The structure of two generalization of the symmetr

arXiv:math/0602623v2[math.GR]20Aug2008TwogeneralisationsofthesymmetricinversesemigroupsGannaKudryavtsevaandVictorMaltcevAbstractWeintroducetwogeneralisationsofthefullsymmetricinversesemi-groupIXanditsdualsemigroupI∗X–inversesemigroupsPI∗XandPI∗X.BothofthemhavethesamecarrierandcontainIX.BinaryoperationsonPI∗XandPI∗XarereminiscentofthemultiplicationinIX.Weuseaconvenientgeometricwaytorealiseelementsfromthesetwosemigroups.ThisenablesustostudyefficientlytheirinnerpropertiesandtocomparethemwiththecorrespondingpropertiesofIXandI∗X.2000MathematicsSubjectClassification:20M10,20M20.1IntroductionOneofthemostnaturalexamplesofproperinversesemigroups(i.e.,exceptgroups)isthesymmetricinversesemigroupIX.Besidepurecombinatorialinterestinthissemigroup,itplaysanimportantrolefortheclassofallinversesemigroupssimilartothatplayedbythesymmetricgroupSXfortheclassofallgroups.ForsomefactsaboutsemigroupandcombinatorialpropertiesofIXwereferthereaderto[5].Seekingforfurthernaturalexamplesofinversesemigroups,FitzGeraldandLeech[4],usingcategoricalmethods,introducedthedualsymmetricinversesemigroupI∗X.Usingmoregeneralcategoricalapproach,I∗Xalsoappearedin[10].Thissemigroupalsohasausefulgeometricrealisation,whichwasex-ploitedin[3,12]tostudysomeinnerpropertiesofI∗X.Inarecentwork[9]therewasfoundanew,representationtheoretic,linkbetweenIXandI∗X.Inaddition,bothI∗XandIXbelongtotheclassoftheso-calledpartitionsemigroups[14,19]andarecontainedinthe“biggestpartitionsemigroup”CX(seeSection2fordetails).Thelattersemigroupwasstudiedmainlyinthecontextofrepresentationtheoryandcellularalgebras[6,8,13,19].SomepuresemigroupaspectsofCXwerestudiedin[6,14].InthepresentpaperweaimatconstructingtwoinversesemigroupsPI∗XandPI∗X,whicharestronglyrelatedtoIXandI∗X,thoughhavemuchmorecomplicatedstructure.Wegivetransparentgeometricdefinitionsforthesetwosemigroupsandthenstudytheirinnerproperties,focusingoncombinatorialaspectsandtheirresemblancetoIXandI∗X.ThesemigroupsPI∗XandPI∗Xarenaturalalsofromtherepresentationtheoreticpointofview:PI∗Xiscontainedinabiggersemigroup,the“defor-mation”ofCX,whosesemigroupalgebranaturallyarisesintherepresentation1theory,see,e.g.,[6].Someotherrepresentationtheoreticaspects,wherePI∗XandPI∗Xappearednaturally,canbefoundin[9].WenotethatbothsemigroupsPI∗XandPI∗Xadmitrealisationsassemi-groupsofdifunctionalbinaryrelations.Thesespecialrelationshavebeenstud-iedinaseriesofworks[1,2,16,17].UsingthisrealisationPI∗Xhasalreadyappearedin[18].2DefinitionsThroughoutthepaperforasetXwewilldenoteby′abijectionfromXontoitselfsuchthat(x′)′=xforeveryx∈X.2.1CXandI∗XFirstwedefineCX.ThecarrierofCXisthesetofallpartitionsofX∪X′intononemptysubsets.Werealisethesepartitionsasdiagramswithtwostrandsofvertices,topverticesindexedbyXandbottomverticesindexedbyX′.Forα∈CXtwoverticesofthecorrespondingdiagrambelongtothesame“connectedcomponent”ifandonlyiftheybelongtothesamesetofthepartitionα(noticethattheremaybemanydifferentwaysofpresentinganelementα∈CXasadiagram,wetreattwodiagramscorrespondingtothesameαasequal).Themultiplicationisdefinedasfollows:givenα,β∈CXweidentifythebottomverticesofαwiththecorrespondingtopverticesofβ,whichuniquelydefinestheconnectionoftheremainingvertices(whicharethetopverticesofαandthebottomverticesofβ).Wesetthediagramobtainedinthiswaytobetheproductαβ.Theformaldefinitionoftheproductαβisasfollows:Letα,β∈CX,and≡αand≡βbethecorrespondentequivalencerelationsonX∪X′.Thentherelation≡αβisdefinedby:•Fori,j∈Xwehavei≡αβjifandonlyifi≡αjorthereexistsasequences1,...,sm,meven,suchthati≡αs′1,s1≡βs2,s′2≡αs′3,andsoon,sm−1≡βsm,s′m≡αj.•Fori,j∈Xwehavei′≡αβj′ifandonlyifi′≡βj′orthereexistsasequences1,...,sm,meven,suchthati′≡βs1,s′1≡αs′2,s2≡βs3,andsoon,s′m−1≡αs′m,sm≡βj′.•Fori,j∈Xwehavei≡αβj′ifandonlyifthereexistsasequences1,...,sm,modd,suchthati≡αs′1,s1≡βs2,s′2≡αs′3,andsoon,s′m−1≡αs′m,sm≡βj′.Wewillcallthismultiplicationofpartitionsthenaturalmultiplication.AnexampleofmultiplicationofelementsfromC8isgivenonFigure1.Aone-elementsubsetofX∪X′willbecalledapoint,andasubsetAintersectingwithbothXandX′—ageneralisedline.AgeneralisedlineAwillbecalledalineif|A|=2.ByI∗XwedenotethesubsemigroupofCXwhoseelementscontainonlygeneralisedlines.OnFigure2wegiveanexampleofmultiplicationoftheelementsofI∗8.2••~~~••UUUUUUUUUUU••••••••×••••@@@•••OOOOOOO•••••••••••••••~~~•OOOOOOO•=••••••••••••@@@Figure1:ElementsofC8andtheirmultiplication.••~~~•~~~••@@@•~~~•@@@•~~~••••×•••••@@@••@@@•iiiiiiiiiii~~~••~~~•~~~@@@•~~~••••••••••••=••••ooooooo••••••••Figure2:ElementsofI∗8andtheirmultiplication.2.2PI∗XLetPI∗XbethesetofallpartitionsofthesetX∪X′intosubsetsbeingei-therpointsorgeneralisedlines.ThesetPI∗XisnotclosedunderthenaturalmultiplicationofCXastheexampleonFigure3shows.However,wecandefineanassociativemultiplicationonPI∗Xasfollows.Letx/∈X.Foreveryα∈PI∗Xsetα∈I∗X∪{x}tobetheelementsuchthatitsblocksaretheblocksofαplusonemoreblockconsistingofx,x′andallpointsofα.Denotebyϕtheinjection,whichmapsα∈PI∗Xtoα∈I∗X∪{x}.Observethatγ∈I∗X∪{x}belongstotheimageofϕifandonlyifx≡γx′.ThisenablesustodefineanassociativemultiplicationonPI∗Xasfollows:α⋆β=ϕ−1(αβ).Intermsofthe