Chapter-4

SEMIGROUP COMPACTIFICATION OF PRODUCTS

AND PROJECTIVE LIMITS

Introduction

Let ( SaJ. be a family of topological semi-aEA

groups with semigroup compactifications{ Aa .a€A

We discuss in this chapter about the corresponding

semigroup compactification of P[ S J In 1961,a aeA

K. Deleeuw and I. Glicksberg [D-G] observed that the

product of Bohr compactifications of a collection of

abelian topological monoids is the Bohr compactification

of their product. They showed by an example that the

identity is not necessary. This work was further

extended and supplemented in [BE]. Here the distinction

between the Bohr compactification and its topological

analogue the Stone-6ech compactification PX is more

pronounced, since PX does not generally have the product

property even for a finite number of factors. A necessary

and sufficient condition for the equality

P P X = P Ox was given by I.Glicksberg in 1959 [GLI].aeA a a€A a

In this chapter, in Section 4.1, we prove that if

{Sa} is any family of topological semigroups withaeA

77

78

semigroup compactifications { A } , then P ^A }a acA a e A a

is a semigroup compactification of P ^ Sa}. Alsoa cA

we consider the family of topological semigroups

(3) with Bohr compactifications £B } and thea acA a acA

latticesof semigroup compactifications[Kl(Sa) I o ThenacA

we show that P At K1(Sa)}C Kl( P Is a J ) is aaCA aEA

,complete lattice.

In Section 4.2,'we discuss semigroup compacti-

ficationr,,Bohr compactification and lattice of semigroup

compactificationsof the limit of a projective system of

topological semigroups.

4.1 Semigroup Compactification of Products

Theorem 4.1.1.

Let [Sal be a collection of topologicalaEA

semigroups with semigroup compactification (yaAa) for

each a e A. Define y : P{ Sa} acA

P {Aa} byaEA

y(x)k - 1k Pk(x), where Pk : P ISa} - Sk is projectionaeA

for each k e A. Then ( y, P € AaJ ) is a semigroupa cA

compactification of P [Sa j .aEA

79

Proof

Let S = P{S } ,A*= P{A a}a aeA a e A

Define y : S > A* by y(x) k = yk Pk ( x). It is known

that A*, being the cartesian product of a family of

compact semigroups with co-ordinatewise multiplication

and Tychnoff topology, is a compact semigroup [C-H-K1].

A straight forward argument shows that y is a continuous

homemorphism. Again y is dense for,

Let y = (yk) be any element of A*,wherek eA

yk E Ak, k C A. To show that A* has a net in y(S)

converging to y.

Since ', kISk) = Ak and yk e Ak for each k e A,

Ak has a net in yk ( Sk) converging to yk, k c A.

i.e., for i E Ik' there exist (xk4) E, Sk such that

yk(xkl) ? yk' for each k e A. where ( Ik' < k)ielk

is a directed set for each k e- A. Then (P £ Ik: k eA}, \< )

is a product directed set by defining i \< j < ) ik -< jk

(i.e., i(k) 4 j(k)) for each k e A. Also we have

A x P fIk:k e A} is a directed set by defining

( k,i) < (k' 9 j) <--7-,> k -,< k' and iy < jy for every ye A.

80

and (Yk (x ki)i ) is a net in y(S)

kEA

such that ( Yk(xk )i)k -> (yk)k = y

y(S) contains a net converging to 'y'.

i.e., y^S) = A*

(y,A*) = (y, P £ Aa} ) is a semigroupaeA

compactification ofaPP Sa3

Next we consider the quotients of Bohr compacti-

fications and prove the following theorem.

Theorem 4.1.2.

Let [S.} be a collection of topologicalaEA

monoids with Bohr compactification (P3a, Ba) . ThenaEA

(y, P {Aa} )is a semigroup compactification ofaEA

P {' Sa j , where A., = Ba/Ra for each a e A.aEA

And P {Ba} /R is topologically isomorphic toaEA

P{ AaJ , whereaEA

R = S((a a)a c- A

, (ba) a c- A)e (P CB a } a x P fB a }aEA) :E A

(aa, ba ) E Ra for each a E A}

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Moreover, any semigroup compactification of P {Sal}

aE A

is a quotient space of P {B a}a LA

Proof.

Let S = P [S a}A,aC

B = P {Ba}aCA

A*= P €Aa }acA

and

Define y : S > A* by Y(x)k = Yk Pk( x), where Pk

is projection and Yk = Ok Pk' where 0k

and Pk : Sk > Bk for each k e A.

: Bk > Ak

Then by theorem ( 4.1.1), we have ( y,A*) is a

semigroup compactification of S. Using product theorem

on Bohr compactification [D-G] (ji,B ) is the Bohr

compactification of S, where 0: S -> B defined by

P(x)k = Ok Pk ( x), where Pk is projection. By the

definition of Bohr compactification there exists a

continuous homomorphism h : B > A* such that

hs = y. Moreover , h is a quotient map and h determines

a closed congruence

R = [((a a) , (b ) )CB x B :aEA a aCA

h((aa) ) = h((ba) ) }ac A aE A

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i.e., T((aa) A, (ba) ) E B x B:(h(aa))a =(h(ba))aac aEA

for each a E A)

i.e.,f ((aa) a €A, (ba) a e A ) G B x B : ha(aa) = ha(ba)

for each a C A }

i.e., R = [ /k ((aa) a EA, (ba) a c A ) G B x B : (aa,ba) E R a

for each a e A 3

Define P : B > B/R, the natural map, then (PP, B/R)

determines a semigroup compactification of S [1.2.4].

Then by induced homomorphism theorem [1.1.13] and first

isomorphism theorem [1,1.14], there exists a topological

isomorphism T) : B/R > A* such that the diagram

commutes

B P B/R

h I n

83

Moreover, if (a,C) is any semigroup compacti-

fication of S, then it is the quotient space of B

follows from [1.2.3].

Theorem 4.1.3.

Let ^ Sa J be a family of topologicala EA

semigroups with lattices of semigroup compactificationa

{K1(Sa) } , for each a e A. Then

P f K1(Sa) } CK1( P {S(x}) is a lattice ofaEA aEA

semigroup compactification.s of P I Sa j.a CA

Proof

Define P { Ki(Sa )}= { ( aa) : Pa ((aa ) ) = as }aEA a ,kEA

where Pa , projection to the ath factor and

as EK1( Sa ) _ {aakEA for each aEA.

Since f Sa} is the family of topological semigroupsaEA

with families cfsemigroup compactification

{ {a ) k CA}= ^K1(Sa

where,

L as ^C, k E A is the family of semigroup compactifica-

tiorSof Sa , for each aEA. Then ( aa ) is a semigroupa,keA

compactification of P { Sa}for each a , k e A, by theorema CA

[4.1.1].

84

i.e., {(aa ) I is the family of semigroup

a, k EA acA

compactificationS of P {acA}

P {k,(Sa )} C Kl( P {Sa} )a EA - ac.A

Moreover P [ Kl( S a )J is a partially ordered set bya EA

defining an order (a S) \< (aa) 4=-> as ,< asacA a CA

for each a c A, s,t E A.

Also (a PA ( Ki(Sa )J, < ) is a complete lattice with join

and meet defined by

( aa) ^ (aa (aanaa)a s EA a tEA aEA, s,t EA

and (as) V (aa) = (aa vaa)a s EA a ,t EA a EA, s,tCA

for each ac-A.

Note.

If [S a^ be a collection of topological monoidsaEA

with Bohr compactifications[ 3a, Ba } and lattices ofaEA

semigroup compactifications {Ki(S a) I , determineda CA

by quotients of Ec for each a e A and

P( a P A E1x ) / R a E A( B a/

85

Then P A [KI(Sa)} KI a P Sa^) is a completeaC A

lattice.

4.2 Semigroup compactification of Projective Limits

In this section, we consider the projective

system of semigroup compactifications of a topological

semigroup S and show that projective limit itself is a

semigroup compactification of S.

Theorem 4.2.1.

Let f (ila ,Aa) : Oa J a Q E D be a projective

system of semigroup compactifications of a topological

semigroup S, where 71a = Oa n, for every pair a

in a directed set D . Then Jim ( rya, Aa) , itself is a

semigroup compactification of S.

Proof.

By the definition of semigroup compactification..

rya: S > Aa is a dense continuous homomorphism for

each a e- D and Aa is a compact semigroup. Moreover, each

bonding map 0a is surjective,

86

for, when a < 0

Aa = fla(S) _ 0a rl^(S)

0a(n S

0ais continuous and

closed, being a continuous map

from compact semigroup to

Hausdorff space)

Hence the system { Aa, Oa }< is a strict projective

system of compact semigroups. Then A* = `Gl im { Aa, 0 }

exists and is a compact semigroup [C-H-K1].

So it'is enough to show that there

exists a dense continuous homomorphism from S into A*.

Define,

r1:S -> P CAa }acA by rl(

x) (a) = rla(x)

for each a C D, x c S.

then ri actually maps S into A*.

for,

if x CS and a< €D

rl(x)(a) = rla(x) _Oa

rl^(x)

OO(rt(x)(P))

Then rl(x) E A*.

87

Since each r; is a continuous homomorphism, so is r).

Claim. iS) = A*

for this , we show that each non-empty basic open set

in A* contains points of r)(S). Since the system is

strict projective, the restricted map fa = Pa l: A* > A(xA*

is surjective for each a e D.

Given r1S) = Aa for every a C Da

Let U be an open set in Aa containing points of TI- a (S).

i.e., ?al ( U) contains points of f1 r)a(S)

i.e., ^l( U) contains points of r)(S) for every a e D.

each non-empty basic open set in A* contains points

of -n(S), since ff.-l(U) / all a, all open U C Aa } forms

a basis for A* [EN]

i.e., T) = A*

.'. (VI,A*) is a semigroup compactification of S.

Theorem 4.2.2.

Let [Sa, Oa ) be a projective system ofa C D

topological semigroups with projective system of semigroup

88

compactification L (r)aAa) , ea } , wherea< CD

ea 11^ = T) Oa for every pair a E D such that

S'3 =<lim Sa exists and /XIX : Pal:S* > Sa is51

surjective for each a e D, where Pa is projection.

Then <^ im (1)a, Aa) = A* is a semigroup compactification

of S*.

Proof.

Since [(11a, Aa), ea} is a projectiveCD

system of compact semigroups,

A* =l im Aa exists and is a compact semigroup [C-H-K1] .

A ey A eaY Aa

n

11 Y r)P 11 a

S O^ S0 a SY a

Define r) : P LSa J C A -> PfAa} aCA by (r)(x))(a)=11a(x(a)).

Then 11 actually maps S* into A*.

i.e., if x P [ S } is in S*, then r)(x) E, A*.a a€A

89

For,

S a 71a Aa

>n

since x E S*, when a-( p, x(a) = O (x(p))

(^1(x))(a) _ 1a(x(a)) _ Ala 0a(x(a))

= ea 71p(x(P))

= ea(i) (x)(3))

Again it:S* --> A* is a continuous

homomorphism such that the diagram

is commutative.7\a

71

for,S* A*

P {SaJ P€Aa)

'Pa T1(x) = ( rl( x))(a) a E D aE D

= T)a(x(a))

_ Ala( /a(x) )

Ta(x) ( '• ' /Aa is surjective)

This is true for all x E S*.

aT) = 'I a a

and fa = Pal:A* > Aa is surjectiveA*

90

for,

A(x = T)a(Sa) Aa is the semigroup

compactification of Sa)

^a ,a S*)( is surjective)

JPan C8*-)

also

Ja(A*) C Aa

fa TI = 1a /\a)

)a is a closed map)

U fa : A* -> Aa and

rI(S*) C A*

i.e. ^ S*) C A*

^a) C (A* )

. ' . `Pa(A*) = A. for each a E D

Claim.

TI(S*) = A*

Since rya Aa S*) = rya Sa ) = Aa for each a C D,

each non-empty basic open set in Aa contains points of

T1 a Aa (S* )

91

Let U be an open set in Aa

U contains points of r)a Aa (s*)

i.e., )Pa-1(U) contains points of Tal rya) (S*)

i.e., each non-empty basic open set in A* contains

points of f(S*)

( since 'Fl(U)/ for all a, all open

U C Ba forms a basis for A*)

• ^) S* ) = A*

(rI,A*) is a semigroup compactification of S*.

Speci;lise to Bohr compactification, we have

the following theorem.

Theorem 4.2.3.

Let € Sa, 0a } be a projective system ofa^<P E D

topological semigroups with Bohr compactifications

{(Poa' Ba )} such that S* = 1 im Sa exists andaCD

= P a I S*:S*--> Sa is surjective for each a e D, where

Pa is projection . Then J im {Ba } is a Bohr compactification

of l im {'S a } = S*

Proof.

Since (Ppa, 9a ) is a Bohr compactification of S.

and oa Oa :S > Ba is a continuous homomorphism for

92

each a . P E D, there exists a unique continuous homo-

morphism ea : B > Ba such that the diagram commutes

[1.1.23].

B

i.e.,

Pop

ea Ba

m >T

SP

9) a

Poa

> Sa

ea 0o0 - boa 0a for all a < E D and satisfies

(i) ea = 1Ba , identity function on Ba

( ii) ea o e = ea for all a ^< y

Thus we have { Ba, eat} as a projective system of compact

semigroups. Then B* _ `im Ba exist is a compact semigroup.

Define : P ^Sa} > P fBa} byU C D aeD

(3(x) (a) Poa(x(a))

Then 0 actually maps s* into B* is a dense continuous

homomorphism.

0 : S* > B* is a dense continuous homomorphism.

( Proof is same as that in theorem 402„2).

93

To complete the proof if g: S* > T is a continuous

homomorphism of S* into a compact semigroup T. We need to

exhibit a continuous homomorphism f : B* > T such that

the diagram commutes.

Define

*T)(X Sa > S* so that

o -n* (x(a)) = x(a) Baaa

0 rla( x(a)) = Oax(a)

for all k,<a e D. PoaK

`Pa B*

S* T9

*Then rla :Sa -> S* aPfS )is a continuous homomorphism,

since it is composite with Pa is a continuous homomorphism.

Then g o r1a: Sa > T is a continuous homomorphism and

since (poa, Ba) is a Bohr compactification, there exists

a unique continuous homomorphism ga: Ba > T such that

the diagram commutes for each a e D.

i.e., ga oa. g rla for each a e D .

Then define f: B* > T by f = ga pa, for each a 6 D

and is a continuous homomorphism such that fp = g

94

for,

fp(x) = ga tap(x)

= ga poa Aa(x)

gapoax(a)

= g TI*ax(a) = g(x) for all x E S*.

Also f is unique, since p(S*) is dense in B* and fp = g.

( p,B*) is a Bohr compactification of S*.

Theorem 4.2.,4.

Letf(Kl(SIA), ^ ),I}^ <kED be a projective system

of latticesof semigroup compactifications of {S/\}D with

0 ,s as lattice isomorphism. Then K ( S*) = im [KA I <_ l(SA

is a complete sub-lattice of P { Kl(SA)JAED

K1(S*) _(AY) E PED K1(SA) PA (AY) _ 0^ (Pk(A1 ))

for all) ^< k E D )

and K1(S*) 4 c. Since {C } is an isomorphism)¢.k e D

there exist (AY) E P {K (SA)} such thatA AeD 1

AA _ k (AY) k C- D.

K1(S*) is a subset of P {Kl(SA)}A ED

95

Again K1(S*) is a partially ordered set by defining an

order

A;) < (A,) E=> A, < A,A,s cD A,tED

where AT = P ((AY))

for each /QED, Y ,< t ED.

If (A' ) t (AA) C Kl (S*), then both (A,)A(AA), (A,) V (AA) EK1(S*).

for, since (A;), (A,) E Kl(S*)

When A < k c- D, AS , _ O, (Ak) s C D

t k tA; _ c&, (Ak)tC- D

Since P fKI(SA)1 is a complete latticeAED

(A,) /A (A,) and (A, ) V (A/\) E P [ Kl(SA)}A C D

and when /\,,< k

we have

^, (Pk (A>) V (AA) )

_ ^, Pk (A,)V OA Pk(A,

= PA (A,) V PA(A,)

= PA ((A,) V(A,)) for each AED and s, t E D.

96

Then (A,) V (A,) E K1(S*) for every A E D.

Similarly,

(A,, ) A (A, ) E K1(S* )/\,s ED A,tED

.°. Kl(S*) is a sublattice of P {K1(S^)}QED

Similarly we can prove that V and /A\ exist in K1(S*)

for every non-empty subset of K1(S*).

K1(S*) is a complete sublattice of P (K l(SA)}ACD