On the semigroup of square matrices

arXiv:math/0510624v2[math.GR]15Dec2006OnthesemigroupofsquarematricesGannaKudryavtsevaandVolodymyrMazorchukAbstractWestudythestructureofnilpotentsubsemigroupsinthesemigroupM(n,F)ofalln×nmatricesoverafield,F,withrespecttotheoperationoftheusualmatrixmultiplication.Wedescribethemaximalsubsemigroupsamongthenilpotentsubsemigroupsofafixednilpotencydegreeandclassifythemuptoisomorphism.WealsodescribeisolatedandcompletelyisolatedsubsemigroupsandconjugatedelementsinM(n,F).1IntroductionThestructureandcombinatoricsoftheclassicalfinitetransformationsemigroupsisnowrelativelywellunderstood.AlotofinformationaboutsuchsemigroupsasthefullfiniteinversesymmetricsemigroupISn,thefulltransformationsemigroupTnandthesemi-groupPTnofallpartialtransformationson{1,2,...,n}canbefoundforexampleinthemonographs[CP,Hi,Ho,La,Li]orothernumerouspapersstudyingtransformationsemigroups.Twodirectionsofstudyfortransformationsemigroups,whichdevelopedoverthelast15years,isthestudyofconjugatedelementsandnilpotentsubsemigroupsinthesesemigroups,whichresultedintoseveralnicestructuralandcombinatorialresults,see[GK1,GK2,GK3,GK4,GTS,GM,KM].Certainly,thepassagetoinfinitetransformationsemigroupscompletelychangesthepic-ture.However,itisstillpossibletoobtainsomeinformationunderreasonable“finiteness”conditions.Oneofthemostclassicalexamplesofaninfiniteobjectpossessingseveralprop-erties,inherentinfiniteobjects,isthealgebraofalllinearoperatorsonafinite-dimensionalvectorspaceoverafield.Forgettingtheadditionoflinearoperatorsonegetsasemigroup,isomorphictothesemigroupM(n,F)ofalln×nsquarematriceswithcoefficientsfromafieldFwithrespecttotheoperationofusualmatrixmultiplication.Thissemigrouphasalsobeenstudiedbymanyauthors,butsofarnotasintensivelyastheclassicaltransfor-mationsemigroups.ManyinterestingresultsaboutM(n,F)canbefoundintherecentmonograph[Ok2]andinitsreferences.TheaimofthepresentpaperistocontributetothestudyofM(n,F)withresultsinseveraldirections.ThemainemphasisismadeonthestudyofconjugatedelementsandnilpotentsubsemigroupsofM(n,F).Intheappendixwealsoaddresstheproblemofthestudyofisolatedandcompletelyisolatedsubsemigroups.Thesedifferentdirectionsare1notimmediatelyrelatedwitheachother,however,whencombinedtogether,theygiveaveryniceillustrationforthefactthatinthepassagefromtheclassicaltransformationsemigroupstomatrixsemigroupsoneshouldexpectthatsomeresultswouldbetransferedtoverysimilarresults,whilesomeotherresultswouldsoundquitedifferently.Andeveninthecaseofsimilarresults,thepassagetomatrixsemigroupssubstantiallyincreasestheleveloftechnicaldifficulties.Thenotionofconjugatedelementsingrouptheoryisveryimportantandhasalotofapplications(forexampleforthestudyofautomorphisms,charactersorrepresentations).Thereareseveralwaystoextendthisnotionforsemigroups.Twomoststraightforwardgeneralizationsare:conjugationwithrespecttoaninvertibleelement,andthetransitiveclosureoftheab∼barelation.Boththesenotionsprovidesomeinvariantonsemigroupsandhencecanbeappliedforthestudyofautomorphisms,endomorphismsandrepresen-tation.Thealreadyexistingliterature,wherethesenotionswerestudied(seeforexample[GK1,Li,KM])suggeststhatthesecondgeneralizationismoreinterestingthanthefirstone.ForM(n,F)thedescriptionofequivalenceclasseswithrespecttotheconjugationbyinvertiblematricesisaclassicalproblem,theanswertowhichisgivenbytheJordannormalformofamatrix(inthecaseofanalgebraicallyclosedfield).InSection2ofthepresentpaperwedescribetheequivalenceclasseswithrespecttothetransitiveclosureoftheab∼barelation.Itturnsoutthatthesearegivenbythe“invertiblepart”oftheJordannormalform.Thisisverysimilartotheresultsobtainedin[GK1,KM]fortransformationsemigroups.TheresultsofSection2ofthepresentpaperandtheresultsof[GK1,KM]werethemainmotivationfortheabstractapproachtothestudyofconjugationforsemigroups,developedin[Ku].ThemajorityofthepaperisdevotedtothestudyofnilpotentsubsemigroupsofM(n,F).Tostartwith,wewouldliketoremarkthatthenotionofanilpotentsemi-grouphasbeenusedintheliteratureinatleastthreedifferentsenses.ThemostclassicaloneisthenotionofnilpotentsemigroupinthesenseofMaltsev,[Ma],whichisdefinedbymeansoftheidentitiesfornilpotentgroups,rewrittenwithoutg−1terms.Thematrixsemigroups,nilpotentinthesenseofMaltsev,wererecentlystudiedin[Ok1].However,inthispaperwearegoingtouseanothernotionofanilpotentsemigroup,whichcomesfromtheringtheory.Asemigroup,S,withthezeroelement0iscallednilpotentofnilpotencydegreend(S)=kprovidedthata1a2...ak=0foranya1,...,ak∈Swhilethereexistb1,...,bk−1∈Ssuchthatb1...bk−16=0.Thisnotionisalmostasoldasthefirstoneandgoesbackatleasttill[Shev].Fromnowonwewilluseonlythelastnotionofnilpotentsemigroups.Thestudyofnilpotentsubsemigroupsofcertainsemigroupsofpartialtransformations,inparticular,ofthesemigroupISn,wasoriginatedin[GK4].Ithappenedthatthecom-binatorialdata,describingthemaximalnilpotentsubsemigroupsoffinitetransformationsemigroups,isusuallyacertainpartialorderontheunderlinedset,onwhichthesemigroupacts.Thisphilosophywassuccessfullyusedlaterin[GK2,GK3]andgeneralizedontheinfinitecasein[Sh2,Sh1].In[GTS]thecombinatorialdescriptionofmaximalnilpotentsubsemigroups