Topological m-Groups


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Topological m-Groups
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iv, 65 leaves : illus. ; 28 cm.
Richardson, R. L ( Robert Lee ), 1937-
University of Florida
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Group theory   ( lcsh )
Topology   ( lcsh )
Mathematics thesis Ph. D
Dissertations, Academic -- Mathematics -- UF
non-fiction   ( marcgt )


Bibliography: leaves 63-64.
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Manuscript copy.
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Thesis - University of FLorida.
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Full Text






August, 1966

Digitized by [ie Inlernel Archive
T62 l&iggr.fding from
University of Florida. George A. SmaIners Libraries wi[n support from LYRASIS anrid [ne Sloan Foundation


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-


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.














. iii

. 1

. 26

. 37

. 51

. 63

a .

. .


In this paper some of the theory of topological

m-groups will be developed with some new contributions to

the theory of algebraic m-groups. The main result is that

any topological m-group can be considered as the coset of

an ordinary topological group. While almost no work has

been done in the field of topological m-groups, an exten-

sive theory of m-groups has been developed through the


In 1928, W Dornte [4] introduced the concept of an

m-group as an extension of a group or 2-group 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ner in an unpublished paper, but his work indicated that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 3-group in Dornte's ter-

minology, apparently without knowledge of Dornte's work

and proceeded to develop a theory of these triplexes.

In 1935, G. A. Miller [13] obtained a result for

finite polyadic groups stating that every finite m-group

is the coset of an invariant subgroup of some ordinary

group. Unfortunately, he makes the tacit assumption that

the set of elements in the m-group comes from an ordinary

2-group initially.

A major advancement in the field of m-group theory

was achieved by E. L. Post [15] in 1940 when he proved that

any m-group (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 2-group concepts, with the

notable exception of Sylow's theorem, can be extended to


In 1952, H. Tvermoes [25] introduced the concept of

an m-semigroup and did a little work on them, but his main

interest was again in m-groups.

In 1963-65, F. M. Sioson introduced the concepts of

topological m-semigroups and topological m-groups. In pa-

pers by him [17], [18], [19], [20], [21], [22], [23], var-

ious generalizations of many theorems in 2-semigroups to

m-semigroups have been achieved. In a paper with J. D.

Monk [24], it is shown that any m-semigroup can be embedded

in an ordinary semigroup in such a way that the operations

in the m-semigroup reduce to those in the containing semi-

group. A representation theorem for m-semigroups then re-



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].



In this chapter we will develop some elementary

ideas concerning algebraic m-groups including a new proof

of the Post Coset Theorem [15]. With the exception of [16],

very few results on m-groups with idempotents have been

published; however, some new theorems concerning m-groups

with idempotents are obtained in this chapter.

One of the important problems arising in the study

of m-groups is the question of which congruences determine

a sub-m-group. 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 will be denoted by x. and if
x. = x = ... = xi+k = x simply as x k+1 denoting the

number of times the element x occurs.

Definition 1.1: An m-semigroup is a set A and a

function f: Am--> A such that for all al, ..., a2m-1 E A,

f(f(al, ..., am), am+l, ..., a2m-1) = f(al, f(a2, ..., am+ ),

am+2, *., a2m-1) ... = f(al, a2, ..., am-1, f(am' ."'

a 2m-1)). Following customary usage, we shall write

[a1 ... am] = [am] for f(al, ..., am) and (A, []) for (A,f).

Proposition 1.2: For any k = 1, 2, ..., (A, []) is

also a k(m-l)+l semigroup (A, ()), where (ak(m-)+l =
[m .. ] k(m-l)+l
[[...[a ]...a(k-1) (m-1)+1 +
For example, a 2-semigroup A is a k(2-1)+l = k+l

semigroup for any k = 1, 2, ... If k = 2, then A becomes

a 3-semigroup by defining (ala2a3) = [[a a2]a3].
Definition 1.3: An m-semigroup (A, []) is an m-

group iff for each i and for all al, ..., ai-1, ai+1, ...,

am, b E A there exists uniquely an x E A such that

[a xam I = b.

The following are :ome examples of m-groups.

Example 1.4: Let R be the set of negative real

numbers and define []:(R-) 3-> R- by [xyz] = x-y.z, the

ordinary product of x, y, and z. Then (R []) forms a


Example 1.5: Let SI, 2, ..., S m be any m-1 sets

of the same cardinality. Let F(S1,..., Sm-1 ) be the family

of all bijective and subjective functions f: U-1 S.-

U m S. such that f(S.) = S where p is any fixed per-
i= 1 1 p(i)
mutation of 1, 2, ..., m-l. Let []: Fm--> F be defined as

the composition of m functions, i.e. [f ...... fm (x) =

(f..... (fm(x))). (F,[]) is an m-group since it is clear

that [] is m-associative and if x E S., f (x) E Sp(i),

f-1 (fm(x)) E S (p(i)),...... (f "' (fm(x))) which is an

element of S where pm(i) is the mth permutation of i.
Pm (i)
m-i th
p (i) is i, so the m permutation of i is p(i). Hence,

[f1 ... f : S--> Sp(i) so that [f ...1 f ] E F. Unique

solvability follows from the fact that the functions are

bijective and surjective.
Example 1.6: Let Z be the integers. Then (Z, [])

forms an m-group when []: Zm-> Z is defined by (xl ... x ]
= x + ... + x + h for any h E I.

Example 1.7: More generally, if (G,.) is an ordi-

nary group and if h is in the center of G, (G,[]) will be
an m-group if []:Gm--> is defined as [x ... x] =

xl'*x2 **..m.h

Example 1.8: Let 2, 2m-1 = I be any complex

(m-1)th roots of unity. Define []:m -> t by [a1 ... am] =
aI + a2 + ... + a m (C,[]) forms an m-group for it is

clear that all conditions except m-associativity are satis-

fied, and that condition being satisfied is apparent if the

following two expansions are studied: [[a ]a 2m-1] = [am] +
m- i-+ I
am+l + ... + a21m-l = a1 + a2t + ... + ai + ai+1 +
m-1 i+1 m-1
... + a+ 1+ + + a m+i+l + ... + a m .
iim 1i-1 i+m i
[air i+m 2m-] = a + a + ... + ai + [a i+mi
l i+1 i+m+l 1 2 + i+1

a i+m+l + + ... + am-l 1 = aI + a2 + ... + aiI +
i+m+i1 i+m+l-i- m-
ai+(m-i)i+m-i-1 + ai+(m+l-i) + ... + a2m-l -
Equality is apparent by noting that a i+(m+li) =

a E1m = a M+ -1 ?= a F
m+1 r+1 am+l.

Example 1.9: Let Zodd be the odd integers under the

operation [x1 x2 x3] = xl+ x2 + x3. Then Zodd is a 3-group.

*Example 1.10: Let V1, V2, ..., Vm1 be finite

dimensional vector spaces of the same dimension n. Let

L(V1, ..., V m1) be the set of all (m-1)-tuples of non-

singular linear transformations (Al, A2, .... Am 1) where

A.:V --> Vp(,) for the permutation p = (12...m-l). L is
1 1 2
an m-group under the operation [(A1, ..., A _-)(A2, ...,
2 m m 1 2 -m 1 2 m-2
Am-1...(A, ..., Am-1] = (A1 A2 ...A1, A2 A3 ...Am-1
m- m 1 1 2 3 mn
A1 A2 ... ,A A2 A ... Am ). Associativity is clear,
1 2 m-1 1 2 m-1
and unique solvability follows from the condition that the

linear transformations be non-singular.

Definition 1.11: An (m-l)-tuple (el, e2, ..., e -1)

of elements e. E A is a left (right) m-adic identity iff
for all x E A, [em-' x] = x ([xe1 ] = x). When

(el, e2, ..., em-1) is both a left and right m-adic iden-

tity it is simply called an m-adic identity.

Proposition 1.12: For any a, el, e2, ..., em-1 E A,

if [el- a] = a([ael-1] = a), then (el, e2, ..., em_) is a

left (right) m-adic identity.

Proof. Let x E A be arbitrary and [el- a] = a.

Choose a2, ..., am E A such that [aam] = x. Then x =

[aa2] = [[el a]a2] = [e-1 x]. The other part is proved

in a similar fashion.

Proposition 1.13: Every left m-adic identity in an

m-group is also a right m-adic identity and conversely.

Proof. Let (el, e2, ..., e m_) be a left m-adic

identity. Note that [ae e-2[e el]] = [a[eml1 e e J-1]
1 m-1 1 1 1 2

= [aem-l]. From the definition of an m-group it then fol-

lows that [e em-11 = e which by Proposition 1.12 im-
m-i 1 m-1
plies that (el, e2, ..., em_ ) is also a right m-adic iden-


Proposition 1.14: If (e,, e2, ..., em_1) is an -

adic identity, then for each i, (el, ..., em1, e1, ...,

e _I) is also an m-adic identity in the m-group.

Proof. If (el, e2, ..., em_-) is an m-adic identity,

then [e1 e2...em-1 e11 = eI and hence by Propositions 1.12

and 1.13 it is also true that (e2, ..., e -1, el) is an m-

adic identity. By a repetition of this argument, the re-

sult follows.

Definition 1.15: An element x in the m-group will

be called an idempotent if and only if [xm ] = x.

Proposition 1.16: If x is an idempotent in an m-

group A, then (x, x, ..., x) (m-1) times is both a left

and right m-adic identity.

Proof. (x, x, ..., x) (m-l) times is a left and

right identity on x; hence, by Proposition 1.12, for all

z E A. By Proposition 1.13, (x, ..., x) (m-1) times is

both a left and right m-adic identity.

Definition 1.17: The inverse of an (m-2)-tuple

(xl, x2, ..., Xm-2) of elements from an m-group is the

unique element x m1 also denoted by (x x2, ..., x m_2)

such that (x x2, ..., xmI1) is an m-adic identity. We

note that such an element always exists by the definition

of an m-group and Proposition 1.12.

Definition 1.18: Let S1, ..., Sm- be any (m-1)

sets. An m-adic function on S, ..., Sm-1 is a function

m-1 m-1
f: U S. -> U S.
i=l 1 i=l 1

such that f(S S (i) where a = (12...m-1).

Proposition 1.19: The family of all surjective and

bijective m-adic functions on sets Sl, ..., S m-1 of the same

cardinality forms an m-group.

Proof. See Example 1.5.

Definition 1.20: Two k-tuples (k < m) of elements

from an m-group A are equivalent, i.e., (a i+, ..., ai+k)

(bi+1 '.., bi+k), iff for all x ..., x, +k+, ...,

xm E A (0 < i, i+k < m),

[i i+k m ] r i i+k m m
x1ai+1xi+k+1 = [xli+i i+k+1

Note that by the above definition (al, ..., am) m (bl ...,-

bm) if and only if [am] = [b ].

Proposition 1.21: (ai+1, ..., ai+k) (bi+l' .

b i+k) iff there exists c ..., ci, c++. ..., c E A such

i i+k m i i+k m
[clai+lci+k+l] = c i+ici+k+1

Proof. Let di+, ..., dm, d2, ..., di+k 6 A such

that (di+1, ..., dm_1, cl, ..., c ) and (ci+k+l, ..., cm'

d ..., di+ ) are m-adic identities of A. Then for each

xsl, -., x xi+k+l', -, xm E A,

[xi i+k m
xlai+ i+k+1
1 i m- i i+k-1 m i+k m
= i+clai+ i+2 i+kCi+k+l2 xi+k+l

r i[m-lr i i+k m rr-i+k m I
= [xI di+1 clai+lci+k+ilEd2 xi+k+1

Sim-1 i i+k-lr m i+k

= [xtdi b d]bi+k-1 [b+cm di+k xNM
S i+l i+l i+2 i+k i+k+12 +k+l
Sii+k m
= LX1bi+iXi+k+l-

Proposition 1.22: a is an equivalence relation.

Proof. That a is reflexive and symmetric is clear.

Suppose (ai+1, ..., ai+k) (bi+1 **..., bi+k) and (bi+1'

..., bi+k) a (ci+l, ..., 'i+). By Definition 1.20, for
i i+k m
any xl, ..., xi, xi+k+l ..., xm t A, [x a i+xi+k+l

Sib i+k m i r i+k m i
xlbi+ixi+k+l = [xlci+xii+k+I1. Hence, (ai+, ..., ai+k)

S(ci+l, ..., Ci+k).

Let S. = Ai /. Note S = A.

Theorem 1.23: Any m-group is isomorphic to an m-

group of bijective and surjective m-adic functions on dis-

joint sets S1, ..., Sm-_.
Proof. Let F(S, ..., S -1) be the family of all

surjective and bijective m-adic functions on Si = A /a,

m-1 m-1
i = 1, ..., m-l. For each a E A, define L : U Si--> U S.
i=l i=1
such that La((Xl, ..., x )/a) = (a, xl, ..., xi)/2, i = 1,

..., m-2, and La((xl, ..., xm_1)/*) = [ax -1
a 1 ix i

This is well defined since if (xl, ..., xi) a

(y' ..., yi), then also (a, xl, ..., xi) u (a, yI, ...,

Suppose La((xlp, ..., x)/) = La((Y "... Yi)/)

so that

(a, xl, ..., xi) a (a, y ', ..."' yi)
i m-1 i m-1
and hence [ax ai+1] = [aylai+1] for some ai+, ...,

a -1 E A. Thus, by Proposition 1.21, (xl, ..., xi)/E

= (y' ... yi)/e; that is to say, La is bijective.
Let (y,l ..., Yi+)/ E Si+I. Then for ai+2, ...,
a E A, there exists uniquely (by definition of an m-group)

an x E A such that
i m i+1 m
[ay2xai+2 = [y1 ai+2 ].
This means that La((y2, -... yi, x)/a = (yl' ...," yi+)/'"

Thus La is also surjective.

Define f: A -> F(SI, ..., Si ) such that f(a) = La-

Note that

([am], xl, ..., xi) a (al, [amxl], ..., x )

[a2x...., ..., x. ) (a2, [a.3x ], ..., xi)

([ax-I ], x) (ai, [a x ]),

([ai+1xl]) a (ai+ ..., am, xI ...o, x ).
L[am]((xI, ..., xi)/s) = ([am], xl, ..., x )/a

=La([ax], x2, ... x)/s) = ...

= L a2 ..L ((x ..., xi)/-).
1 2 ^m

Whence f([am]) = L = LL ...L = f(al)f(a2)...f(am
[al ] 1 2 m

f is also one-to-one. For, if La = f(a) = f(b) = Lb,

then La((xl, ..., xm-_)/m) = Lb((xl, ..., Xm-_)/a) and

therefore [ax -] = [bxl -] or a = b by Definition 1.3.

Next we shall prove the Post Coset Theorem. Other

proofs may be found in Bruck [3] and Post [15]. The ana-

logue of this theorem for m-semigroups has been proved by

Los [ll] for m = 3 and later for arbitrary m by Sioson and

Monk [24].
Theorem 1.24: (Post Coset Theorem). Let (A,[]) be

an m-group. Then there exists a group (G,-) and normal

subgroup N of G such that G/N is cyclic and A = xN with

[am] = a *a2* *an for all a,,a2,...,am E A. In fact, G/N

= tA,...,A -1), N = Am-1 and the order of G/N divides m-1.
Proof. By the representation theorem for m-groups,

every m-group is isomorphic to an m-subgroup of the m-group

of surjective and bijective m-adic functions on Si,...,

Sm-_. Let G be the group generated by A.

Note that for a fixed a E A and any element b = E Am- there exist uniquely x, y E A such that
Sm-1 i
[aI x] = a1...a x = a

[yam ] = = a.

Thus every element of Am-1 can be expressed as ax- or y- a

for any fixed a E A and x, y E A. Thus, if b1 = y1 a,

b2 = y2 a are any two elements of A then blb2 =

-1 -1 -1 -I -1 -1
(yl a)(y2 a) = yl1 aa y2 = yl y2 is also an element of

A This means A is a subgroup of G. Note also that
rM-1 rm-1 -l rn-1 ,-i
for each a E A, aA m-1= A = A a. Thus a A a = Am- =

aA a for each a E A. Since A generates G, then any

g E G may be written as g = all...a in for a. E A and i =
n im

-1 or 1. Then g A g = A and A is a normal sub-
r-i -la m-i -1
group of G. From aA = A we obtain a aA = a A =

A m-. Thus a Am-l = (a A)Am-2 = A m-2. Similarly, for
-l -1 m-1 -l m-2 nm-3
al, a2 E A, a1 a2 A = a A = A and so on. Thus

G/Am-1 = A U A2 U A3 U ... U Am-1.

Some of the Ai's may be equal. In any case, the order of

G/Am-1 is a divisor of m-1.

Definition 1,25: G will be called the containing

group of A and A m- the associated group of A.

If the order of G/A is exactly m-1, 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 m-group by

considering a free group generated by the elements of the


Tne following theorem due to Sioson [20] will prove

useful in the sequel.

Theorem 1.26: The following conditions for an m-

semigroup are equivalent:

(1) A is an m-group;

(2) For all i = 1, ..., m, for al, ..., ai-1,
ai+l, ..., am, b E A, there exists an x E A such that

[a-l xa i+ = b, i.e., [ai-1Aa = A;

(3) For some i between 1 and m, for al, ...,

a b E A, there exists an x E A such that [a xa. ] =
i 1 i+l
b, i.e. [a i Aa+] = A.

(4) For each al, a2, ..., am_1 E A, [am- A] =

A = [Aa-1];

(5) For each a E A, [aAm-1] = A = [Am-la];

(6) For all a., ..., am-2 E A, there exists

a -1 E A such that (al, ..., am-1) is an m-adic identity.

Proof. (1) implies (2) implies (3) implies (4) im-

plies (5) are obvious. (2) implies (1) by definition of m-
i-1 m i-I m
group; for, if x, y E A such that [aI xai+l] = [aI Yai+l'

then for some elements b., ..., bm_ c2, ..., c. E A,
i m-1 21
(bi ..., bmn-_l, al, ..-, ai-l), (ai+1, _-., am, c2, ..., ci)

are m-adic identities and hence x = y.

(5) implies (2). Let (5) hold and al, a2, ...,

a E A. Noting A = A, then
Si-1 m i-1 2m-1 m m
[a Aai+l = [a A ai+
Si-2 rm-li rm-i m
= [a1 [a A ]A[A a+ 4a ] ]
= ie1ai+l i+2]

= [a-2 A a+2] = ... = Am = A.

(2) implies (6) as we have already seen.

(6) implies (4). We shall show the existence of an

x such that [xa ] = b for any a2, ..., am, b E A. Since
such thtxa

[Am] = A, then we may write the above equation as
rrr 1 1 1b 22 b2 m-2 m-2 i.nm-2
[[...[[xb b]bb2...b m-1...]b b2 ...b ] = b.

By applying (6), then
m-2 m-2 m-2 -1 2 2 2 -I 1
x = [b[bl ,b2 ,...b m_-1.. .[b,b2,...b m_-1[bl,b21...

b1 .

Definition 1.27: A subset S of an m-group (A, [])

is called a sub-m-group iff S is closed under the same

operation [] in A and for each xl, x2, ..., Xm-2 E S, (xl,

x2, ..., Xm-2) E S.

Proposition 1.28: If S is a sub-m-group of an

m-group A, then H = S is a subgroup of the containing

group, in fact, of the associated group of A.

Proof. Let A be an m-group with containing group G.
2 M-1
As before, we may assume A c G = A U A u .... U Am Let

S be a sub-m-group and H = Sm-i c G. Since S is a sub-m-

group of A, for each x x2, ..., m2 S, there exists

(x x, ..., x ) = x 1 E S such that (x x2, ...,

x m-) is an m-adic identity. Hence x x2 ... Xm-1 as an

element of G is the identity and is also an element of H.

Next, let x1, x2, ..., xm-1 be m-1 elements of S. Then for

any y, y2l ...' Y m-2 6 S there is a ym-1 E S such that

([xm-y]' Y2' ... Ym-_) is an m-adic identit-" which im-

plies then that (xlx2 ... Xm-1 (y12 .. Ym_-1) is the

identity of H. Hence yly2 ... ym-1 is the inverse of

x x2 ... x -1 an arbitrary element of H. Also note that

H.H = Sm-1 Sm-1 = Sm Sm-2 = Sm-1 = H. Thus H is closed

both under the binary operation in G and inversion and

hence is a subgroup of G and hence of A m-, the associated

group of A.

Definition 1.29: A sub-m-group S of A is called in-

variant iff [a' S ai+.] = S for each m-adic identity (al,

a2, ..., am_1) of A and each i = 1, 2, ..., m-2 and also

[a1 [ a 2]] = S for each a a2, ..., a2_r2 such that

([am], a ..., a ) is an m-adic identity.

Definition 1.30: Let S be a sub-m-group of A. If

the associated subgroup of S in the containing group G of

A is invariant in G, then S is called semi-invariant.

Proposition 1.31: Every invariant sub-m-group is


Proof. Let S be an invariant sub-m-group of A

with the associated subgroup H = S in the containing

group G = A U A2 U .... U Am-1 of A. Let x C G so that

x = aa2...a. for some i < m-1 and al, a2, ..., a, E A.

If i < m-1, then there are a.i+, ..., a m- E A such that
(a,, a2, ..., am_1) is an m-adic identity. Thus x =

a. .... a -i is the inverse of x in G. S is invariant,
i m-i
so aI S a i+ = S. Since the group operation gives the m-
Srm-1 1 m-i Im-l
group operation, then H = S = (a S a )i+

nm-1 i m-2
(a a .... a)(S a1 a)-2 S(ai+ .... am) =

rm-1 -i -i
x S x = x H x If i = m-1, then there exist

a ..., a E A such that ([am], am+,, ..., a2m_2) is

an m-adic identity and hence also

(a2m2, [am], a +, ..., a2m). Note also that then
2m-2a1 m+1 2m-2

([a2m2am-]am, ..., a2m-3) and (am, ..., a2-3

[a am-1 ]) are also m-adic identities. Then
x = a a ... a_, x = a a m+ .... a 2m2 and

Sm-1 m -1 2m-2 m-l (
H = S -= (arI (S am )) = (a1 a .... ar-) "

( 2m-3 m-2 sm-i -1
(Sam (a a )) S (a... ) = xS x
Sm 2m-2 1 am 2m-2) =
x H x Thus in both cases H is invariant in G.

Proposition 1.32: A sub-m-group S of an m-group

A is semi-invariant iff [a Sm-1] = [Sm-l a] for all a E A.

Proof. Suppose [a Sm- 1 = [Sm-1 a] for all a E A.

Let H = S m-1 be the associated subgroup of S and G the
containing group of A. Thus in G we have a H a = H.

Since a E A, then a = (a2, a3, ..., a m1) such that

(a, a2, ..., a ) is an m-adic identity. Since G is

generated by A, then H is normal in G and the result fol-


Theorem 1.33: If S is a semi-invariant sub-m-group

of A, A/S exists and is an m-group.

Proof. Let H be the associated subgroup of S and G

the containing group of A. First note that A = U(y yExH].

Consider the set [xH xEA). Since H is normal in G, if

x 1 x2, either x 1H = x2H or they are disjoint. An m-ary

operation may now be defined on the set (xHIxEA) by defin-

ing (x1H x2H ... xmH) = [xm]H. This operation is associ-

ative for ((x H ... xmH)x m+H ... x2m-1H) =

([x ]H x H ... x H) = [[x] x2m-1 ]H = [x [x i+]
1 m+1 2m-1 1 m+1 1 i+x

[x [x i+mxm- 1H = (x H ... x i[x+mH ... x H)
Xi+l ]xi+m+r"l' xiLxi~1H +*i+m+H X2m-1H)

= (x1H ... x H(xi+1H ... xi+mH)x i+m+H ... x2m-1H). Next,

it is necessary to show that for any xEA there exists aEA

such that for x,, x2, ..., xmEA, (x H ... xH a H xi+H ...

x H) = xH or [xi a X+l ]H = xH. Since A is an m-group,

this can clearly be done. The set (xHIxEA) is A/S.

As opposed to a 2-group, a congruence relation on an

m-group need not determine a sub-m-group. Consider the con-

gruence relation of equality. This will determine a sub-m-

group if and only if the m-group 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 sub-m-group.

Definition 1.34: A relation R on A is a subset of

A X A.

Definition 1.35: The domain TT,(R) of a relation R

on A is the set nl(R) = (x: (x,y) E R for some y E A) and

its range is the set "2(R) = [y: (x,y) E R for some x E A).

Notation 1.36: If U is a subset of A, denote

RU = n (R n (A x U))

UR = 12(R n (U X A)).

Definition 1.37: A congruence relation R on an m-

group A is a relation which is reflexive, symmetric, and
transitive, and such that if (xl,yl) E R, ..., (x ,ym) E R

then ([x1,[y1]) E R.

Theorem 1.38: Let R be a congruence relation on an

m-group A. Then R* = (A x A) U [(x,y) x,yEA for some i = 2,

..., m such that x = e ...e. x', y = e ...e. y' implies

(x',y') E R) is a congruence in the covering group of A.

Proof. Let (el2,e2, ..., e m_) be an m-adic identity

of A. Note that each x E G is either an element of A or can

be uniquely written in the form

x = e1e2...ex'

for some x' 6 A and i = 1, 2, ..., m-2. For each x, y E G

define (x,y) E R* if and only if either x, y E A or

x, y E A for some i = 2, ..., m-1 such that x = ele2...e ix'
and y = e1e2 ...eily' with (x', y') E R. Reflexivity, sym-

metry, and transitivity of the relation R* are clear. Sup-

pose (xt, yt) E R* (t = 1, 2) so that either xl, y, 6 A or

x = ele2 ...ei- 1x Yl = eie2...e yl with (x', Yl) E R

and either x2, y2 E A or x2 = le2e...e ej-x2, Y2

ele2...e jlY with (x,' y) E R. Obviously, x x2, y1y2 Ak

for some k = 1, 2, ..., m-1. Thus it suffices to consider

the case when k is greater than 1. Let x lx2 = ele2...ek_-x'

and yly2 = e e2. *.ek-ly. We shall show (x', y') E R. Ob-

x [m-1 k-i ,1 m-1 m-1 i-1 j-1
x = [ekel x ] = [ek x1x2 = [ek el xel x ]

m r o-1 k-1 m-1 m-1 i-1 j-*
and y' = [e- el y'] = [ek 1Y2] = [ek el'lyeJly].

Since (xi, y) E R, (x2, y2) E R and R contains the diagonal

of R, then ([emk ei x'e x2], [e-e ei. Ye1 Y12]) E R.

Whence (x', y') E R; and (x x2, y1y2) E R*. Lastly, we

also show that (x, y) E R* implies (x y ) E R*.

Suppose (x, y) E R* so that either x, y E A or

4 = e ...e. ix' and y = el...ei-lY' with (x', y') E R. We

shall consider two cases. In case i = m-2, so that

x = ele2...em-3x' and y = ele2...em-3y' with (x', y') E R,
-l -1
then x = x' E A and y = y' A. Thus

m-1 ,-1 1 m-3 ,y
x = [e xo] = [[e xoJe y'y ]

m-1 m-lx m-3
Y = [e ly'] = [e [xem x yo]].
Since (x', y') E R and R contains the diagonal of A, then
r m-i m-3 m-x1 m-3 -1 -1
([eI xoel x yo], [el xe Y'y o) E R. Whence (x y )

= (x', y) E R. Otherwise, in case i A m-2, let x =
ee ...e *xm and y- = e e ..e As in the pre-
e2.emi_2xO ad ye12 m2i_2
vious case, from the fact that
m-1 m-i-2 i-1 m-i-2 ,1 r m-1 m-i-2 i-i m-i-2
([emi e ye x '], [emilel Ye y1 x)
mn-i-l 1 o1 1 o 'e ]m-i-) o 1 1 o
E R and
S m-1 m-i-2 i-ix m-i-2x] and
Yo em-i-le [ye1 'el1
xo = em-i m-i-2'] = em- rem-i-2 i-ly e m-i-2
S em-i-lel o =em-i-1 iel Yoel y'el~e2 xo
rm-I m-i-2 i-i, mi-i-2
S[em-i le 2oel y1 e1 xo]], it follows that
-l -1
(xo', y) E R and hence (x-, y ) E R*.
Since el ..em-1 is the identity in G and R* is a

congruence in the group G, then e e2. .e -1R* is a normal

subgroup of G. Observe also that e e2...e -1R* =

(ele2...em-2)-(em-1R). If x E (ele2...em-2)-(em-1R) so

that x = ele2...em-2x' where (x', em-_) E R, then

(x, e E R*. Thus, the right side of the above

equality is contained in the left. If (x, ele2...e m1) E R*,

then x = ee2...em-2x' for some x' E A with (x', em-1) E R

and hence also x E (ele2...e _m2)(em-1R). The above equal-

ity is thus demonstrated.

Theorem 1.39: Let R be a congruence relation on an

m-qroup A. The congruence class em-1R of R is a sub-m-

qroup of A if and only if there are elements el, e2, ...,

em_2 E e m-R such that (el, e2, ..., em_,) is an m-adic


Proof. If emlR is a sub-m-group of A, then obvious-

ly it contains elements el, e2, ..., em_2 such that (el, e2,

..., em) is an m-adic identity.

Conversely, suppose em,-R contains elements el, e2,

.., em2 such that (el, e2, ..., eml) is an m-adic identi-

ty. By the preceding theorem, R and (el, e2, ..., em,)

determine a congruence relation R* on the covering group

G of A whose congruence class of the identity e e2*...em1R*

= (ee2...e m2)*(e m1R) = N is a normal subgroup of G. We

shall show that em-_R = (em-le...e m-2). (e mR) = e m-N is

a sub-m-group of A. It is clearly a subset of A. The

closure of e ,R = e ,N is also clear, for, if x. E e1 R

for i = 1, 2, ..., m, so that (xi, em,1) E R, then

([xm], [(e )m]) E R. Since also (e., e ) E R (i = 1,
1 m-l i m-1
2, ..., m-1), then (e [(e )m]) = ([em-le
S m-l 1 m-1
[(e )m ]) E R. Hence ([xm], e ) E R and therefore
m-1 1 m-1
[xm E e R. Finally, let x, x2, ..., x m e R
1 mI- 1 2m ..., m-) a m-
e 1N. Then x = e1 n (i = 1, ..., m-l) and

[(em_ ln) (em-1n2)...(em-lnm-l)(em-1N)] = em-ln lem- n2

e n e N = m[(e )m]nn ^.n- N for some n', n, .,

n'- E N. From the steps above recall (e ,, [(e m_)m]) R
or [(em-_)m] E em-1R = e-1N so that [(e )m] = e -1n for

some n E N. Hence [X1X2...xm-1(em-1N)] = [(em-ln)(em-1n2)

...(em-i nml)(e m-N)] = [(eml )m]nn'...n' m- =

e m (nni'n ...n' ,N) = e mN. In an analogous manner
m-1 1 2 m-1 m-1i
[(e _1N)x 2...x _] = e N. This completes the proof

that e -1N = em LR is'a sub-m-group of A.

Corollary 1.40: If R is a congruence relation on an

m-group A with idempotent e, then eR is a sub-m-qroup of A.

Proof. The proof follows immediately from 1.38.

Under certain circumstances it may be possible to

reduce, say, a 5-group to a 3-group 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 n-i tuple (a,, a2, ..., an-1)

will be said to be commutative with an element x if and

only if

(x, al, a2, ..., an-l)s(al, a2, ..., an-1, x).

Proposition 1.42: An m-group is reducible to an n-

group, m = k(n-1) + 1, k a positive integer, if and only

if there exists an n-i tuple (al, a2, ..., an_1) which com-

mutes with every element of the m-qroup and such that

(al, a2, ..., an_-, al, a 2, ..., an-_, ..., a, a2, ..., an-),

with the n-1 tuple (al, a2, ..., an_-) repeated k times, is

an m-adic identity.

The n-adic operation may be defined as (x1.. .x) =

n-i n-i
[xl...x al .. aI ] where the right-hand side of the

above equation is the ordinary m-group operation, and an-

occurs k-i times. (k-l) (n-l)+n = kn-n+1-k+n = kn-k+l =

k(n-l)+l so the operation is properly defined.

Proof. If there exists an n-i tuple (a,, a2, ...,

an-) satisfying the stated properties, Condition 2 of

Theorem 1.26 is immediately satisfied since the m-group

has this property. Furthermore, the operation is associa-
n 2n-l n) 2n-1 n-1 n-l
tive for ((x ) x = [(x) x+ a ...a ] =

n n-1 n-i 2n-1 n-1 n-l [ [Xn+1 n-1 n-l
L[x a ...a1 ] x1 a ...a ] = Lx[x a ...a
1 1 1 n+1 1 1 1 2 1 1
2n-1 n-i n-1 n+l 2n-1 "n-1
xn+2 al ...a = (X(X2 ) n+2 = =
[2n-l n-1 n-l n-1 n-li n-1 2n-l
[x a ...a a ...a = (x (x )
n i 1 1 i n
Next, if the m-group is reducible to an n-group, the

n-group has an n-adic identity. (al, a2, ..., an-, al, a2,

... an-1, a a2, an_-, ..., al, a2, ..., anl) k times

will then be an m-adic identity and that (al, a2, ..., an,)

commutes with every element of A is clear by Proposition

1.13 and the fact that A reduces to an n-group.

Corollary 1,43: Every commutative m-group with idem-

potent reduces to a 2-group.

Proof. It is immediately apparent that the condi-

tions of the preceding theorem are satisfied.

Example 1.6 is an example of an m-group, m > 2, that




In this chapter we derive some of the properties of

topological m-groups and quotient m-groups.
The following definition is due to F. M. Sioson.
Definition 2.1: A topological m-group (A, [], .)

is an m-group (A, []) together with a topology T on A under
which the functions f and g defined by f(x1, x2, ..., xm)

= [xm] and g(x x2, ..., Xm2) = Xr where (x1, x2, ...,

Xm_ ) is an m-adic identity, are continuous.
Proposition 2.2: If (A, [], 7) is a topological m-
group, then for any k = 1, 2, ..., (A, [], T) is also a
topological k(m-l)+l group (A, (), 7) when we define

(ak(m-1)+l) = [[.*[am]*.]ak(l)m-1)+l
1 1 (k-1)(m-)+l1
Proof. The proof is clear.
Example 2.3: Let R be the set of negative real
numbers with []:(R-)3-> R defined by [xyz] = 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 3-group.

Example 2.4: Let S' = (zlz = ex 0 A x < 2r ).
Define []:(S')m--> S' as follows: If zl, ..., zm E S' and

ix ix
z = e ..., = em then [zl] = e .(X1+--- If
1 1
T is the usual topology on S', then (S', [], T) is a topo-

logical m-group.

Example 2.5: If (G, *, 7) is an ordinary topologi-

cal group, and if h is in the center of G, (G, [], 7) will

be a topological m-group if []: Gm--> G is defined as [xT]

= X *x2 Xm h.
1 2 m
Theorem 2.6: The function h: A A defined by

h(x) = [ai-1xa+ ] is a homeomorphism for each choice of

i = 1, ..., m and elements al, ..., am E A (where by con-

vention [a i- xam is [xam] when i = 1 and [am-lx] when
I xai+l 2
i = m).

Proof. The function h is the restriction of the map

f to the subset [al)x...x(ail)X A X (ai+]x...X(a m of

A x A X...X A (m times) and hence is continuous. Let bl,

b2, ..., bm_1 E A such that (al, a2, ..., ail, bi, ...,

b _) and (bl, b2, ..., bil, ai+l, ..., am) are m-adic
Si-i m-l
identities and define k(x) = [b1 xb 1]. As before, k is

continuous. Note however, that hk = identity = kh. Thus

h and k are inverses of each other and are both bijective.

Whence, h is a homeomorphism.

Corollary 2.7: Every topological m-group is homo-


Proof. If a and b are any two elements of a topo-

logical m-group, then for elements al, ..., am in the m-

group such that [al 1aai+l] = b, the map h(x) = [al xai+]

is a homeomorphism that takes a to b.

Proposition 2.8: Let A be a topological m-group

and let Al, A2, ..., A be any m subsets of A. If A. is

open for some i, then [A,, A2, ..., Am] is open. If A1,

.., Am are compact, then [A1...Am] is compact.

Proof. If Ai is open, then by Theorem 2.6

[al..-a.i Ai Ai+l *am] is open, and [A.**.Am =

U[[al''ai_- A ai+l' am]lai Am) is open. since
f:(x1, ..., xm) -- [x] is continuous, if each Ai is com-

pact, A1 x A2 x..-x Am is compact, so [A1A2...A ] being

the continuous image of a compact set is compact.

Proposition 2.9: Let A be an m-group with idempo-

tent e. Then for any neighborhood U of e there exists a

neighborhood V of e such that [Vm] c U.

Proof. Since f:(e,...,e) ->e is continuous if U
is any neighborhood of e, there exists U...UUm neighbor-

hoods of e with [UI...Um] c U.
Let V = n U. and we see that [Vm] c U.
Proposition 2.10: Let A be a topological m-group

with S a sub-m-group of A. Then S with the relative topo-

logy is a topological m-group.

Proof. The mappings f and g in Definition 2.1 are

continuous and hence the restrictions of f and g to S are


Proposition 2.11: Let A be a topological m-qroup

and let A1, A2, ..., A be subsets of A. han

(i) [Al...A] C [A1...A]
S-1 -1-
(ii) (A', ., A 2) c ((A1, ..., Am-2)
Si-i -
(X ..A-T [Xi1 A. X.'
(iii) [x ..A x ] = [x1 A x
1 j 1+1 1 3 1+1

Proof. (i) It is known that for any continuous

function f, f(A) c f(A).

(ii) Same as (i).
i-i m
(iii) By Theorem 2.6 f:x -- [xI x Xi+1] is a


Proposition 2.12: If H is a sub-m-semigroup, sub-m-

group, or semi-invariant sub-m-group of a topological m-

group A, then H is, respectively, each of these.

Proof. Let H be a sub-m-semigroup of A. Then

[H] c H and by Proposition 2.11 (i) [(H-)m] c [Hm]- c H-.

Next, let H be a sub-m-group of A. Then, as in the

first part of this proof, [H"]- c H. Since (H, H, ... H)-1

c H, by Proposition 2.11 (ii) (H H ... H )-1 c

((H, H, ... H) 1)- c H-.

H is defined to be an invariant sub-m-group iff for

each m-adic identity (el, e2, ..., em_),[e H e ] H

and by Proposition 2.11(iii) [e H e +] = [e H e

= H The proof is similar for semi-invariant sub-m-groups.

Proposition 2.13: A sub-m-group H of a topological

m-group A is open iff its interior is not empty. Every

open sub-n-qroup is closed.

Proof. Suppose the interior is not empty and let

em-1 be an interior point of H. Then there exists an open

neighborhood V of em-1 with V c H. Let e, ..., em-2 be
a collection of m-2 elements from H such that (el, ...,
em-) is an m-adic identity. (Such a collection exists

since H is a sub-m-group.) Hence for any h E H,

[h m-1] = h, so [Hm-1 V] = H and by Proposition 2.8, H is
open. By Theorem 2.6, [xm-1 H] is open for any choice of
xl, ..., xm-_ E A. Hence, U ([x1m- H]lh 4 [x-1 H] for
any h E H) is open and is A\H. Hence, H is closed.

Proposition 2.14: A sub-m-group H of a topoloqical

m-qroup A is discrete iff it has an isolated point.
Proof. Suppose H has an isolated point x. Then
there exists an open set U c A such that U n H = (x). Let

y E H. Since H is a sub-m-group of A, there exists xl, x2,
..., xm-1 E H such that [x1~ x] = y. Since U is open
about x, [xm~- U] is open about y by Theorem 2.6. Since

U n H = (x), (y) 6 [x1-l U] n H. If y $ y E [x1 U] n H,

y E [x-I U] which implies that for some x E U, Y =
[x-1 x. Now x, x2, ..., Xm E H, y E H and H being

a sub-m-group implies that xo E H, and therefore, that

x E U n H. But x 0 x by uniqueness and the choice of

y 0 Yl,a contradiction. Hence, [xm1I U] n H = [y) and H
is discrete. The converse is clear, for if it is discrete,
all of its points are isolated.
Definition 2.15: A relation R on a topological

space A is lower semicontinuous iff UR is open for every
open set U in A.

Definition 2,16: A relation R on a topological

space A is said to be closed iff it is a closed subset

of A X A under its product topologv.

Definition 2.17: A relation R on a topological

space A is upper (lower) semiclosed iff xR = [x)R (Rx =

R(x)) is closed for every choice of x E A.

The following theorem has been proved for general

algebraic systems in which the congruences commute by

Mal'oev [12, p. 136]. It can be shown that the congru-

ences in an m-group commute but we will prove the theo-

rem directly.

Theorem 2.18: Any congruence R of a topological

m-group A is lower semicontinuous.

Proof. Let U be any open subset of A and suppose

UR is not open; that is to say, there is an x E UR such

that for any neighborhood V of x, we have V n (A\UR) 0].

Since x E UR, then there is a y E U such that (x, y) E R.

Let x2, x3, ..., Xm-1 be elements in A such that (x, x2,

..., Xml) is an m-adic identity so that [xxm-ly] E U. By
m-1 2y
the continuity of the m-ary operation on A, then there

exists an open set V containing x such that

[vx-ly] c U.

By hypothesis, since UR is not open, then there is a v E V

such that v ( UR. Since (x,y) E R, (v,v) E R, and (x, x i)

E R for all i = 2, ..., m-l, then ([vxmy- [vxm-x]) R.

However, (x2, ..., Xml, x), being a cyclic permutation of

an m-adic identity, is also an m-adic identity so that

([vx -y], v) E R. Note [vxm-ly] E U and hence

([vx2 y], v) E R n (U X A). This implies that

n2([vx y, v) = v E UR, which is a contradiction. There-
fore, UR must be open.

Definition 2.19: Let (A, [], 7) be a topological m-

group and let R be an equivalence relation on A. Define

n: A -- A/R by n(a) = aR for each a E A. n will be called

the natural map. Let U be the family of subsets of A/R

defined by U E 21 iff nl (U) is open in A.

Remark 2.20: If R is a congruence and U E 2 then

U may be expressed as (xRlx E T E T) for if U E 2 set
T n- (U). Conversely, if T is open in A, n(T) E 9 for
n (n(T)) = TR which is open in A since R is lower semi-


Theorem 2.21: The family of sets z in Definition

2J19 is a topology for A/R. The mapping n is continuous

and 2 is the strongest topology on A/R under which n is


Proof. Let (uRju E T X TX E T be an arbitrary

collection of sets in S. Then U (uRlu E T ) =
(uRju E U T ) E since U T is open in A. If

(uRju E Til.= Ti E T is a finite collection of members of

n n
s, n (uRju E Ti] = (uR u E n Ti) which is in U since
i=l e=l
n is open in A. Hence 1 is a topology on A/R. It is

clear that n is continuous. Next, let a be another topolo-

gy on A/R such that s c a. Let F E B be an I-open set
which is not %-open and suppose n is continuous under the

topology 8. Then n- (F) is open in A,so suppose n-(F) =

T E T. Then n(T) is an element of 9 by the remark, so n(T)

is X-open and n(T) = n(n-(F)) = F, i.e., F is a-open, a

contradiction. So n is not continuous under the topology B

and is the strongest topology under which n is continuous.

Proposition 2.22: The natural mapping of A onto A/R

is open.

Proof. By Remark 2.20, if T is open in A, n(T) is

open in A/R.

Proposition 2.23: Let A be a topological m-qroup

with congruence relation R on it. If aR is compact for

some a E A, xR is compact for all x E A.

Proof. Let x E A and choose a, ..., am_1 E A such

that [am-1 a] = x. Observing that [am-1 (aR)] =

= [al a]R = xR we see that xR is compact by Theorem 2.6

being the continuous image of a compact set.

Proposition 2.24: Let A be a topological m-group

with a congruence relation R on A. Then A/R is discrete

if and only if aR is open in A for some a E A.

Proof. Suppose aR is open for some a E A. For any

x E A, let a,, ..., am_l be chosen so that [aI-1 a] = x.

Then [am-1 (aR)] = xR which is open by Theorem 2.6. Hence,
if aR is open for some a E A, xR is open for any x E A.

By Proposition 2.22, (yRly E xR) = xR E A/R is open. Hence,

if xR is an open subset of A, (xR) is open as an element

of A/R, so A/R is discrete. If A/R is discrete, (aR] is
open in A/R, so n-1 aR) = aR is open in A.

Proposition 2.25: If an equivalence relation R on

a. T topological space A is closed, then A/R is a T, too-

logical space under its quotient topology.

Proof. From Wallace's Algebraic Topology Notes [26,

Cor. 3, p. 8] we know that if Y is compact, then the pro-

jection map n: A x Y -> Y is closed. If R is closed, then

R n (A x (y)) is also closed for any y E A. Hence,

Tl(R n (A x (y])) is closed and A/R is T .

Theorem 2.26: If h: A -- A is a topological (con-

tinuous) homomorphism between two topoloqical m-groups and
A is T 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 being T implies it is

in fact T2 and hence A is closed in A X A. Hence, h-h-1 is

also closed. By the same reasoning as in the proof of Pro-

position 2.25, Ry = nl((h X h)-1 (W) n (A x (y))) and yR =

r2((h x h)-1 (A) n ((y] x A)) are closed for each y E A and
hence R is both lower and upper semiclosed.

Theorem 2.27: If h: A A is an open continuous

epimorphism of T0 topological m-groups, then A/R where R =

h*h-l is iseomorphic to A under the natural mapping.

__ ~_

Proof. The algebraic isomorphism between A/R and A

follows from general algebra. A/R and A possess precisely

the same open sets which are images of open sets in A.

Hence, they are homeomorphic.

Theorem 2.28: Let A be a compact (locally compact)

topological m-group. If R is a congruence relation on A,

then A/R is compact (locally compact).

Proof. Since n is continuous, A/R is the continuous

image of a compact set and hence compact. Next, let

(xR) E A/R and let U be an open neighborhood of x in A such

that U is compact. Then n(U) is open about (xR) in A/R and

n(U) is compact since n is continuous [8, 3.13].

Definition 2.29: A topological space X has the fixed

point property if and only if for each continuous map

h: X -> X there exists x E X such that h(x) = x.

Theorem 2.30: Let A be a topological m-group. Then

A does not have the fixed point property.

Proof. Define a function h: A -> A by choosing

x1, ..., Xm- E A such that (x ..., Xm- ) is not an m-adic

identity and letting h(a) = [xm-1 a]. By Corollary 2.7,

h is continuous and suppose h(a) = a for some a E A. Then

[xT-1 a] = a, so by Propositions 1.12 and 1.13, (xl, ...,

Xml) is an m-adic identity contradicting the choice of

(xl, ...;, Xm-l)
Remark 2.31: If A is a topological m-group, then A

is not homeomorphic to [0,1] x...x [0,1] = [0,1]n for any

n, nor is A homeomorphic to the Tychonoff cube since both

have the fixed point property (5, p. 301].

Theorem 2.32: Let A be a topological m-qroup, and

for some x E A, let C be the component of x. Then, if

[x-1 x] = y, [xlm- C] is the component of y.

Proof. Let x and y be given as in the statement of

the theorem and let xl, ..., Xm-1 E A be chosen so that

[x11 x] = y. Then y E [xl1 C] and [xm1 C] is connected

and closed by Corollary 2.7. [xm-1 C] is the component of
y, for if not, let K be the component of y. Then [x1I C]

c K. Let x i, ... i,m-2 E A be chosen such that (x, ...,

Xim-2, xi) is an m-adic identity, i = 1, ..., m-1. Then

[Xm-1,1l.''m-l,m-2 1'Xl,1 '''l,m-2 X1 2"...m-1 C] = C

X-,1l... Xm-l,m-2...xl, 1.. .xl,m2 K] which is closed and
connected, contradicting the assumption that C is the com-

ponent of x.



In this chapter we will prove the topological ver-

sion of the Post Coset Theorem by showing that each topo-

logical m-group 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 m-group with
A as its associated group according to the Post Coset

Theorem. If (el, e2, ..., em_1) is any m-adic identity of

A so that el, e2, ..., em E A, then for each x E A1

and any fixed index i = 1, ..., m-i, there exists an ele-

ment a E A such that

el...eilaei+...em1 = x.

Proof. Since x E Am-1, then x = ala2...a m- for

some al, a2, ..., am_1 E A. For any am E A, there exists,

by definition of an m-group, a unique a E A such that

[e lae +am] = [al]. Considered in the containing group

of A, the equality above reduces to el. ..eilaei+1...emlam

= ala2...a -a = xa and hence, e ...e. aei+. ...e- = x.

Proposition 3.2: Let (el, e2, ..., em-1) be an

m-adic identity of a topological m-group (A, [], T) with

S(e -_) a local open basis at em_1. Then S(ele2...e -) =
(ele2...em-2U U E6 (emI)) is a local open basis of the

identity e = ele2...em -1 of the containing group of A under

some topology.

Proof. It is sufficient to show closure under inter-

section. If U1, U2 E 1(eml), then clearly we have

ele2...em2U1 n ee2...em-2U2 = ele2...em-2(1 n U2).

Remarks. Preparatory to the demonstration that

2(ele2.. .em ) defines a topology that converts the con-

taining group of A into a topological group, we shall prove

a series of Lemmata.

Lemma 3.3: Let (el, e2, ..., em_1) be an m-adic

identity of a topological m-qroup (A, [], 7). For any

i = 1, ..., m-2 and each open set V containing ei, there

exists a basis element U E 9(em1 ) such that ee2*...em_2U c

el. 1 em-i Vei+l ... em-l1
Proof. For each v E V and fixed element x1 E A,

there exists, by definition of an m-group, an element w E A

such that
m-2 i-1 m-1
[e1 wxl] = [el vei+lxl]
i-1 m-1
Define two functions h and k by h(v) = [e1 vei+xl] and

k(w) = [e-2wxl]. Let (xl2,x, ..., Xm ) be an m-adic


m-2 i-i m-i
W = (w [e wxl] = [el ve'i+x ] for some v E V) =

i-i m-1 m-i
(wI [eml [e ve i+x ]x2 = w for some v E V) =
i-1 m-i m-1 m1 -.1
[em-i Ve x- x-] = [e h(V)x -] = k-h(v)
m-1 1 i+X1 12 m-1 2
is an open subset of A containing em-1 since both h and k

are homeomorphisms by Theorem 2.6.

Since 21(em_ ) is a local basis at em_1, then there
i-1 m-i
exists a U E 1(e ) such that U c W. Whenc e [ei Ve" xl] =
Si+m- 1I

[e-2WxI] D [em2 Ux1]. Considered in the containing group

this gives e ...e. iVe. +...e xml e ...e m2Ux1 and hence

e 1...ei-1Vei+l...em_1 3 e...em-2U.

Notation. In Lemmas 3.4 and 3.5 let f and g be the

functions of Definition 2.1.

Lemma 3.4: For each U E N(em-1), there exists a

V E (em-_l) such that (e e2...e _2V)(ele2...em_2V) c

ele2 erm-2U

Proof. Let U E 9(e -1) and define a function
h: A x A -> A by h(x,y) = [xel y]. Being a restriction

of the continuous function f on A X (e1) X...x (em_2) X A,
h is therefore also continuous. h- (U) is thus an open

subset of A X A. Since the first and second projection

maps Vl and V2 are open maps, then V = nlh (U) n T2h (U)

is also open. It is also non-empty, since h(em_-,em,) =

[em-le-1] = em E U and therefore em E V. We now claim
(le2.2 for, if 2 E V, then
(ee2...e m-2V)2 c ee2...e m-2U; for, if vl, V2 E V, then

h(v1,V2) E U and hence (ee2.. .em-2V1) (ele2 ...em2v2) =

ee2 ...e [lel 2] = ele2..em-2 h(v, v2) E e e2..em2U.

Lemma 3.5: For each U E 1(eml ), there exists a
V ( M(e_ ) such that (e e2...e mV) c ee2...e _U.
m-1 2 m-2 1 2 m-2

Proof. By definition of a topological m-group,
g (U) is open in A X A X...X A (m-2 times). Since

U 6 5(em- ) and therefore em_1 E U, then (el, e2, ..., em-2)

E g (U). Thus, there exist W1 E 2l(el), W2 E S(e2), ...,
Wm_2 E f(em2) such that W1 W2 X... W 2 g (U). Let
W = W W ..W Observe that W c U in the containing
1 2* m-2
group. Thus
-1 -1 -W- -1
(We2...e )1 c (We )- = e-W = e "e ...e U.
1 2 m-1 m-1 mL 1 2 m-2
By Lemma 3.3, there exists a V E 9(em_-) such that

ele2...em-2V c Wle2...e -1

and therefore

(ele2.. .em-2V)-1 ce ele2..em-2U.

Lemma 3.6: For each U E N(eml ) and x E ele2.* .em-2U,

there exists a V E W(em_ ) such that x(ele2...em2V) c

ele2" .em-2U.

Proof. Let x = ele2...em_2u for u E U. Define the

function h such that h(v) = [ue-2 v]. Since this is a

homeomorphism by Theorem 2.6, then h-l(U) = W is an open

and non-empty subset of A. Inasmuch as em_1 E W, then

there is an element V E %(e -) contained in W. Hence

x(ele2...em2V) c x(ele2...em2W) = (ele2...em-2u).

(ee2 ...e W) = ele2...e_2 ue mW = ee ...e h(W) =
1 2 m-2 1 2 m-2 1 1 2 m-2

ele2" .em-2U.

Lemma 3.7: For each U E (em- ) and any element x

in the containing group of A, there exists a V E M(em1)
such that x(e e2...em_2V)x- c ele2...e mU.

Proof. Recall that the containing group of A is

given by A U A U...U Am-l. We consider two cases:

Case I. Suppose x A-1 so that x = a a2

Let a i+, ..., am E A such that aa2 ...aiai+l...am_ is
the identity element in the containing group. Denote x =

ai+. ..am_ For any fixed element a E A, let

W i i e m-2 ] m-1 m-2
1W = w [e wai+1 ai+2a] E [e Ua]J.

Since [e -2Ua] is an open subset in A (in fact a basis ele-

ment at the point a E A), then, being the inverse under a

homeomorphism, W is also open. Since em-1 E W, there exists

a V E U(e -1) such that W contains V and hence

r m-2 m-1 i m-2 m-1 m-2
[al[e vai1 ]ai+2a] c [al[e1 Wai+]ai+a] = [el Ua].

Thus, after simplication in the containing group of A, we
obtain x(ee2...e m2V)x = (a ...a.)(e e2...e V)*

( c ele2...em-2U.

Case II. Suppose x E Am so that x = a

for al, ..., am 1 E A. Let am, .., a2m-2 E A such that

([a], am+l, ..., a2m-2) is an m-adic identity and denote

x = a a ....a 2. For any fixed element a E A, let
Sm-i m-2 2m-2 m-2
W w[[ai [e wa]] am+1 a] E [e Ua]. The rest of

the proof proceeds as in Case I.

Theorem 3.8: Any topological m-group 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 2-group.

then G is disconnected.

Proof. Lemmas 3.4 to 3.7 showed that 8l(ele2...em-_)

is an open basis at the identity of the containing group G

of A which converts G into a topological group [8, 4.5] with


(xU: x E G, U E W(ele2...em-_)) or

(Ux: x E G, U E (ele2 ...em-1).

Since A in the containing group is clopen, then each coset

of G by N is also clopen (being homeomorphs of A). If m > 2,

then G is disconnected and G/N is discrete under its quo-

tient topology.

It thus remains to show that the topology defined in

G by )(ele2...em-1 ) gives rise to the same topology on A

when restricted or relativized. By homogeneity of A, it

suffices to consider the basis at any point, say em_1 E A.

If U is any basis element of A containing em_1, then el

ele2...em2U is a basis element in G of e = ele2...e m2em- .

Hence, em-l(ele2...em-2U) = U is also a basis element of G

at e Thus, every basis element of A at e 1 (and hence

at any point) results from a basis element of G. This

means that the topology induced on A by the topology in G

coincides with the original topology on A.

Corollary 3.9: A_ T topological m-qroup is always

completely regular and hence Hausdorff.

Proof. By Theorem 3.8, any topological m-group A

is the coset of a topological group modulo a normal sub-

group N which is homeomorphic to A. Since A is To, then

N is T and hence completely regular. Thus A is also

completely regular.

Theorem 3.10: Any compact T topological m-qroup A

is homeomorphically representable as an m-group of m-adic


Proof. Observe that the topological m-group is in

fact completely regular and hence Hausdorff (Corollary 3.9).

By the representation theorem for algebraic m-

groups, the given m-group A is isomorphic to a sub-m-group

of the m-group of m-adic functions F(S1,S2, ..., Sm-1) on

the sets S. = A x...x A/m (i times). Since A is compact

Hausdorff, then A x...x A (i times) is compact Hausdorff

and hence also Si under its natural quotient topology.
Thus U Si is also compact Hausdorff under its sum topology.
Any bijective continuous function on U S. is also
a homeomorphism and thus the collection of all such func-
tions H( U Si) is a group (of homeomorphisms). By a result

m-1 m-1
of Arens [1, p. 597], since U Si is compact, then H( U Si)
i=l i=l
is a topological group under the compact open topology and

the operation of composition. Since F(S, S2, .,Sm-_) c
H( U Si), then for any fl f2' .*.. f E F(Sl, S2', .' Sm-_),
the functions

(fl f2' fm) fl f2"''fm [f= ]

-1 -1 -1 -1
(fl' f2' "'.. fr-2) -- (fl.f2*f...m_2)- 2 2 3"f
"m-2 1^2* m-2m-2 m-3*f 1

are continuous. Thus F(S1, .*, S1 ) is a topological m-

group under the compact open topology relativizedd).

Consider now the regular representation
h: A -> F(S1,S2, ..., S-) c H( U S)
defined by h(a) = La. The following Lemma will be needed.
m-i m-1
Lemma 3.11: The function k: U S -> U A = G such
i=l i=l
that k((al, a2, ..., ai)/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 E U. By the continuity of the multiplication in

G, then there exist open subsets in G, a1 E VI, a2 ( V2'

..., a E Vi such that

k(V1 X V2 X...X V /a) = VIV2.Vi c U.

Thus, h is continuous. That h is open is obvious.

Let L E (C, U) or
{(a,al, ..., ai)/aJ for some i = 1, ..., m-1 and

(al, ..., a )/m E C3 c U. Applying k on both sides, we


(aala2...ail for some i = 1, ..., m-i and

(a1, ..., ai)/u E C) c k(U).

By Lemma 3.11 k(C) is compact and k(U) is open.

Thus, for any fixed index i = 1, ..., m-1 and

(al, ..., ai)/! E C, by the continuity of the multiplica-

tion in G, there exist open subsets

a E Va ...a a V, a2 E Va, ..., ai E Va

such that

aa .ai Vala2 .aVa Vl c k(U).


a V ...Vai k(C).
a, ..., ai E C

By compactness of k(C), then there exists a finite number
1 1 n n
of i-tuples, (al, ..., a ), ...,(aa, ..., a) such that

U V k k...V k 3 k(C).
k=l aI a2 ai

Let U = n v k k'** k which is non-empty since it con-
k=1 ala2...a

tains a. Then

n n
U UV kV k'V k = U( U V kV k...V k) c k(U)
k=l aI a2 ai k=l al a2 ai

and hence

U U X V k X...x V k / U.
k=l a1 a.

This means,
U(C) C L( U V k X...X V k / a) c U,
k=l a ai

in other words,

h(U) = LU c (C, U).

Whence h is continuous and the final result follows.

Corollary 3.12: Let (A, [], t) be a topological m-

group, F a compact subset of A, and U an open subset of A

containing F. Then for each m-adic identity (el, ..., em1)

of A there exists open subsets U1, ..., Umr1 with ei E Ui,
i = 1, ..., m-1 such that [FUm-1] U [U--1F] c U. If A is

locally compact, then Ul, ..., Um 1 may be chosen so that

([FUm-l] U [fU-1F])- is compact.

Proof. Let G be the containing group of A. By a

result in Hewitt and Ross [8, 4.10] for topological groups,

for F, U as above there exists an open set V containing

the identity of G such that FV U VF c U. Then e = el...em-1

E V and by the continuity of the operation in G there exist

open subsets U1, ..., Um-1 with i E Ui, i = 1, ..., m-1

such that U ...Ul c V. Thus [FUm-1] U [Um-1] =
S...Um-1 1 1
(FU ...U M) U (U...U mF) c FV U VF c U.

If A is locally compact so that its containing group

is locally compact, the second part of [8, 4.10] states

that V may be chosen so that (FV U VF) is compact. Hence,
([FUm-1] U [Um-1F]) as a closed subset of a compact space
is compact.
Proposition 3.13: Let Al and A2 be two m-groups
such that A1 is iseomorphic to A2. Then their covering
groups G1 and G2 are iseomorphic.
Proof. Let f: Al1 A2 be an iseomorphism and G =
m-1 m-1
Al U...U AL G2 = A2 U...U A2 .
Define g: G1-- G2 as follows. If x E G1, x = x -...
xi with x1, ..., xi E A1 let g(x) = f(xl)...f(x ). Then if

y E G1 and y = yl...Yk with yl, ..". yk E A1, we have g(xy)
= g(xl...x iy...yk) = f(xl)...f(xi)f(yl)...f(yk) = g(x)g(y)
so g is a homomorphism. Next, suppose g(x) = g(y). Then
f(x1)...f(xi) = f(yl)...f(yk) so i = k and if Zi+l,.

zm E A1 we have [f(x1)...f(xi)f(zi+) ...f(zm)] = [f(yl)...

f(yi)f(zi+l)...f(zm)] and f[x zi+1] = fly i+l]. f is bi-
izm izm
jective, so [x 1 i+] = [yz i+l] which implies that xl...xi

= Y'..Yip i.e., x = y and g is bijective.
Now let U be an open neighborhood of el the identi-
ty of G1. Then e1 = X1 ...xM1 with (x1, ..., xm ) an m-
adic identity, xi E A i = 1, ..., m-1. U can be expressed
as U = x ...X2 U' with U' open in A1 and x m U'.

g(U) = g(x1...xm2 U') = f(x1)...f(xm_2)f(U') and f(U') is
open in A2 so f(x1)...f(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.

Theorem 3.14: If a topological m-group A is compact,
locally compact, a-compact, 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 m-1 costs of G, F compactness is clear.

Proposition 3.15: Let A be a topological m-group

and let Al, A ..., A be subsets of A. If A is T and

[xP'mj) = [xT] for any permutation p and any choice of

xi E A, then [xP(m)] = [xm] for any choice of xi E Ai.
i i t p(1) 1
Proof. Let e be the identity of the containing

group G of the m-group A. Let
-1 -1
H = ((al, ..., am) E G x...X Gal.amap( )..ap(m)

= fell.
Since A is To, G is T and (e) is closed. H is the in-

verse image of a closed set under a continuous function

and hence is closed in G X...x G. Now A is closed as a

subset of G, so H n (A X...X A) is closed. It is clear

that A1 x...X A c H. Hence (A X...X A )c A- X...X A H.
1 m 1 m 1 m
Proposition 3.16: If A is T topological m-qroup

and H is an abelian sub-m-semigroup or sub-m-qroup of A,

then H is (respectively).

Proof. By Proposition 2.12 H is a sub-m-semigroup

or sub-m-group if H is. By Proposition 3.15, H- is abelian.

Lemma 3.17: Let A be a topological m-group with

congruence relation R on it, If aR is compact for some

a E R, xR is compact for all x E A.

Proof: Let x E A and choose al, ..., am-1 E A
m-1 m-1
such that [a 1a] = x. Then [al (aR)] = xR is compact

since by Theorem 2.6 it is the continuous image of a com-

pact set.

Lemma 3.18: Let Al, ...Am be a collection of sets

such that A1, ...Am_1 are compact and Am is closed. Then

[A1...A] is closed.

Proof. Since in a group the product of compact sets

is compact, we have A ...Am1 compact in the containing

group so [A1...Am] is closed in A [8, 4.4].

Theorem 3.19: Let A be a topological m-group with

a congruence relation R on A. If A/R and aR are compact

for some a E A, then A is compact.

Proof. By Proposition 2.23, since aR is compact,

xR is compact for any x E A. Let el, ..., e -1 E A such

that (el, ..., em-1) is an m-adic identity. As shown in

Theorem 1.37, N = el..em-2(em_1R) is a normal subgroup of

the covering group G. Since G is a topological group,and

em1R is compact, N is compact. We next show that G/N is

compact. Since A/R is compact and A/R = (xNlx E A),

(xNlx E A) is compact. Hence, in G, y.[xNlx E A) is com-

pact for any y E A and y.-xNlx E A) = (yxNlx E A) =

(zNjz E A ). Continuing this process, we see that for any

i = 1, ..., m-1, (xNlx E A ) is compact so

m-1 i
G/N = U (xNIx E A ) being a finite union of compact sets
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, a-compact

topological m-qroup. Let f be a continuous homomorphism of

A onto a locally countably compact To topological m-group 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 a-compact,

G also is. Since A' is locally countably compact and To,

so is G'. Thus f is an open mapping from G onto G'

[8, 5.29]. Since A and A' are open subsets of G and G',

respectively, the restriction of f' to A (which is f) is




In this chapter it will be shown that each arcwise

connected, locally arcwise connected, and locally simply

connected topological m-group with idempotent has an arc-

wise connected, locally arcwise connected and simply con-

nected universal covering m-group [14, p. 232].

Definition 4.1: A regular, To, and second countable

space A is arcwise connected (locally arcwise connected) if

and only if for each pair a, b E A there exists a continu-

ous function c: [0,1] -- A such that cp(0) = a and t(1) = b

(for each a E A and every neighborhood U of a there exists

a neighborhood V of a contained in U such that for all x E V

there is a continuous function :p [0,1] --> U such that C(0)

= a and (Il) = x).

Definition 4.2: A space A is simply connected (lo-

cally simply connected) if and only if for each a E A (for

each neighborhood U of a there is a neighborhood V of a

contained in U) such that for any continuous function

cp: [0,1] -> A (p: [0,11 --> V) such that p(0) = c(l),

then p is homotopic to 0 in A (in U).

Lemma 4.3: If f: [0,1] -- A is homotopic to zero

and cp: [0,1] -> A is an arbitrary continuous function.

then e*f is homotopic to cp, where c*f is defined as fol-

Scp(2t) for 0 < t <
(Op*f)t) = -
f(2t-1) for < t < 1.
Proof: Note f(0) = p(l). Then the homotopy is
effected by the following function:
I 2t for 0 t < l+s
9(o +s 2
F(s,t) =
f(0) for 1 s t 1.

Theorem 4.4: For each arcwise connected, locally
arcwise connected, and locally simply connected topological
m-group (A, [...], T) with an idempotent element e, there
exists an arcwise connected, locally arcwise connected, and

simply connected universal covering m-group (A, [...], T)
which is locally iseomorphic to (A, [...], T) and such that
if 8: A -> A is the covering homomorphism, then A/e8e-1 is
iseomorphic to A.
Proof: Let A be the family of homotopy classes of
continuous functions p: [0,11 -> A such that p(0) = e.
The homotopy class containing p will be denoted by $.

Consider any 9 E 6 E A with cp(l) p E A. If V is
an open basis of the topology T on A, then for each U con-
taining p in 8, let

U = ($? f: [0,1] --> U such that f(0) = p)
and I = (I" U E 8. It is easy to see that if c is re-

placed by any q E U E U, then U and 4 will determine ex-
actly the same 0 [14, p. 221].

(1) i is an open basis for some topology i
on A.
If U, V E E, so that U, V E Z, then U n V E U. It
is not difficult to show that U n V = 0 n V.
(2) (A, ;) is a To topological space.
Consider any pair p1, c2 E A such that pl 2. Let

cpi E pi (i = 1, 2) such that cpi(l) = Pi.
If p. 0 P2' then since A is To there exists a neigh-
borhood U of pl such that p2 L U. In this case, l E ri but

P2 P U and hence A is also T.
If p, = p2, then since A is locally simply connected,
there is a neighborhood U of p, = p2 such that every con-
tinuous f: [0,1] -> U with f(0) = f(l) = p, is homotopic
to zero. If U = ([~ | f: [0,1] -- U with f(0) = pl),
then ;2 P U; for, if c2 E U, then there is an f: [0,1] --> U
with f(0) = p, such that cpl*f E ,2' where

*f, (2t) for 0 f(2t-1) for j < t < 1.
Note: cp(1) = f(0) = = (Cpl*f)(l) = f(l). Thus
f is homotopic to zero and l = cp*f = 2, which is a con-
(3) If for ci E Pi E A (i = i, ..., m) one de-
fines [m~2...cpm](t) = [cl(t)c2(t) ... pm(t)] and [152"'..m] =

[pl2''...], then (A, [...], l) is a topological m-group.
Since [](0) = [cpl(0)2(0)...cp(0)] = eem] = e
and [cpP2...p m] is a continuous function on [0,1] to A, the
above operation on A is clearly well defined. Associativity

follows from the following relations
M 2m-1
r [ilm1+ ( [m[+ ](t)' -m ]

=[[CPl(t)2(t). .. m (t) ]m+l (t) ... cp2ml -(t)]

= [C(t)... i-cp t) [0Pi(t).. Pi+m-l(t)]

cpi+m (t)'...2m-I(t) = [ l (t) -tpi-l(t)
[pi(t) .. pi+m-l (t) ]Pi+m(t) .. .2m-l (t)]

= [ t) i(t) +m-(t)ci+(t) ..2m-l (t)]
i-lr i+m-1 2m-1
1 i+m J(t)
which holds for each i = 2, ..., m. For each ci E qi
(i = 1, ..., m-2), let cpm-: [0,1] -> A be the function
such that cPm- (t) = (qp(t), ..., c ,2(t))- for each
t E [0,1]. Since [em] = e, so that (e, ..., e) is also an
m-adic identity, then pm_1(0) = e. Define (Cl' ",' cm-2)-l
= $m-,. Then for each c,
[CpCP *m-cp =
Thus far, we have shown that (a, [...]) is an m-group.
Next, we show that the functions
m) [ .~. ]
(' ''-2) M-2 1' -2
are continuous. Let cpi E i (i = 1, ..., m) such that
pi(l) = p.(i = 1, ..., m). Let V be any neighborhood of
[19 2 ..~Pm] = [1p2 pm] so that every element of 9 is of
the form [cp 2"]*f for some f: [0,1] -> A such that

f(o) = p = [P]P2"- pm = ['l(1)'2(1).."m(1)] = [ m](1) v.
By the continuity of the m-ary operation on A, there exist
neighborhoods U. containing pi (i = 1, ..., m) such that
[U U ...U ] C V.
Then [U1U2...Um] c V, for, if 'i E Ui (i = 1, ..., m), then
every 'i E i is of the form .i = pi*fi (i = 1, ..., m) for
some fi: [0,1] -> U. such that f.(0) = p.. Thus,
i(t) i(2t) for 0 < t <

f.(2t-1) for a t < 1.

[1l(t)2(t) ..m(t) ] = [cp(2t)2(2t) ....(2t) ]
= [cp](2t) for 0 < t < and [l(t)2(t)..*. m(t)]
= [fl(2t-)f(2t-...f(2t- = [2t-1 (2t-) [ t-1) for
< t < 1.

Since fi(t) E Ui, then [fl]: [0,1] -> V. Also, if [(f](0)

= [fl(0)f2(0)] = [PIP2..'.m = p E V, then
[5l = 2[2 = pm ]*[fm] E V.
S ~ ~ -1
Next consider (1,' C2' ".''' -2) = m-1 where

pi(l) = pi (i = 1, ..., m-2) so that (Pi, P2,' .'' Pm-2)-~
= Pm-l. Let V be a neighborhood of cPm- with cp m(1)

= Pm-_i V. By the continuity of the inverse operation on
A, there exist neighborhoods V. of pi (i = 1, ..., m-2)
such that
(V1 X V x ... x V 2)- c V.
Again, we claim
(V X X ... _2)- c V.

For, if (i, 2' "'' m-2) E V1 X V2 X ... Vm-2 and
iE 6 (i = 1, ..., m-2), then i = cpi*fi(i = 1, ..., m-2)
for some fi: [0,1] --> Vi. If fm-: [0,1] -> V is the
continuous function such that fml(t) = (f(t),

f -2(t)) then f i(1) = (fl(l), ... fm 2(1))-
= Pm E V. Whence
(' 2' **' m-2) = l 2 'm-2

= (1p P2, "'-I m-2) *(f1' f2' "*' fm-2

= m-l m-1
This completes the proof that (A, [...], ) is a topological

(4) (A, D) is a second countable space.

If IS1 to, then also Iji .O-
(5) The covering function 8: --> A such that

80() = gc(l) = p is an open continuous map which is locally
a homeomorphism.

If U is an arbitrary neighborhood of p, then

e(U) c U, obviously, so that 9 is continuous.
Let c E A and U be a neighborhood of c defined by
the neighborhood U of (p() = p. By local arcwise connected-
ness there exists a neighborhood V of p contained in U such
that for any x E V there is a continuous function f: [0,1]
-> U such that f(0) = p and f(l) = x. This implies that

cp*f E U and 9(cp*f) = (cp*f)(l) = f(2*1-1) = f(l) = x. Whence
V c 8(U), and 8 is open.

Next, let c E A and (cp) = p(1) = p E A. Since A is
locally simply connected, there is a neighborhood U of p
such that every continuous f: [0,11 -> U such that f(0) =
f(l) = p is homotopic to zero in A. 8 is one-to-one on U,
for, suppose 6e(,) = 2(2) so that for some fi: [0,1] -> U
we have fi(0) = p(i = 1,2), and cp*fl E _l and cp*f2 E 2'
If f{(t) = fl(l t) so that f2*f{: [0,1] -> U is the con-
tinuous function such that

(f2*f{)(t) = f (2t) for 0 < t < _
(f2*fY)(t) =
lfi(2t-1) for t < 1,

then (f2*fi)(0) = f2(0) = e and (f2*f')(1) = f(1) = f1(0)
= e. This means then that f2*fi is homotopic to zero and
therefore c*fl is homotopic to p*f2 or 11 = 12'
Since 9 is continuous, one-to-one, and open on U it
follows then that it is a homeomorphism on U.

By virtue of this, then

(6) (A, 7) is also locally arcwise connected,
locally simply connected, and regular.

(7) (A, <) is moreover arcwise connected.
Consider any $ E A and 1 E A where w contains the
null path. By homogenity of a topological m-group, it suf-
fices to show that W and p are connected by a continuous
path. Let cp E cp and cPs: [0,1] -> A be the continuous
function such that cps(t) = q(st). For any fixed s E [0,1],
note that ps(0) = cp(0) = e and p s(1) = cp(s) = p E A; and
hence cs E A. Define now 4: [0,1] -> A such that 0(s) = s.
s s

Then I is continuous [14, p. 223]. Moreover, (0) = 0
= and 1(1) = -1 = p. Whence A is arcwise connected.

(8) (A, ;) is simply connected.
As in the previous, let E A be the homotopy class

containing the null path. Consider any continuous function

$: [0,1] -> A such that (0) = w = (1). Define cp such
that cp(t) = 9((t)) where 9 is the covering map. Define

also for each s E [0,1], s: [0,1] -> A such that p (t) =

cp(st). Note that ps also depends continuously on t and

Csp(O) = c(0) = 8(I(0)) = 8(;) = e so that Vs EA.
Firstly, observe that 8((s)) = cp(s) = e(cs) for

each s E [0,1].

We wish to show that $s = i(s) for all s E [0,1].

For s = 0, the equality obviously holds:

P = (0) = = = (0).
Let U be a neighborhood of = (0) for which 9 is a homeo-

morphism. Since both c and (s) are continuous functions
of s by an argument used in (7), then for some k sufficient-

ly small we have

([Ok)) c 0 and ([O,k)) c U.

For each x E [O,k) so that ;x, 1(x) E 0, then since (qx) =

08((x)) and 9 is one-to-one on U, then cx = (x). This
shows that Ps = (s) for all s less than k and hence by the
continuity of the function ps and I(s) we obtain

k = lim cp = lim i (s) = 4(k).
s-->k s--->k

Repeating the process now for k instead of 0, we should

eventually show that the above relation holds for all

s E [0,11. For s = 1, in particular, we have

qP = C'l 4(1) = (o) = W
so that q is homotopic to zero. Suppose that this homo-

topy is effected by the continuous function F: [0,1]x [0,1]

--> A so that F(0,t) = c(t), F(l,t) = cp(0) = e, F(s,0) =

p(0) = e, F(s,l) = p(1) = C(0) = e.

For each fixed s and t, define

G(s,t): [0,1] -> A

to be the function such that G(s,t)(x) = F(s, tx). Since

F(s, tx) is continuous in s, t, and x, then G(s,t) is also

continuous in s and t. For each fixed s and t, note also


G(s,t)(0) = F(s,0) = V(0) = e

G(s,t)(l) = F(s,t) = p.

Thus for each s and t, G(s,t) E A and G(s,t) also depends

continuously on s and t. The following relations now show

that G(s,t) effects the homotopy of I and 0:

G(0,t)(x) = F(0,tx) = pt(x) or G(0,t) = Ct = (t)

G(l,t)(x) = F(l,tx) = p(0) = or G(l,t) =

G(s,0)(x) = F(s,0) = cp(0) = or G(s,0) = 0 = 1(0)

G(s,l)(x) = F(s,x) = p = S = 1(1).

(9) 8 is a homomorphism of A onto A.
It is clearly onto. Let ci E i E A such that

cpi(1) = p(i = 1, 2, ..., m). Then

9([P1cp2...' ]) = e( 2lP2 m] = [c tp.P2.-n 1

= [cp(1)C2(L)] = [plp2"pm] =
= [e(cp )o( tc )...(Cm) ].

The following theorem proves that A is the universal
covering m-group, i.e., A is unique up to iseomorphism.

Theorem 4.5: Let h:A'-> A be a continuous, open.
local homeomorphism, homomorphism of an arcwise connected,
locally arcwise connected and simply connected topological

m-group (A', [], U) with idempotent e' such that h(e') = e

onto the given topological m-group (A, [], t). Then A' and
A are iseomorphic.

Proof. Let 8:A -> A be as previously defined and
define a map k:A'- A as follows: Let p' E A' and let

p:([0,1] -> A' such that cp(0) = e, c(l) = p' with p a con-
tinuous function. Then h o cp defines a curve in A with
(h o p)(0) = e and (h o p)(1) = h(cp'). Since A is simply
connected,the choice of curve p(t) connecting e' and ep' is
unique up to homotopy, and since h is continuous, the image

under h of any two such choices will be homotopic. The
function h o cp[0,1] -> A and hence defines a unique point

h o p in A. Define k(cp') = h o c. It is clear that k is
well defined.

Now let U' be an open subset of A' and let cp E U'.
Let U = (cplp:[0,1] -> U', cp() = cp' and each cp is continu-
ous) and let # be continuous with #:[0,1] -> A such that

#(0) = e', *(1) = cp9. Since U' is open in A', h(U') is

open in A and A being locally simply connected implies

that there exists an open set V in A about the point h(cp'),

V C h(U') such that any closed curve in V is null homo-

topic, or that any two curves contained in V beginning at

qp and ending at the same point are homotopic.

Letting W = (h o q1|p E U, h o ~:[O,1] -- V such

that (h o p)(0) = h(cp)), we see that

(1) W c h o U = (h o Ucpp e U),

(2) **(h o cp)h o cp E W) defines an open set

in A, precisely the open set V, and

(3) Vc k(U').

Now, (1) is clear and (2) is clear if we note that the

choice of V gives only one function (up to homotopy) con-

necting h(cp) and any point in V and this function will be

homotopic to the h-image of the corresponding function in

A'. V c k(U') by the simple connectedness of A' and the

definition of k. Hence, k is an open function. Next we

must show that k is continuous. Let cp E A' and let V be
a basic open set in A about k(pO). Since e is an open map,

9(V) is open in A and A is locally simply connected so

there exists a simply connected open set W in A such that

W c e(V) and 9(k(co)) E W. Then h- (W) is an open set in
-1 -
A' about cp' such that k(h (W)) c V. To see this, let

*:[O,1] -> A' such that *(0) = e', ((1) = cp and # is con-
tinuous. Let X = [cpcp:[O,l] -> h-1(W) c(0) = *(1) and

each c is continuous). Since A' is simply connected,

*,X = [(*cpCP:[O,l] h-(W), cp(0) = *(1) and each cp is

continuous) represents all continuous functions

i:[0,1] --> A' such that f(O) = e' and f(l) E H (W).

Since W is simply connected, h o X = (h o cplp:[0,1] -->
h (W) cp(o) = (1) and each cp is continuous) represents all

continuous function g:[0,1] -> W such that g(0) = (h o cp)
-1 --- / -- _
(0) = h(cp). Hence, k(h (W))= h o (*X) = (h o *)*(h o X)

=e 9CV.

Next, let E', c' E A' and suppose k(') = k(cp'). If
k(*') = k(cp'), the functions *:[0,1] -> A' and c:[0,] -->

A' must be homotopic and hence, #(l) = (p(l), i.e., = '.

Hence, k is 1-1.

Now let tp, ...cp E A' with associated functions
pi:[0,1] -> A' such that pi(l) = ci, i = 2, ..., m
1. i = 27 ... Y ll.
Then k[cp{ ... ps] = h o [1 cp ] : [h o cp "... h o cp]

(since h is a homomorphism) = [h o p, ... h o pm]

[k(p') ... k(cp)].. Thus, k:A' -> A is an iseomorphism.

Recalling that the iseomorphism of A/8.-1 to A was

demonstrated in Theorem 2.29, we see that the truth of

Theorem 4.4 has been demonstrated.


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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 half-time interim instructor to do

further graduate work, remaining until August, 1966 when

he received his Doctor of Philosophy degree.

Robert Lee Richardson is married to the former

Eleanor Rita Dundon of Orwell, Vermont. He is the

father of three children, Robert Lee, Jr., Mary Margaret,

and Patrick Joseph.

This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Arts and
Sciences and to the Graduate Council, and was approved as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.

August, 1966

Dean, Colleg 'ot r s and Sciences

Dean, Graduate School



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