Tight representations of semilattices and inverse semigroups

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1、TIGHTREPRESENTATIONSOFSEMILATTICESANDINVERSESEMIGROUPSR.Exel*ByaBooleaninversesemigroupwemeananinversesemigroupwhosesemilatticeofidempotentsisaBooleanalgebra.WestudyrepresentationsofagiveninversesemigroupSinaBooleaninversesemigroupwhicharetightinacertainwelldefinedtechnicalsense.Thesere

2、presentationsaresupposedtopreserveasmuchaspossibleanytraceofBooleannespresentinthesemilatticeofidempotentsofS.Afterob-servingthattheVagner–Prestonrepresentationisnottight,weexhibitacanonicaltightrepresentationforanyinversesemigroupwithzero,calledtheregularrepresen-tation.Wethentackleth

3、equestionastowhetherthisrepresentationisfaithful,butitturnsoutthattheanswerisoftennegative.Thelackoffaithfulnessishowevercompletelyunderstoodaslongaswerestricttocontinuousinversesemigroups,aclassgeneralizingtheE∗-unitaries.1.Introduction.WeshallsaythataninversesemigroupSisaBooleaninver

4、sesemigroup,ifE(S),thesemilatticeofidempotentsofS,admitsthestructureofaBooleanalgebrawhoseordercoincideswiththeusualorderonE(S).Booleaninversesemigroupsarequitecommon,awellknownexamplebeingthesemi-groupI(X)ofallpartiallydefinedbijectionsonX.ThesemilatticeofidempotentsofI(X)coincideswith

5、theBooleanalgebraP(X)ofallsubsetsofX,thisbeingthereasonwhyI(X)isaindeedaBooleaninversesemigroup.GivenaninversesemigroupSonemightliketostudyhowfaritisfrombeingaBooleaninversesemigroupbyconsideringhomomorphismsσ:S→B,intosomeBooleaninversesemigroupB.Simplyrequiringσtobeasemigrouphomomor-a

6、rXiv:math/0703401v1[math.RT]14Mar2007phismcompletelysidestepstheissuesince,incaseSitselfhappenstobeaBooleaninversesemigroup,ameresemigrouphomomorphismhasnoreasontorespecttheBooleanalgebrastructuresinvolved.Todealwiththissituationweproposetoconsideraspecialclassofhomomorphismscalledtigh

7、trepresentations(seeDefinition(6.1)),whichappliestoeveryinversesemigroupwithzero.IncaseSisaBooleaninversesemigroupweproveinProposition(6.2)thattightrepresentationsarepreciselythosewhichrestricttoahomomorphismσ:E(S)→E(B)inthecategoryofBooleanalgebras.2000MathematicsSubjectClassification