x) = p = 5 = $(1).
(9) 9 is a homomorphism of A onto A.
It is clearly onto. Let cp^ â‚¬ â‚¬ A such that
cpi(l) = pi(i = 1, 2, ..., m) . Then
â€¢ * â€¢ 9
60
e([cp1cp2.. cpm]) = e( [qpxqp2   ) = [cpLcp2.. . cpm] (i)
= [cp1(l)cp2(L) . . .cpm(l) ] = [p1P2...pm] =
= [ 9 (cpj^) 9 (cp2 ) . . . 8(cpm) ] â€¢
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 1â€”> 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 A1 and
M#
A are iseomorphic.
Proof. Let 0:A â€”> A be as previously defined and
define a map k:A'â€”> A as follows: Let cp1 â‚¬ A* and let
cp:[0,l] â€”> A' such that cp(0)  e' , cp(l)  cp' with cp a conÂ¬
tinuous function. Then h o cp defines a curve in A with
(h o cp) (0) = e and (h o cp)(l) = h(cp'). Since A is simply
connected, the choice of curve cp(t) connecting e' and cp' 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,l] â€”> A and hence defines a unique point
li~'o~~cp in A. Define k(cpâ€™) = /h~~o~'cp;. It is clear that k is
well defined.
Now let U1 be an open subset of A1 and let cp^ â‚¬ U1.
Let U = {cp  cp: [ 011 ] â€”'> Uâ€™, cp (0) = cp^ and each cp is continuÂ¬
ous] and let ij/ be continuous with ijf:[0,l] â€”*> A such that
1^(0) =eâ€˜, ij/(l) = cp^. Since Uâ€˜ is open in A1 , h(U') is
61
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(Uâ€™) such that any closed curve in V is null homoÂ¬
topic, or that any two curves contained in V beginning at
cp^ and ending at the same point are homotopic.
Letting W = {h o cp  cp â‚¬ U, h o cp: [ 0,1 ] â€”> V such
that (h o cp) (0) = h(cp^)}, we see that
(1) W c h o U = {h o cp  cp 6 U},
(2) \JÃ*(h o cp)h o cp 6 Wl 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) conÂ¬
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' â‚¬ A' and let V be
o
a basic open set in A about kicp^) . Since 0 is an open map,
0(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 Q(V) and 9(k(cp^)) 6 W. Then h ^(w) is an open set in
A' about cp^ such that k(h ^ (W) ) c V. To see this, let
if:[0,l] â€”â– > A' such that \Ji (0) = e', \r (1) = cp^ an<3 â™¦ is conÂ¬
tinuous. Let X = {cpcp:[0,lj â€”> h ^(W) cp (0) = (1) and
each cp is continuous}. Since A' is simply connected,
62
\!*X = [ ii *cp  cp: [ 0,1 ] â€”> h ^"(W), cp(0) = ijf(L) and each cp is
continuous} represents all continuous functions
i/:[0,l] â€”> A' such that f(0) = e' and f(l) 6 H ^(W).
Since W is simply connected, h o X = {h o cpcp:[0,l] â€”>
h L(W) cp(o) = \Jr (1) and each cp is continuous] represents all
continuous function g:[0,l] â€”> W such that g(0) = (h o cp)
(0) = h(cp^) . Hence, k(h ^(W)) = 'h o (ty*X) = (h~ o \i) * (h oâ€™ X j
= R C V.
Next, let cp1 â‚¬ A' and suppose k(iji ' ) = k(cp'). If
k(i)i') = k(cp'), the functions iji:[0,l] â€”> A' and cp: [ 0,1 ] â€”*>
A' must be homotopic and hence, i/(l) = cp(l), i.e., f ' = cp â€™ .
Hence, k is 11.
Now let cp[, . .., cp^ 6 A' with associated functions
cp i : [ 0,1 ] â€”> A' such that cp, (1) = cp â€¢ _ 9 _
/ ) ^ )
Then k[cp ... cp^ ] = ho [cp1 ... cpml = [hocp^ ... hocpm]
(since h is a homomorphism) = [h o cp^ ... ho cpm] =
[k(cp) ... k(cp^)]. Thus, k:A' â€”â– > A is an iseomorphism.
^ ~ / 1
Recalling that the iseomorphism of A/0*0 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 horaeomorphism groups,
Amer. J. Math. 68 (1946), 593610.
[2] Boccioni, D. "Symmetrizazione d'operation naria,"
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[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. Foundations of Algebraic ToÂ¬
pology . Princeton University Press, 1952.
[6] Gluskin, I. M. "Positional operatives," Dokl. Akad.
Nauk SSSR, 157 (1964), 767770.
[7] Gluskin, I. M. "Positional operatives," Mat. Sb.,
68 (1965), 444472.
[8] Hewitt and Ross. Abstract Harmonic Analysis.
SpringerVerlag (Berlin), 1963.
[9] Hosszu, M. "On the explicit form of mgroup operaÂ¬
tions," Publ. Math. Debrecen, 10 (1964), 8792.
[10] Lehmer, D. H. "A ternary analogue of abelian groups,"
Amer. J. Math., 54 (1932), 329338.
[11] Los, J. "On the embedding of models I," Fund. Math.
42 (1955), 3854.
[12] Mal'cev, A. I. "On the general theory of algebraic
systems," Amer. Math. Soc. TransÃ. (2) 27 (1963),
125142.
[13] Miller, G. A. "Sets of group elements involving
only products of more than n," Proc. Nat. Acad.
Sci. 21 (1935), 4547.
63
64
[14] Pontrjagin, L. Topological Groups. Princeton
University Press, 1939.
[15] Post, E. L. "Polyadic groups," Trans. Amer. Math.
Soc. 49 (1940), 208350.
[16] Robinson, D. W. "mgroups with identity element,"
Math. Mag. 31 (1958), 255258.
[17] Sioson, F. M. "Cyclic and homogeneous msemigroups,"
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[18] Sioson, F. M. "Generalization d'un theoreme d'Hop
kinsBrauer," C. R. Acad. Sci. 257 (1963),
18901892.
[19] Sioson, F. M. "Ideals and subsemiheaps," to appear
in Rev. Math. PurÃ©s Appl. 11 (1966).
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t
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Japan Acad. 39 (1963), 283286.
[23] Sioson, F. M. "Remarques au sujet dâ€™une Note anter
ieure," Acad. Sci. 257 (1963), 3106.
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groups, and function representations," to appear
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[25] Tvermoes, H. "Uber eine verallgemeinerung des Grup
penbegriffs," Math. Scand. 1 (1953), 1830.
[26] Wallace, A. D. "An outline for a first course in
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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 ColÂ¬
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 UniverÂ¬
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.
65
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
s and Sciences
Dean, Graduate School
SUPERVISORY COMMITTEE:
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UNIVERSITY OF FLORIDA
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