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University Microfilms International 3 0 0 N. Z E E B R O A D . ANN A R B O R . Ml 4 8 1 0 6 18 B E D F O R D ROW. L O N D O N W C 1R 4 E J . E N G L A N D
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8013688 t
t [
PECK, JOHN EDWARD L. THE EMBEDDING OF A TOPOLOGICAL SEMIGROUP IN A TOPOLOGICAL GROUP AND ITS GENERALIZATIONS, AND AN ERGODIC THEOREM FOR A NONCOMMUTATIVE SEMIGROUP OF LINEAR OPERATORS Yale University
University M icrofilm s International
PH.D.
300 N. Zeeb Road, Ann Arbor, MI 48106
1950
18 Bedford Row, London WC1R 4EJ, England
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! i
TEE EMBEDDING 0? A TOPOLOGICAL SEMIGROUP IN A TOPOLOCICAL GROUP AMD ITS GENERALIZATIONS,
i
I i !
and
(
 ! I
All ERGODIC THEOREM FOR A 11OilCOMMUTATIVE SEMIGROUP 0? LI11EAR OPERATORS.
$ J (

John S.L.Peck
A Dissertation presented
to the Faculty of the
. Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy
1950
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SUHE&RY
This dissertation consists of two chapters which are not related, except that there is a theorem of topological algebra which is common to both. The first chapter deals with generalizations of the problem of embedding a topological semigroup in a topological group.
First, both the commutative and the associative laws
are replaced by the weaker law of alternation.
We are
therefore concerned with alternation topological groupoids. Then while retaining the associative law, the commutative law is replaced by a law which, roughly speaking, postulates the existence of common multiples on one side.
Some results
are also obtained for compact semigroups. Xn the second chapter it is proved that a totally bounded semigroup of linear bounded operators, acting on a Banach space, and satisfying certain generalized commutativity postulates, is an ergodic semigroup and every element of the Banach space is ergodic with respect to this semigroup.
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The author cashes to thanh Professor Dinar Hille, and other members of the Department of Mathematics, uho by their encouragement and advice, have contributed to the preparation of this dissertation.
i
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TABLE OP COHTEHTS
Chapter 1.
The embedding of a topological semigroup in a topological group and its generalizations page
Chapter2.
1
An ergodic theorem for a noncommutative semigroup of linear operators.
Bibliography
..
•.
•*
••
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page
40
page
53
CHAPTER 1 .
THE EMBEDDING OP A TOPOLOGICAL SEMIGROUP II'I A 6 E
TOPOLOGICAL GROUP AHD ITS GSHSRALIZATIOUS.
K I ? I; 
Introduction.
We shall use the term groupoid to
denote a set of elements S such that, to any two elements a t
and b in S there is defined a unique product ab in S.
An
I I
element a in a groupoid S is a left (right) canceller if for
I
any pair of elements x and y in S, the equation (xa*xy)
implies that
x — y.
a x = ay
An element is a canceller
if it is both a left and a right canceller.
An element a
in a groupoid S is left (right) proper if to any element b in S there is a uniquely defined element c in S such that ac=b
( ca = b ).
and right proper.
An element is proper if it is both left A groupoid is associative if the relation
(ab)c^a(bc) holds between any three elements a,b,c. associative groupoid is called a semigroup.

1
An
A groupoid

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in which, all elements are cancellers is called a cancellation groupoid, and an associative cancellation groupoid is called a cancellation semigroup*
A groupoid in which every element
is proper is a quasigroup and an associative quasigroup is a group in the usual sense.
For quick reference these notions
may be conveniently tabulated thus:
an operation
groupoid all elements cancellers
cancellation groupoid
all elements proper
quasigroup
an associative operation semigroup cancellation semigroup group
Any one of the above will be referred to as a system. Unless it is stated otherwise, we shall use the term topology to mean a Hausdorff or separated topology Jl: p32j . We shall introduce topologies into .these systems in the fol lowing manner.
A systemwithtopology is a system which
is also a topological space, but this topology is not neces sarily related to any operation between elements of the system. A topological groupoid S is both a groupoid and a topological space, in which the correspondence
(a,b)>ab
is a contin
uous function from the product space SxS into S.
A topological
quasigroup is both a quasigroup and a topological space in

2

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which any one of the three elements a,b,c & IK
i
£• U
related by
ab = c
is a continuous function of the ordered pair formed by the other two in SxS.
A touological semigroup is an associative
topological groupoid, while a topological group is an associa tive topological quasigroup.
This coincides with the ortho
dox definition of a topological group. It is well known that a commutative cancellation semi group may be embedded in a group.
In fact, [2: p27] a com
mutative semigroup may be embedded in another semigroup in which the new set of cancellers forms a group containing the original set of cancellers. £ K: £ s tv r fvi'
I

Malcev £53 has shown that a
noncommutative cancellation semigroup may not, in general, be embedded in a group, and he has also discovered complicated necessary and sufficient conditions [4j].
Ore L 5 3 has given
a construction of the embedding process in the case where the cancellation semigroup, has common multiples on one side, and a similar construction is made by Dubreil [6: pl42j in the case of noncommutative semigroups, where not all elements are cancellers and with a weaker common multiple property. She lander [73 has proved a similar theorem without assuming the associativity.
He defines an alternation groupoid to
be a groupoid where the relation
(ab)(cd)= (ac)(bd)
satisfied between any four elements.
is
We observe that this
is a weakening of both the associative and the commutative laws.
We say that an element a is a '"'canceller if every — o 
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power [7] of a is a canceller.
Sholander proves that an
alternation groupoid may be embedded in another alternation groupoid, which is such that the new set of ""'cancellers is a quasigroup, which contains the original set of ""'cancellers. In particular then any alternation cancellation groupoid may 
I I  f I
be embedded in a quasigroun. In the well known case of embedding a commutative cancellation semigroup S in a group G, we recall that the first step is to set up an equivalence relation in the product semigroup SxS defined by
(a^,ag) ~ (b^,bg) if an only if aj_fbg =. ag + b^.
I f
Using square brackets to denote the equivalence class generated by a pair, we define the operation between equivalence

classes by
Eal,a2l
Eal + ^ ^ a g + bgj.
® ie se^
r

equivalence classes with this operation forms the group G

v/hile the correspondence

of the semigroup in the group.
 I:  tr' V §:i !■ F 
s>[a,a+s]
defines an embedding
If S is also a topological semigroup then there is a natural tocology in the product semigroup SxS, while the “* ^ equivalence relation introduces a topology, in a natural way, in the set of equivalence classes G, viz. the finest
£•
I
topology for v/hich the correspondence
 tt : r.‘
SxS to G is continuous.
p

topology.
j[
a topological group.
(a,b) —
Ea»'°3
from
Ihis topology is not necessarily
separated, but in a sense we have that G is a groupwithIt is therefore natural to ask whether G is Since the topology in SxS is sym
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metric and since the inverse of
in G is [a^ajl, it
is clear that the inverse operation in G is continuous.
For
iE;
G to be a topological semigroup it is sufficient that the
£
equivalence relation be open.
fe
what restrictions are required of the original topological
f e K1
It is not immediately obvious
i:V
semigroup.
We propose to discuss this problem, but in a
more general way as follows Section 2 is concerned with some preliminary topological theorems, some of which are common to all the later sections. In section 3 we replace both the associative and the commutative laws by the weaker postulate of alternation and base our topological investigations on the work of Sholander M . In section 4 we keep the associative law, but weaken the commutativity and use the algebraic results of Dubreil [6]. These two sections therefore constitute generalizations in two different directions. In section 5 we show that in the case of a compact topological cancellation semigroup, there is no problem since it is already a topological group. Since we shall quote the work of Dubreil and other French writers, it is well to observe that their demigroup is our semigroup> and their semigroup our cancellation semigroup.
 5 
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2.
Topological Groupoids We shall say that a topo
logical space S is embedded in
a topological space S ’, if
S is homeomorphic with a subset of S* using the relative iX
i> I;
% >':.ftt,ts)lt£ T}  9 
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£
2 .6
from S into TxS is open in (2)
TxS.
every element of I is
relative to T, if and only
a continuous left canceller,
if the
mapping s — ^£(t,ts)lt £ T j
from S onto E is open in E, and I?is a subset of the left cancellers. Proof:
(1)
Let every element of 2? be a continuous left
divisor relative to T, and let A be an open set in S. B be its image under the mapping considered. in B.
Let
Let (t,ta) be
Then v/e may find E(t)xE(ta) in TxS such that if
(tVs*) is in E(t)xE(ta) then there is an a.' in A such that s ^ t ’a*.
This proves that ( t S s ’) is in B and that B is
open in TxS.
Conversely let the mapping be open and let B
be the open image of the given open set N(a) in S.
For
every t in T ve have that (t,ts) is in B and T/e may find E(t)3E25"(ta)£ B.
This means that if (t*,s*) is in E(t)xE(ta),
then there is an a* in E(a) such that s ’  t * a ’, and this proves that t is a continuous left divisor relative to T. (2)
We observe that the expression (t,t
for an element of E is not unique in s unless t is a left canceller.
If every element of T is a continuous left can
celler relative to T and if A is an open set in S and B its image in K, then let (t,ta) be in B.
Then we may find
E(t)xE(ta) in TxS which is such that if (t’jt’a 1) is in E(t)xE(ta)aE in E.
then a* is in A.
This proves that B is open
Conversely, let a,E(a) in S and t in T be given,

10

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I
where T Is a set of left cancellers and the image of 3S(a)

is open in K.
Since (t,ta) lies in the image of K(a) we
may find N(t)xS(ta) in TxS such that if (t’jt’a 1) lies in K(t)xN(ta)nK, then (t'^t’a 1) is an image of an element of N(a).
But the a ’ in this expression is unique, so that
a 1 is in U(a), which proves that t is a continuous left can celler relative to T. Corollary 2.7 .
If every element of T is a continuous
left divisor relative to T, then for any t in T and any open set A of S, tA is open in S. Proof:
Since the set
{(t,tA)Jt£T}
is open in
TxS, if we fix t, then the section. tA is open in S. Continuous divisors thus have the important property of translating open sets into open sets. Corollary 2.8 .
If T is a set of left cancellers,
andcontinuous left divisors,
relative to T, then it is a
set of continuous left cancellers relative to T. Proof:
If the correspondence
s ^4.('k*'ks) 11 € T} is
open in TxS, then it is certainly relatively open in K, so we obtain the corollary by combining both parts of theorem 2.6
. Theorem 2.9 .
If S^ and Sg are two topological groupoids
whose elements are continuous left divisors, then every ele ment of the topological groupoid

11
S^xSgis a continuous left

1
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2 .1 0
divisor. Proof:
This follows either directly, or from the
previous theorem and the fact that if from A^ to B^, i=l,2, A 1xA2
to
then f^xfg
f^
is an open mapping
is an open mapping from
BjxBgLet every element of the groupoid
Theorem 2.10
withtopology S he a continuous left divisor and let S ’ be a continuous open homomorphic image of S.
Then every element
of S ’ is also a continuous left divisor. Proof:
The theorem may be proved either directly
or as follows. S onto S 1.
Let f be the continuous open homomorphism of By theorem 2.6, we must show that the mapping
y 1 — ^ x * ,x’y ’)[x* £ S f} of S ’ into S ’x S 1 is open. this is the result of three mappings,
y*
However
f“^(y’),
y^?x,xy)j x€ s} , where y (f(x),f(xy)) = U ' j X ’y'), where x ’ =f(x); from S 1 to S, from S toSxS and from SxS to S ’x S 1, each of which is open. We now investigate the embedding process which is fun damental to each of the cases "under discussion.
If we have
a set S in which is defined an equivalence relation, then the *v*
set of all points in S, each of which is equivalent to some point of a given set A, will be called the saturation of A and will be denoted by contained in satA
satA.
The set of equivalence classes
will be denoted by [A].

12
In a like manner

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2 .1 1
the equivalence class containing the element s is denoted by [s] and the set of points therein by [[satA]] = [a ] £ £s ]
and
sat s.
[sat a3 = £a]]€£sj .
Thus We shall say
that an equivalence relation is open if satA is open whenever A is open. The natural topology induced on the set of equivalence classes [l: p52] is the finest topology which is such that the natural mapping
a^£al
is continuous.
[sj are the sets £a J' where
satA
Thus open sets in
is open in S.
It does
not in general follow, that the topology in [S] is separated. We shall verify that this is so in each particular case. Theorem 2.11 .
Let S be a topological groupoid and
let T be a set of left cancellers in S.
In the subset K
of TxS, consisting of the elements of the form (t,ts),t€T, MWJMJt! >»WWWiai^A W IISW )IMHgmi>a»MW»WM!B'Wiro(t,ts), and (t,ts)>
[t,ts}, from S to 2 and from K to [Kj.
Thus the corres
pondence is a homeomorphism if and only if the mapping s sat(t,ts) is open and this, by theorem 2.6, and since sat(t,ts) = {(t,ts) t€T_[ is equivalent to the statement that 
every element
\
to T.
of T is a continuous
left canceller relative
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3
Alternation Tooological Grouuoids. If in the
Euclidean plane we denote by ab the midpoint between any two points a and b, then it Is clear that this operation has the property of alternation, because of a simple geomet rical property of a quadrilateral.
In fact if the plane
is in no way restricted, it. becomes an alternation topolo gical quasigroup under this operation.
Any convex set in
the plane will be a topological cancellation groupoid.
If
we take, for example, a circle of unit radius in the plane, then in order that the equation ax = b
should have a solution
when a and b lie inside the circle, we must adjoin the points inside a concentric circle of radius three.
If we repeat
this process a countable number of times, we obtain the whole plane, and In this way we have effectively embedded a topolo gical groupoid in a topological quasigroup.
In the general
case one might therefore expect that an alternation cancel lation groupoid will generate a quasigroup, by means of a countable number of extensions.
This is in fact the case.
iiany more interesting examples of alternation groupoids are given by Sho lander £7]. We shall be concerned here with the more general case where it is not assumed that every element is a canceller. We begin by recalling some theorems of Sholander If S is an alternation groupoid, let L (R) denote its set of left (right) cancellers and Lw (R*) its set of left (right)
 15 
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"cancellers.
The same notation will be used with subscripts
also. Theorem 5.1 .
{7: 4,2  4.1C>]
If S is an alter
nation groupoid and if we set up the relation (a1,ag) in L^xS whenever (a^a^)b^ = (a^b^)a^,then this is an equivalence relation, and the set [L"x S3=S j_ of equivalence classes be comes an alternation groupoid under the operation £ai,a23[bi,b23 = [ajb^agbgl . [u,uaf],
S is embedded in
Under the correspondence a — ^ and If"' is embedded in
if a is in If* and b in S, the equation solution
ax = b
while
has a unique
x = [a,b^ in S^.
Sj_ is called the first left extension of S. vacuous then
S
and
is vacuous.
first left extension of 2^, n ^ 1,
If If* is
If
is the
we obtain an increasing
sequence of alternation groupoids, whose union S«j has more interesting properties mentioned in Theorem 5.2 .
[7: 4.ll
An alternation groupoid
S may be embedded in an alternation groupoid manner that If" is embedded in
in such a
and every element of L^"'
is left proper. is called the left extension of S. may find a right extension of S.
Similarly we
Sholander proves that
the left extension of the right extension is isomorphic with the right extension of the left extension, and he calls either
 16 
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3.3 one S' cjooO
*
Tills t reives jJ
Theorem 5.5 .
[7: 5.4j
An alternation groupoid
S may be embedded in an alternation groupoid manner that the ‘"'cancellers of Soooo
in such a
form a quasigroup con
taining the ‘"cancellers of S. In particular we have
[7: 5.5] that an alternation group
oid S may be embedded in a quasigroup if and only if every element is a canceller. As might be expected we can find a "workable'* topology in the extensions only by putting further topological restric tions upon S, which are carried over to S^, and which make the equivalence relation in L*"*xS open.
One set of conditions
is the openness of If"* and the continuity of divisors.
This
is the main concern of Theorem 5.4 .
Let S be an alternation topological
groupoid in which every element is a continuous divisor and L* is an open set. (1) (2)
Then:
the equivalence relation in L‘"xS is open, is an alternation topological groupoid in which
(3)
every element is a continuous divisor,
(4)
1^*" is open in
(5)
the mapping (a,b)>x = [a,b} where a x = b , from L'"xS
and
to. S^ is unique, continuous and open. (6)
Also
L" is embedded in
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3 .4
(7)
S is embedded in
(8)
S is open in
Proof: lation (a^ag) L*'*‘xS is open. (bi,bg)e satA. b ^ a , or
and
.
We prove first that the equivalence re (b^jbg) if and only if (a^a^Jbg = (ajbj^ag in Let A be an open set in L*xS.
Let b =
Then there is an a (a^,ag)€ A
(a^a^)bg = (a]_b2 )3 .3 *
such that
Since Lw is open in S, we
may choose HCa^)* H(ag) in S, where iKaj^^L* and ECa^xETCagJc A. Consider the expression (a^bj^agnc.
Since ajb^ is a con
tinuous left divisor, we may choose I'Ua^b^) such that if x 1 is in lUa^b^) then there is an Slow choose I'Ub^) ^ L^
such that
in ^(ag) with x ’ag's; c. a^KCb^)£ ^(a^b^).
consider the expression (a^a^Jbgsc.
The set a^lKa^) is an
open srt of S containing a^a^, by corollary 2.7. is a continuous right divisor, we may find IT(bg)
Since bg in S such
that if bg* is in ITCbg) we may find an aj_rin lUa^) (a^a^Obg* = c.
Also
Put I'I(b) = Etb^JxiUbg) .
with
Then if b ’^ C b ^ b ^ )
is in U(b) we have that a^b^1 is in a^h'Cb^) and we may find a^* in Nta^) and a2 * in IT(ag) such that ( a ^ b ^ a g 1=. c = (a^a^1)bg* This proves that b ' ~ a ' 6 A L7: 4.33,
bhat satA is open,
so that the equivalence relation in L^'xS is open, and this is (1). We observe that since L* is open then L* is also an alternation topological groupoid element is a continuous divisor.
(7: 3.lQ in which every Thus by theorem 2.9 and
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3 .4
its dual, the same is'true of L*xS. that the natural mapping of L “xS onto
But we have just proved S^
is a con
tinuous open homomorphism, so that Sj_, by theorems 2.5 and 2.10 is an alternation topological groupoid with continuous divisors, if we can verify the separation axiom.
This we do as follows.
Let \ei\ and [b] be elements of S,, where £aj * £b] . Thus if a =.(a,,a2 ) and b = (b^bg) then (aiai)t>2 = ^alb l^a2* But vie may find H{(a1a^)bg^ and IT ajb^)a2\ in S with a vacuous intersection.
Since S is a topological groupoid, we may
choose H(a.,) and IT(b^) in L"* and l\T(a2),N(bg) in S, such that
{H(a1)LT(a1)} L'(b2)^ U{(aLa1)b2}and ^(a^UCbj^lKag) £ ll£(a1b1)a2^. Thus £lT(a, )xlT(az)j , ^2f(b t)xN(bz)j are open sets in S^ contain ing £a] and £b] respectively, because the equivalence relation is open, and they have a vacuous intersection, because if we suppose to the contrary, then we may find (a^’jag*)
in
iT(a^)xH(a2) and (b^’jbg*) in IT(b^)xlT(b2) such that a ’^ b 1 or ^a^ tai ,)^2I “ ^al ,^l* ^a2*
"kkis ^s a c033*1*adiction.
We
have therefore established (2) and (3). Since Lj*' = Ob'bcL^ L"xL
fV: 4.6]
is open in L"xS, so that
and Lv is open in S, then is open in S^_, which is (4).
If a is in I?c and b in S then the unique solution x in Sn of the equation ax But the mapping
b
may be written x = £a,b}
Jj7; 4.103 *
(a,b)> ja,b] is continuous and open from
L'“xS
to S^_, since the equivalence relation is open, so we
have
(5).
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5 .5
The correspondence with a subset of
a^£u,uaj is an isomorphism of S
[7: 4.9^ .
This is also a homeomorphism
of S with {u,uS^ because by corollary 2.8, each element of L’“ is a continuous left canceller, use theorem 2.11 & : 4.8] .
relative to IT, so we may
Thus S is embedded in S^.
proves (7) and therefore (5).
This
Since every element of L’"* is
a continuous divisor, it follows from theorem 2.6 (1), that the set
§u,uS)\ uCli^j
is open in L'icS
is open in S^, which is
and hence that S
(8) .
We observe that parts (2),(3) and (4) of this theorem show that the topological restrictions imposed upon S are inherited by the first left extension S^. apply the same theorem to
and so on.
We may therefore 3y applying this
theorem n times we obtain Theorem 5.5 .
If S is an alternation topological
groupoid with continuous divisors and If'* open, then the same is true for each S^, while each subset of
n > 0 , S0 ~S.
If we put Sgf,
is embedded as an open
xOO \J,
as in theorem 2.1.
then we may define a topology in Because each
is open in
then So,, is an alternation topological groupoid, from theorem 2.4 j7: 4.lj .
It may be verified directly that every element
of Soo is a continuous divisor. since L* — \JIn*
Also
*6l»

is open in S^,
each 1^* is open In Sn .
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5 .6
Thus ue have Theorem 5.6 .
If S is an alternation topological
groupoid with continuous divisors and L* open, then S may "be embedded in an alternation topological groupoid S^ , ■which is such that every element of
is a continuous divi
sor, L^' is open and every element of The groupoid
Is left proper.
is called the left extension of S.
In a similar way, If R* is open in S, we may find the right extension of S.
Y!e may also prove that the right extension
of the left extension is isomorphic and homeomorphic with the left extension of the right extension.
For this we
need the following theorem, in which we employ the notation of Sholander £7: 5.1]. Theorem 5.7 .
If S is an alternation topological
groupoid with continuous divisors and both L*' and R*' open, then the first right extension of the first left extension (Si q )o i
is isomorphic and homeomorphic with the first left
extension of the first right extension (Sq x ^i o * Proof:
Sholander has established the algebraic
part of this theorem [7: 5.1].
Since the result is trivial
if either L** or R* is vacuous, we consider the case where both are nonvacuous and hence L5,:‘= R*
[7 : 5.7].
sets up a correspondence between x in (S^q )q ^_ and y
Sho lander in
(Soi)io> by means of the sequence of four correspondences between the spaces (S^o^Ol* RlO*xSio > L*::'xSxL*xS = R*xSxR*xS ,

21

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3 .7

V  £■■ 
i
SqjsRq^,
(SqiJ^o
defined in the following way.
in (S^o^oi* ciloose (**»p) in Rio^xSlO suck tbat xr = p, then jj,
choose (a,c,b,g) in L^xL^xL^xS such that ar = c, b p = g , then

(s,q) in ^ o i ' ^ O l is determined by
I
^01^10 ^7
[•
t
Given x
ft
sa = b,ac=g
and y in
Sholander shows that this is an isomorphism of
1(
v
by using the equality L"'*' and R"“ in each group considered.
I:
A
To see that it is a homeomorphism, we need only observe,
£
fe
Si; f t t? 
from theorem 5.4 partC5) and its dual, that each of the con stituent correspondences considered is either a continuous open homomorphism or is the reciprocal of such.
Thus open
sets correspond to open sets in both directions. 7tre now see
[7: 5.2] that if we make m first left exten
sions and n first right extensions, the result is independent C to whithin homeomorphic isomorphism ) of the order in which these extensions are made.
We thus denote the resulting
topological groupoid by
♦
The next step is to prove
that
is independent of the order of the extensions.
This is
done as follows.
in He©o*X"xSx2b2 ^ ' because any two
equivalence classes in the neighbourhoods on the left hand side may be written
[_a^!,ag>3 ,£b^’,bgQ , where a ^ ’^HCa^),
a g ’^lUag), bj* ^KCb^), b g ^ H C b g ) , and therefore there exist xp*£ iUx^), x ^ ^ l U x g )
such that
xi*a2* = ^ ’^l* * so " fc3ia'k
[al ‘,a2 ^ B>l**b 2 0 ~ [ xi lal I>x2 ,b2 ,le LK ^x lal#x2^23 *
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132113
4 .2
is a topological semigroup, if we can establish that it is a separated space. To do this let b = ( b 2 _,b2 ).
[aj ^[b}
in
Thus if we choose
xa^ = yb^, then
is open, we may find
both il(a^)
and
x' £ lUx) a
b^_
N(b^)
with
y* £iUy)
y
in
C
such that
S
By continuity, and since
C
in
and
C
and
N(ag),mbg) in
If(y)IHbg)Q U(ybg).
Since
are continuous right divisors, we may choose such that if
16 hC a^)
x ’a^1 = xa^> and if
with
a~b,
N(xag), NCybg) in
EF(x), h(y)
iJ(x)hT(ag) 4 H(xag)
aj_ and
and
Thus we may choose
with a vacuous intersection.
where
x
a«=(a^,ag),
xag^ybg, for otherwise we would have
which is false. 
S
S^, where
y b ^ y ’b^1.
then there is an
b^’^KCb^) then there is
Then
[H(aj_)xN(ag)l and
are open sets of equivalence classes containing [a}
and
[b"]
respectively, but with vacuous intersection,
for the following reasons.
If we suppose that some equivalence
class is common to both, then we may find iUa^JxNCag),
(b^jbg*) in
However there exist x ’a ^ ^ y ’b ^ ’. x^g^y^Dg*, fact that and
NCb^xUCbg), where
x* 4 K(x)
and
y ’€H(y)
in
a ,~'bI. such that
If we can prove that this implies that then we have reached a contradiction of the
KCxag)/\Er(ybg) = 0.
x ’a^’= y ’b ^ 1
uag’srvbg*.
Thus we prove that
implies that
means that there exist and
( a ^ S a g 1)
u
x ’ag* ssy^g1.
and
Choose
v
r,s

28
in in
C C
a ’/^b1 But
such that so that

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a ’^ b *
ua^1 =.vb^*
ru~sx*.
Then C
rvb^’— rua^1 = sx'a^1 = sy’b ^ ’, whence since
we have
rv — sy ’.
whence since
s
Also
is in
C
3
x^
2
we have
b^* is in
’=rua2 *— rvbg 1 = sy’bg* x ’a g T
contradiction being confirmed, we know that [a 3 and Lbl stay be separated by open sets, and
S^
is therefore a topological
semigroup. The set of cancellers C ^ = [CxC3
and
CxC
is open in
is open in
CxS.
because
Then
C^
logical group because the inverse operation in from the reversal of coordinates in
is a topo
Cj_ is derived
CxC, which is obviously
a continuous operation. That theorem
2
S
is embedded in
S^, follows immediately from
.1 1 , since the embedding isomorphism is
and every element of relative to
C
a>[u,ua3
is a continuous left canceller,
C, by corollary 2.8.
'we proceed to generalize this theorem by showing that we may just as well work with ideals as with the whole semi group.
hore precisely we say that a subset
semigroup
S
is an ideal of
S
if
S0 S< SQ
S0 and
of a SS0£ SQ .
Using this concept, we may weaken our topological postulates. For this we require the Lemma 4.5 S, such that
If SQn C ^ 0
and if
of the semigroup
SQ , then
Proof:
That
SQ
is an ideal in a semigroup CQ
is the set of cancellers
C0 — S0r\C.
CaS0 ^C 0
is easy.
Let
 29 
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c
be in
4 .4
CQ
and let
Choose
u
are in
Sc
a,b in
be elements of
Sq a C , then
so that
S.
Suppose that
cau = cbu.
and
bu
Thus
c
is
Similarly we may show that
c
is
However
au=bu, from which
a left canceller of
S.
a right canceller of
au
c a e cb.
S, and therefore
a = b.
c € C , or
C0 = S 0aC.
This leads to the Theorem 4.4 . SQ an ideal in
Let
S, such that
S^ generated by
S
S
be a Dubreil semigroup and
Sot\C=£0.Then the semigroup
(according to theorem 4.1)
with the semigroup
SQ^
Proof:
generated by
is isomorphic
SQ .
Vie observe first that
SQ
does generate
a semigroup. SQj_ in the sense of theorem 4.1, since also a Dubreil semigroup. find
u,v£ C
such that
lemma 4.3, so are a € S0 , b £ CQ then if
wu
w£C0
then
is
In fact if a , b £ C 0 , then we may ua = vb.
and
we may find
S0
But if
wv, while
wua=wvb.
u4C, v € S
w u g CQ,
w € C Q , by the
so that
w v € SQ , while
Also if ua=vb,
But
wua=wvb.
It will be necessary to distinguish between the equiva lence relations in
CxS
and in
0oxSo , and we shall do this
by denoting the equivalence classes in We set up a transformation from as follows.
Let
w
in
0oxSQ
S^ = [CxSl
diagCox0o
to
be fixed;
find it convenient to denote the coordinates of same letter, S^,
define
i.e. PCa3
w = (w,w).
If
> so that
by
£
SD^ — Cc0xS030 we shall w
by the
£a^
is an element of
F[a]
is an element of
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Sq i 
The function
from
onto
then
[wa]G = [wb]0
F
is single valued and one to one,
S0^> because i.e. in
w
is fixed and if
wa~wb
in
follows that
a^b
CxS.
because with
a = (a^,a2 ), b = (b^,b2 )
x^wa2 = XgWb^
x ^^Cq
C0xS0 , from which it
It is also an isomorphism
P [a] P[ bj = [wa] 0 [wb] 0 — where
FlaJ= P[bj
,
in
CxS
,XgWbg ]Q ,
x^
S ^
and
Fit a][ b3} = P [yia^ygbg] = [ w y ^ , wygbg] 0 ,
where
7ia2 ~ 3 r2b l5
such that
Yi€ C,
rxj_w — swy^
y2 £ S.
Choosing
r,s
in
0o
we have
rxgwb ^ =rx^wag — s w y ^ g ^ swy2b1, whence since
b^C,
rxgTf — swy2 .
[■wy1a1,wy2b2J0=Jx1wa1 ,x2wb2l6 ,
This proves that
and that
P
is an isomorphism.
We may extend this result to topological semigroups in the follov/ing Theorem 4.5 .
Let
semigroup, and let intersects ted
S
C.
S0
S
be a topological Dubreil
be an ideal in
S
such that
Then the semigroupwithtopology
SQ S^
genera
Is isomorphic and homeomorphic with the semigroup
withtopology
So1
Proof:
generated by
SQ .
We use the isomorphism of the previous
theorem, but we observe that the elements in the equivalence class
F(V]= [wa]Q
are precisely
any element which is equivalent to equivalent to
a
in
CxS
i.e.

31
(sat a) wa
in
n C0xS0, C0xS0
F£a] =[(sat a)
o

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because
is also Cq X s J
q
.
Any open set
A^_
of Sj may be written
is a saturated open set of which proves that Bq1
FA^
Bo
prove that
B
CxS.
CQxS0,
w
Conversely, if
C0xS0
and
£b 3 = F“*B0^
CxS.
Let
wb£ B0 .
b
B
so that
b
be an element of
are isomorphic
B.
However the left trans
is an interior point of F
is a
it remainsto
so
is a continuous transformation from
This proves that Sc^
then
A
we may write B o 1 =.[B01 o ={b ,v(Co x So }3
Thus
is open in
w & d i a g C oxG0
lation by
So1
where
FAj=F[.A3 — [Aa (Cox So )1 0
Sq I*
is a saturated open set of
saturated set in
Since
Then
is open in
is an open set of
where
CxS.
A ^ = C a 1,
CxS
B, or
is a homeomorphism, so that
to
B
Sj_
is open. and
andhomeomorphic.
Using this we may reduce slightly the topological assump tions made on the semigroup in theorem 4.2. Theorem 4.6 . group, and let
S0
Let S be a topological Dubreil semi be an ideal in
S, which intersects
Let
SQ
have continuous divisors in
let
CQ
be open in
of CQ
S, relative to CQ .
generates a topological semigroup
Proof:
relative to
SQand let all elements
continuous cancellers in
and such that
SQ
S^_
S
conditions of theorem 4.2,
SQ
S0
S
is embedded
Cj_ is a topological group containing Since the semigroup
SQ , be
Then
in which
C.
G.
satisfies the
generates a topological semi
group, which by theorem 4.5 is isomorphic and homeomorphic with the semigroupwithtopology
generated by
 32 
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S.
Thus
I  * I I
is a topological semigroup and
is a topological
group. We define equivalence classes in CGxS
l
section of the equivalence classes in CxS
I
0oxS.
to be the interwith the space
We denote the equivalence relation thus set up in
I
C©xS
by
[.
thatthe equivalence classes in
(J0xS0
:
of equivalence classes in
with the space
C
Iq*.
p[a] Q =Ea]0 ’. onto
Prom the proof of theorem 4.5,
CxS
we know
are intersections 0oxSo .
This defines a transformation from
£0oxSjo 5 because
SQ
is an ideal.
Put
£C0xS0^ 0
This transfor
mation is continuous, by definition of the equivalence classes and is open because the two spaces are homeomorphic. with
(CoxSl0 ‘.
Thus
and
[0oxSo3 Q
is a homeomorphism of
However by theorem 2.11,
phically embedded in is a
P
[CxS^
S
is homeomor
[C0xS]0 ’, because every element of
continuous canceller in
S
relative to
S is homeomorphically embedded in
[C0xS0]0
CQ .
are connected, then so is
S^_.
and in
Also if
SQ
C0
Therefore
We now remark that if in the above theorem both CQ
[Co^ol 0
S^=[CxS3 SQ
and
has any one
of the properties (a) locally conpact,(b) separated, (c) locally connected, then
has the same property.
• Since any commutative semigroup is immediately a Dubreil semigroup, then the above theorem applies to commutative topological semigroups yielding a generalization of theorem 3.10.
In particular we have
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4.7
Theorem 4.7 .
If a commutative topological cancel
lation semigroup
S
every element of
S0
to
SQ
which is such that
is a continuous divisor in
SQ, relative
SQ ,
and a continuous canceller in
S, relative to
S
generates a topological group
G
then in
contains an ideal
and
S
SQ,
is embedded
G. That the consideration of the ideal
Sc
in
S
does
indeed give us something more, is illustrated in the following example.
Let
S
be the semigroup of real numbers greater
than or equal to one, under the operation of addition, and with the usual topology. greater than one.
Let
Then
SQ
SQ
be the set of real
is obviously an ideal in
Every element of
SD
is a
continuous divisor in
every element of
SQ
is a
continuous canceller in
tive to
SQ .
numbers S.
SQ,while
Vfe know then, by theorem 4.7, that
S rela S
gene
rates and is embedded in a topological group, which is of course, the group of reals under addition. element
1
in
S
However the
isnot a continuous divisor in
S,
so
without this generalization we could not say anything about the topology of the group generated by the trivial semigroup S.
 34 
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5.
Compact Semigroups . In this section we estab
lish. the fact that any compact topological cancellation semi group is already a compact topological group, so that the problem of embedding does not exist. Theorem 5.1 .
For this we need the
Let G be a topological groupoid and
let Q be a subset of G, which is such that Q is an abstract quasigroup, every element of Q is a canceller and Q, is com pact [l: p59} .
Then both Q, and Q are topological quasi
groups . Proof;
Let a and b be any two elements of Q,
Since Q is a compact space, its topology is uniform [l: pl07j so the neighbourhoods of a point p will be denoted by where A
is independent of p.
We may then
choose directed sets {a^ [ot € & }
and jb*\oc e
ging to a and b respectively.
Since G is a topological
groupoid, then the directed set [a^b^  € A } to ah [8], so that ab is in 5
in Q converges
Also the pair a
defines unique elements c ^ jd^ The directed sets {c^joiG A }
in Q conver
,bx
in Q, such that a ^  b ^ c ^ s d ^ b ^ . and fd^ } * £ A]j
are both in
the compact space Q, and therefore have cluster points £93. Let c be a cluster point of {c^iot.^ A.} a^rbc.
Then we may choose Y in A
is vacuous. Thus a
But we may find
and suppose that
so that
(a)n[ys(b)7K (efl
so that c ^
is in V^(c) .
€■ V CC (a)C VxO (a), and bOCcoC € V* (b)Vjwr (c), which is a con
 35 
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5.2
tradiction.
Therefore a=bc, and c is a unique solution
of this equation because of the cancellation law in
Simi
larly there is a unique element d in Q such that a = db and this proves that ^ is a quasigroup. To prove that Q is a topological quasigroup, we repeat the construction of the previous paragraph, except that we allow a^ and b^ to lie in ^ and use the fact that a directed set in a compact space, with a unique cluster point, must converge to that cluster point.
Finally since Q is a sub
quasigroup it is also a topological quasigroup. If we assume the associative law we have Corollary 5.2 
A compact topological semigroup,
which is an abstract group, is also a topological group. We need also the following theorem which will be stated in a general form so that it can be used also in chapter two. Theorem 5.5 .
If T is a compact topological semi
group, and if S is a subsemigroup of T, which satisfies the pseudocommutative conditions (i)
If a,b
are in S
there existx,yinS such that ax = by,
(ii) If a,b
are in S
there existx,yinS with ab = xa = by,
then the set
G  sOtoo S3
is a nonvacuous tooological group. w
Proof:By the hypothesis sets {15)s € A } where vacuous intersection.
A is
(i)the collection of a finite subsetofS, have a non
Therefore, by the compactness
 36 
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[l: p6cf
5 .3
G is nonvacuouc. If we set up an ordering in S, by saying that a > b if and only if there exists a c in S such that a«=bc, then S becomes a directed set under this ordering, because of (i).
The set
G is then the set of cluster points £93, of the directed set
S in the compact space f.
Because T is a compact space, its
topology is a uniform topology, so we denote a fundamental system of neighbourhoods of a point p by {V^CpHe^ 6 A\
with
/\ independent of p.
In the product set
SxA
we set up the ordering (a,
(b, p. ) if an only if a > b and
oC> p
, so that S x A
)>
is also a
directed set. Let a and b be in G. given (s, s’ as,oc£lro s’
111611 as„
s and "bSjoC € T^Cb). choice of ©s, ^ in ^ ( a ) .
in S
Row we make a fresh
in S so that a s> b s,^ bg, ^
and agjet is
By postulate (ii) this means that we may write
 37 
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5 .4
as,cc =fcS,oc b s,K cs,e,=ds , ^ b s,ec'bs,oc where cSj>c, d S5eC are in S.
Again by (ii) b 3>e(.cS}ei > s
directed set point.
and s*
Suppose that a^bc.
Then by continuity, we may choose p in
/\ so that 7/g(a)A[7p(b)Va(efl is vacuous.
given t in S, we may choose
a^ ^
^h©
b Sj0
0
,
If we put
in the strong
={x^,... ,xn ; £.} ,
and denote the set of all oc
= £ Tl T € £. , [Tx^ l< £
,1
=
1
, ...,n}
,
by
the
is a fundamental system of neighbourhoods
of the zero operator in the strong topology. It is easy to verify that topological space. because if in
&
is an
0
such that
lT*l ^ M
for all f e T 1 .
is a fundamental durected set in 3 C we may find
T^x
to verify that in
so that
But {T*x)fc6 T}
which is complete, so
T^x^T^x.
It is a routine matter
is. a linear operator,
llci $ K
T^^Too
. We know also that
is an abstract algebra satisfying
the postulates (1 ) and (2 ) of theorem is a totally bounded subset of therefore bounded in norm Thus if
p> =[2^,... ,xn ; e \
which we have that putting
2
.2 , because (l) if
(X
£, , then it is bounded and
i.e. \ T \£M put
for all
T
in
 { x^ ... jx^;
G.1^ ^ ^y3 ? a*1*! (2 )
ot. ={Txj_,... j^x ^ £.} ensures that
X . , from
if y3 ={5C^,. ••,2 ^; a} ^   ^ 3
•
We shall adopt the following definition due to Eberisin [13J .
 49 
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3 .1
A semigroup
in
$
set of operators
is ergodic if there is a directed
{t * I* € T* }■ ,
in
£,
(a)
T^x is in (con S )x for all V
(b)
there is an M > 0
(c)
if T is in
f
with the properties and all x in 3 £ ,
such that TX( ^ H
for all
V ,
then TimCTT*  T )  0 , lim(T5T  T) = 0,
where the limits are in the strong topology. Also an. element x in 3 C
is erfeodic with respect to this
ergodic semigroup if there exists a that
y€(con^?)x
and
Ty— y
y = Tcox
for all
T
in
in
$
3C
such
.
Our main theorem is Theorem 5.1 .
Let
plicative semigroup in
be a totally bounded multi
S
, which satisfies the pseudocom
mutative laws (i) such
if Sj_>Sg are in
that
such that cf
then there existS3 ,SA
in
^
then there exist
in
«f
S^Sg — SgS^ *
(ii) if sj.,S2
then
$
S^Sg =
are in
“ SgSA ,
is an ergodic semigroup and every element of 3 € is
ergodic. Proof:
We first show that
semigroup.
Let A and 3 be elements of
directed sets A * € A + V^
{A* lot 6 A } and {B^j* &
and
directed set in Vg
is a topological
$
we may choose
€ B+V* . $
Then
}
a
$
in
We may choose x? ,so that
{A^B^jotG A} is a
which converges to so that
.
A3 because,
£ frVg ,
and
 50 i
J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V^B £
given .
3 .1
Then for
©y3
we have that  A3 = A ctB ot  AetB + A^B  AB
SAJT^+V^B s vY ;f
.
Is therefore a topological semigroup. The semigroup
is compact because it is a totally
bounded set in a boundedly complete topological linear space. By theorem 5.3 of chapter, one, the set topological group. P in con ^ by
is a
By theorem 2.2, there is an operator
such that
PG — P =GP
for all
G
Corollary 5.4 of chapter one, if S is in then
SG
and
GS
are in
in $
.
Then
and 5
is in
and
SP S(GP) = (SG)P* P=^ P(GS)= (PG)S  PS. We may therefore put
P — T$
trivial directed set, satisfying order that
x
Sy =■ SPx =■Px ~y
element of
GC
obtain
a
the conditionsrequired in
should be an ergodic semigroup.
$
Finally if and
for all Y and we
is in 3 C for all
then S
in
y=Px $
is in
(conj^x
so we have that every
is ergodic.
We may say a little more about the operator P, as in the following Theorem 5.2 .
The following statements are equivalent.
(1)
P is in
(2)
P = E , the identity of
(3)
. .
consists of one element.
.  51 
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3 .2
(4)
The directed set £
converges in
€• .
(5)
The directed set $
converges in
£, to S = P .
Proof:
(1) implies (2), because P^ = P
can have only
oneidempotent element
which is the identity.
(2) implies (3), because for all
G
in
^
and a group
G =BG = PG ~ P = E
. (5) implies (4), because the directed set
inthecompact
S'
space
so that it converges to
S
has a unique cluster point
S,
E.
(4) implies (5), because if the directed set is