AMALGAMS OF FREE INVERSE SEMIGROUPS

AMALGAMSOFFREEINVERSESEMIGROUPSAlessandraCherubini,JohnMeakin,BrunettoPiochiMarch12,1996AbstractWestudyinversesemigroupamalgamsoftheformSUTwhereSandTarefreeinversesemigroupsandUisanarbitrarynitelygeneratedinversesubsemigroupofSandT.WemakeuseofrecentworkofBennetttoshowthatthewordproblemisdecidableforanysuchamalgam.Thisisincontrasttothegeneralsituationforsemigroupamalgams,whererecentworkofBirget,MargolisandMeakinshowsthatthewordproblemforasemigroupamalgamSUTisingeneralundecidable,evenifSandThavedecidablewordproblem,Uisafreesemigroup,andthemembershipproblemforUinSandTisdecidable.Wealsoobtainanumberofresultsconcerningthestructureofsuchamalgams.WeobtainconditionsfortheD-classesofsuchanamalgamtobeniteandweshowthattheamalgamiscombinatorialinsuchacase.Forexampleeveryone-relatoramalgamofthistypehasniteD-classesandiscombinatorial.WealsoobtaininformationconcerningwhensuchanamalgamisE-unitary:forexampleeveryonerelatoramalgamoftheformInvA[B:u=vwhereAandBaredisjointandu(resp.v)isacyclicallyreducedwordoverA[A1(resp.B[B1)isE-unitary.1IntroductionIfS1andS2aresemigroupssuchthatS1\S2=Uisanon-emptysubsemigroupofbothS1andS2,then[S1;S2;U]iscalledanamalgamofsemigroupsandUisthecoreoftheamalgam.Theamalgam[S1;S2;U]issaidtobestronglyembeddableinasemigroupifthereexistsasemigroupSandinjectivehomomorphismsi:Si!Ssuchthat1jU=2jUResearchofallauthorsissupportedbyagrantfromtheItalianCNR.Therstandthirdauthors’researchwaspartiallysupportedbyMURST.Thesecondauthor’sresearchwasalsopartiallysupportedbyNSFandtheCenterforCommunicationandInformationScienceoftheUniversityofNebraskaatLincoln.1andS11\S22=U1=U2:AsemigroupSisaregularsemigroupifforeacha2Sthereexistsa02Ssuchthata=aa0aanda0=a0aa0:suchanelementa0iscalledaninverseofa.IfeachelementofShasauniqueinverse,Siscalledaninversesemigroup:equivalently,aninversesemigroupisaregularsemigroupwhoseidempotentscommute.Suchsemigroupsmaybefaithfullyrepresentedassemigroupsofpartialone-onemapsonasetX.WereferthereadertoPetrich[18]forthisresultandmanyotherstandardresultsandideasaboutinversesemigroups.Inparticular,wedenotebya1theuniqueinverseoftheelementainaninversesemigroupS.Itiswellknownthatasemigroupamalgam[S1;S2;U]isnotnecessarily(strongly)embeddable.HoweveritwasshownbyHowiein1962[7]thattheamalgamisstronglyembeddableifUisunitaryinS1andS2.Thereisalargeliteraturedevotedtothequestionofwhenasemigroupamalgamis(strongly)embeddable.WerefertothepaperofHowie[8]forsomereferencestotheliteratureonthisproblemandforanintroductiontotheconnectionbetweenthisproblemandthenotionoftensorproductofsemigroupactions.Inparticular,averyimportanttheoremofT.E.Hall[6]showsthatthecategoryofinversesemigroupshasthestrongamalgamationproperty.Thatis,everyamalgamofinversesemigroupsmaybestronglyembeddedinaninversesemigroup.Itfollowsthatsuchanamalgammaybestronglyembeddedinthefreeproductwithamalgamationinthecategoryofinversesemigroups.WeshallbeconcernedexclusivelywithinversesemigroupsinthispaperandwilldenotebyS1US2thefreeproductofS1andS2amalgamatingtheinversesubsemigroupUinthecategoryofinversesemigroups.Thisobjectisdenedbytheusualuniversaldiagraminthecategoryofinversesemigroups.Whilethereisalargeliteratureconcernedwiththeembeddabilityquestionforsemi-groupamalgams,thereappearstohavebeenuntilveryrecentlylittleworkconcernedwiththestructureofsemigroupamalgamsorevenwiththewordproblemforsuchsemigroups.Duringthepastyearseveralimportantdevelopmentsalongtheselineshaveoccurred.In[5]Haataja,MargolisandMeakinshowedhowtheideasoftheBass-SerretheoryofgroupsactingongraphsmaybeusedtostudythestructureofthemaximalsubgroupsofaninversesemigroupamalgamS1US2inthecasewhenUcontainsalloftheidempotentsofbothS1andS2.In[4]Birget,MargolisandMeakinshowedthatthewordproblemforasemigroupamalgamS1US2isingeneralundecidableevenifS1andS2havedecidablewordproblem,themembershipproblemforUinSiisdecidableandUisafreesemigroupwhichisunitaryineachSi.Thusthesituationisverydierentfromthesituationforgroupamalgams,wherethewordproblemistriviallydecidableiftheambientgroupshavedecidablewordproblemandthesubgrouphasdecidablemembershipproblemineachgroup.Recently,Sapir[19]hasprovedthattheembeddabilityproblemisingeneralundecidableforsemigroupamalgams,evenifallsemigroupsarenite.InhisPh.Dthesis,Bennett[1]hasintroducedaclassofinversesemigroupamalgamsandhasdevelopedanalgorithmforconstructingthecorrespondingSchutzenbergergraphs.Weshallmakeuseofhisideasthroughoutthispaperandwillprovideabriefoutlineofpartsofhisworkinthenextsection.AnevenmorerecentpaperbyStephen[21]providesadditionalinforma-tionabouttheSchutzenbergergraphsofinversesemigroupamalgamsandenableshimto2obtainanalternativeproofofHall’sembeddabilitytheoremandconsiderablestructuralinformation.OurconcerninthepresentpaperiswithinversesemigroupsoftheformS=FIS(A)UFIS(B)whereAandBaredisjointnon-emptysetsandUisanitelygeneratedinversesubsemigroupofFIS(A)andFIS(B).HereFIS(A)denotesthefreeinversesemigrouponA.WereferthereadertothebookofPetrich[18]formuchinformationaboutfreeinversesemigroups-i.e.freeobjectsinthecategoryofinversesemigroups.