SOME RESULTS ON COMPACT CONVERGENCE SEMIGROUPS DEFINED BY FILTERS

VOL. 21 NO. (1998) 153-158

SOME RESULTS ON COMPACT CONVERGENCE SEMIGROUPS DEFINED BY FILTERS

PHOEBEHO, PAULPLUMMER,andSHINGSO CentralMissouriState University

Warrensburg,Missouri

(Received June _i0, 1996 and in revised form September 30, 1996)

ABSTRACT.

In

this paper the conceptofconvergence definedby filters isusedandapplied inthe study of semigroups. Special emphasisisplaced^oncompact convergence semigroupsand theirproperties.

KEY

WORDS AND PHRASES: convergence semigroups, convergence groups, ^maximal subgroups, compact,pseudotopological,^dense.

AMS(MOS)

SUBJ. CLASSIFICATION CODES: Primary: 54A20, 54Hll,22M99

1. INTRODUCTION.

Let /be the set ofall subsets ofa non-empty set S which contains

{x}.

^{Then is an}

ultra.filtercalled the principle

ulrafilfer

^In

[1],

^Kent’sapproach ^toconvergencewastoset up amapping qfrom

F(X),

the setofall filterson aset

X,

to

P(X),

^{the power,c* ofX. Then a}

filter

"

^on

^X

^{issaid to q} converge tox in

X,

denoted by 9^r x,ifx 6

q(’).

DEFINITION. A

convergence

space

(X, q)

^isanon-empty set

X’and

amappingq bctween

F(X)

^and

P(X)

^{whichsatisfythe}following conditions:

i) x,

forallx6X;

ii)

^{if ’-xand}

’<_G,

then

Gx;

iii)

^if ’xand

x, then’nGx.

When these properties aresatisfied q is knownas aconvergencestructure. Ifthe conver- gencestructure is fixedforaspecific discussion,asinreferencetoageneral convergencespace, wewillreferto

(X, q)

as"theconvergencespaceX".

A

convergencespace

X

isHausdorffifeach filterconverges toat mostonepointandXis compa..ct if every ultrafilterconvergesinX.

DEFINITION. Let

(X,

q) ^and

(Y, p)

be convergencespaces and f"

X --

^{Y. If}

-

^{is a}

filteron

S,

then

f(.T’)

^willdenotethe filteronYwith

f(F)

F .T" ^asitsbase. The function fissaid tobe continuous atxif

f(’)

p-convergesto

f(x)

^whenever

-

q-convergestox.

DEFINITION. Aclosure operationon aconvergencespaceS isdefined in thefollowing way. IfA ^C Sandx

S,

thenx Aifthereexistsafilter

"

^{such that}

^A "

^and

"

^x.

A

is defined to beclosedifand onlyifA A.

Thefollowingfive lemmasareimmediateresultsfrom the definitions involved.

LEMMA

^1.1. If Ais a compact subset ofX and

f: (X,

q)

(Y,

p) is a continuous function,then

f(A)

^iscompact.

LEMMA

1.2. IfXisacompactconvergence space and

T

aclosedsubset of

X,

thenTis compact.

LEMMA

1.3. If

X

isacompactspace,and2)isadescending family of non-empty closed subsetsof

X,

then N:D }.

LEMMA

1.4. If

X

is aHausdorff convergencespace and

T

acompact subset of

X,

then Tis closed.

LEMMA

1.5. IfA and

B

arecompact subsetsofa convergencespace

X,

then A

xB

is compactinthe product convergencestructureonXxX.

2. MAIN RESULTS.

DEFINITION.

A

convergence semigroupis aconvergencespace S together with^a con- tinuousfunctionm: SS S suchthat:

i)

^Sis Hausdor;

ii)

^mis associative.

Thefollowingnotations areusefulinthediscussionofconvergence semigroups:

i)

^Fora, b

S,

ab

m((a, b)).

ii)

^For

A,

B C_ S,

AB rn(AxB) {ablaAandbB}. In

particular,

A{D}

^{will be}

denoted Ab.

iii) .TxisthefilteronSxSwith{FGIFUandG}

asitsbase.

iv)

^{.T’. is the filteronS}with

rn(.T’{)

^asitsbase.

v)

^’ais the filter with

{Fa If ^Y}

^{asits base.}

LEMMA

2.1. If

"

^{and arefilterson a}convergence

semgroup

Ssuch that

.T"

x

and y,then

.T.

{ xy.

LEMMA

^2.2.

H A

and

B

arecompact subsetsofaconvergence semigroup

S,

then

AB

is compactinS.

DEFINITION.If

f: ^{X X,}

^then

{x X If(x x}

^{is calledthe set of}fixed pointsof

The next lemma showstwo properties of fixed points inconvergencespaces.

LEMMA

^2.3. Let

X

beaHausdroff convergencespaceand

F

theset of fixed points ofa continuousfunction

f: ^X

^{X. Thenthefollowingis true:}

i)

^If

F

9r afilteron

X,

then

f(-) ;

ii)

^IfXis Hausdorff,then

F

is closed.

PROOF. i)

IrA f(’),

then there existsA* ’suchthat

f(A*)

^{C_ A. ButF 9v} implies that FNA*

:fi

^{and FNA*}

^.T’.

^{Now FA* _C}

^F

^impliesthat

^F

^A*

/(FOA*)

^_C

f(A*) c_ A

which implies thatA

E..

Therefore,

f()

C_ ’.

IfAE

’,

thenANF

-

^and

^f(ANF)

^ANF.This implies thatANF

f(").

^NowANF C_

A

implies that

A f(’).

Therefore,

v

^C_

f(’).

ii)

^{Let y F. Then there exists} afilter.T such that F E

-

^and

^"

^{y. Since}

^f

^is

continuous, ^weknow

f(Y) f(y),

and from part i),

f(Y’) Y’,

so

Y"

f(y). But X being Hausdorff impliesthat

f(y)

y. Soy F. Therefore, F C_

F

soF F.

DEFINITION.

An

elementeofa semigroupS is calledan idempotent ife e. We

use

E(S)

^todenotethe set of allidempotentsinS.

DEFINITION. A nonempty subset T of a semigroup S is said to be a subsemigroup ifT.T T C_ T. A nonempty subset Gof S is a subgroup ofS if G is agroup under the multiplicationdefinedonS.

THEOREM 2.1. Ifa Hausdorff convergence semigroupS is compact thenS contains anidempotent.

PROOF.Wewill showthatScontainsaminimalclosed subsemigroup andthat every such subsemigroupconsistsofasingle idempotent. Let Sdenote thesetofclosed subsemigroups of S. Note S

S,

^{so ,5’}

# .

Partially orderS byinclusionandlet beamaximaltower inSby useofthe HausdorffMaximalityPrincipal. Let

T

^fXJ. Then,from Lemma 1.3, T

#

^{0. Let}

x T. Since

T

isanon-empty closed subsemigroupof

S,

Lemma1.2impliesthat

T

iscompact;

hence, byLemma _2.2, xTis compact. Therefore by Lemma1.4, xTis aclosed subsemigroup ofScontained inT.

Now,

by the maximalityof

C,

^weseethat

xT T

and,similarly,

Tx

T.

ThusT isasubgroupofS.Ifeistheidentityof

T,

the maximality ofCensuresthatT

{e}.

Let

f

^{be a}mapping from fromthe semigroup Sinto SxS by

f(x) (x, x)

^{for all x}

e

S, and let

"

^{beafilter on S suchthat}

"

^{x. If}

^{F :,}

^the

^{f(F) {(,x) F}.}

^Now

r,(f(F)) ^F

^for 1, 2, ^so

f(’) (x, x).

^Therefore

f

^is continuous.

Next,

consider the composite function

f

^{orn. Since}

f

^{andrn are}both continuous,weknow

f

^orn is continuousand

f

^{orn" S S by}

f

^o

re(x)

^x

^2.

Then,byLemma2.2thesetof fixed

oints

^of

f

^{o rnisclosed.}

Buttheset offixedpointsof

f

^omis

^E(S),

^thesetof idempotentsin thesemigroup. Therefore,

wehave thefollowingtheorem.

THEOREM2.2. The set

E(S),

^{of all}idempotentsofaconvergence semigroup

S,

isclosed.

THEOREM 2.3. Let Sbeaconvergence semigroupandGasubgroup ofS.ThenGisa

subsemigroupofSwith identity.

PROOF. Letx,y E G. This implies that thereexists

"

and (; such that G iscontained

inY’and,andY’x,--,y. SinceG isagroup, G.G=GsoGffY-and’---,xy.

Therefore,xy G^so C_ G.

Nowifeisthe identityofGand x

G,

then thereexists

Y

such that G

Y"

^and

-

^x.

But since 6 e,^weknow6-Y" ^ex. Alsosince

{e}

E 6 andeG

G,

^wehaveG eG Let F

’.

Then G,F ’which implies that GNF ’and since GFC_G we know

e(GNF)=GCFsoGflFE6..’.

^But Gr3FCFimpliesF6.’whichmeans

So

. _"

^{x. Now Sbeing}Hausdorff implies that ex x,similarlyxe x. Therefore,e isthe identityforG.

THEOREM 2.4. IfSis acompact convergencesemigroup andG asubgroupofS,then Gis alsoasubgroup.

PROOF. By Theorem 2.3, ^it suffices to show if xEG then x has an inverse which is equivalenttothe existence ofanx^-1 E suchthatx-ix xx

-

^{e. Lety}

^.

^{Nowif y G.}

G is agroupsoy hasaninverseandwe aredone. Ify

G,

then there exists afilter

-

^such

that G E

.T"

and

.T"

y. Foreach F C_

G,

let F^-1

{x

^-1

Ix

^E

F}

^and

^’-1

the filter with

F- F

E

Y

and

F

C_

G}

^{asitsbase. Nowthereexistsan}ultra.filter suchthat

--1 <

and, sinceSiscompact,^7-/--,h forsomeh S. NowsinceG

"

^andG-1

^G,

^G

^’-1

^{so G}

and

hE.

By Lemma 2.1, ." hy so consider

HF

.’. Since

’- ^{_< H}

^and

FfG

e ’, (FNG)

^-1 e/ which implies

HN(FNG) -

^{7-/ and}

HN(FG) -1#(3.

^Thus

there exists x

- ^E(FG) -

such that x

-IH

^and

x(FNG)

C_F so

x-x=eHF,

^so

e

HF

forall

HF

E ^?’/.

"

^whichimplies

"H..T" _< .

^{Therefore, hyandweknow}

but thefact that

S

is Hausdorff implies that hy e. Similarly, we can show yh e. Thus, h=y^-1

G.

DEFINITION. Let G be a subgroup ofa convergence semigroup S. If the inversion f" G G defined by

f(x)

^x

-

^is continuous, then G is called a convergence semigroup.

DEFINITION.

A

convergence space

X

is said to be pseudotopological if a filter convergesto somepoint x inXifand onlyif allultraaltersfinerthan

"

^convergetox.

THEOREM 2.5. IfS is acompact pseudotopological semigroup, which isalgebraically

agroup,thenSis aconvergencegroup.

PROOF. Letf" S Sdefinedby

f(x)

x

-

^and

^"

^afilter suchthat

"

^{x. Then}

^f(

"

^{is afilterinS. Let?’fbea}ultrafiltersuchthatT/>

f( " ^).

^{SinceSis compact,?-/y for}

somey inS.Since for everyH ?’/, HN

f(

^9r

-#-

^{} for all}

F .

^Thene

^HF

^forall

^H

^E

and

F

whereeisthe identity of the groupS.

Now -

^{T/ y,}

^"

^?’/<

^,

^{and eimply}

that xy e.

Similarly, yx ^e. Thus y z

-

^{Since S is} pseudotopological,

f( "

^z

-

^{so fis}

continuous. Therefore, Sisaconvergencegroup.

DEFINITION. Asubset

D

ofaconvergencespace

X

is said to be densein

X

ifD X.

DEFINITION. IfS is a convergence semigroupand e

E(S),

^then the largest proper subgroupofSwhich containseiscalled themaximalsubgroupofScontaininge.

THEOREM2.6.

i)

^IfSisacompact convergencesemigroup, theneachmaximalsubgroup iseitherclosedordense inS.

ii)

^{IfSis a}compact pseudotopological semigroup,then either every maximalsubgroupofS isclosedorSis aconvergence group.

iii)

^{If S is a} compact pseudotopological semigroup, thenevery maximal subgroup of S is a

convergencegroup.

PROOF.

i)

^By Theorem2.4,ifGis asubgroupthenso is

. SinceGCCS,isa

group, andGismaximal,eitherG GandG isclosed,orG S andGis dense.

ii)

^{IfSdoes havea}dense subgroup,then G SimpliesSisagroup. SinceSiscompact and pseudotopological,Sis aconvergencesemigroup byTheorem 2.5.

iii)

^LetGbeamaximalsubgroupofS. ThenG iseitherclosedordenseinS.

IfG is closed,then G satisfiesall the hypothesis of Theorem 2.5. so G is a convergence group.

IfGisdense inS,thenby

ii)

^Sisaconvergencegroup. Since the inversion mappingonGis arestrictionoftheinversionmappingon

S,

itmust be continuous. ThereforeGisaconvergence group.

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1.

KENT, D.C., Convergence

FunctionsandRelated Topologies, Fundamenta Mathematicae 54

(1964),

125-133.

2.

WILLARD, S.,

GeneralTopology,Addison-Wesley, Reading,Massachusetts, 1970.

3. PAALMAN-DE

MIRANDA, A.B.,

Topological Semigroups, Mathematical Centre Tracts, 2ndEd., Mathematiche

Centrum,

Amsterdam,1970.

4.

CARRUTH, J.H., HILBEBRANT, J.A.,

and

KOCH, R.J.,

TheTheoryofTopologicalSemi- groups, MarcelDekke,

Inc.,

New York, 1983.

5.

HUSAIN, T.,

Introduction toTopological

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W. B. SaundersCompany,Philadelphia andLondon,1966.

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PLUMMER, P.J., Convergence

Semigroupsin

Convergence

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