Theme and variations on the concatenation product - PDF document

ThemeandvariationsontheconcatenationproductJean-EricPin1?LIAFA,UniversityParis-DiderotandCNRS,France.Abstract.Theconcatenationproductisoneofthemostimportantop-erationsonregularlanguages.Itsstudyrequiressophisticatedtoolsfromalgebra,nitemodeltheoryandpronitetopology.Thispapersurveysresearchadvancesonthistopicoverthelastftyyears.Theconcatenationproductplaysakeyroleintwoofthemostimportantresultsofautomatatheory:Kleene'stheoremonregularlanguages[23]andSchutzen-berger'stheoremonstar-freelanguages[60].Thissurveyarticlesurveysthemostimportantresultsandtoolsrelatedtotheconcatenationproduct,includingconnectionswithalgebra,pronitetopologyandnitemodeltheory.Thepaperisorganisedasfollows:Section1presentssomeusefulalgebraictoolsforthestudyoftheconcatenationproduct.Section2introducesthemaindenitionsontheproductanditsvariants.TheclassicalresultsaresummarizedinSection3.Sections4and5aredevotedtothestudyoftwoalgebraictools:Schutzenbergerproductsandrelationalmorphisms.ClosurepropertiesformthetopicofSection6.HierarchiesandtheirconnectionwithnitemodeltheoryarepresentedinSections7and8.Finally,newdirectionsaresuggestedinSection9.1TheinstrumentsThissectionisabriefreminderonthealgebraicnotionsneededtostudythecon-catenationproduct:semigroupsandsemirings,syntacticorderedmonoids,freepronitemonoids,equationsandidentities,varietiesandrelationalmorphisms.Moreinformationcanbefoundin[1{3,18,35,42,45].1.1SemigroupsandsemiringsIfSisasemigroup,thesetP(S)ofsubsetsofSisalsoasemiring,withunionasadditionandmultiplicationdened,foreveryX;Y2P(S),byXY=fxyjx2X;y2YgInthissemiring,theemptysetisthezeroandforthisreason,isdenotedby0.Itisalsoconvenienttodenotesimplybyxasingletonfxg.Ifkisasemiring,wedenotebyMn(k)bethesemiringofsquarematricesofsizenwithentriesink. ?TheauthoracknowledgessupportfromtheprojectANR2010BLAN020202FREC. 1.2SyntacticorderedmonoidLetLbealanguageofA.ThesyntacticpreorderofListherelation6LdenedonAbyu6Lvifandonlyif,foreveryx;y2A,xvy2L)xuy2LThesyntacticcongruenceofListherelationLdenedbyuLvifandonlyifu6Lvandv6Lu.ThesyntacticmonoidofListhequotientM(L)ofAbyLandthenaturalmorphism:A!A=LiscalledthesyntacticmorphismofL.Thesyntacticpreorder6LinducesanorderonthequotientmonoidM(L).TheresultingorderedmonoidiscalledthesyntacticorderedmonoidofL.Thesyntacticorderedmonoidcanbecomputedfromtheminimalautomatonasfollows.FirstobservethatifA=(Q;A;
;q;F)isaminimaldeterministicautomaton,therelation6denedonQbyp6qifforallu2A,q
u2F)p
u2Fisanorderrelation,calledthesyntacticorderoftheautomaton.Thenthesyntac-ticorderedmonoidofalanguageisthetransitionmonoidofitsorderedminimalautomaton.Theorderisdenedbyu6vifandonlyif,forallq2Q,q
u6q
v.Forinstance,letLbethelanguagefa;abag.Itsminimaldeterministicau-tomatonisrepresentedbelow: 1 2 3 4 0 a b a a;b b a b a;bTheorderonthesetofstatesis264,163and1;2;3;460.Indeed,onehas0
u=0forallu2Aandthus,theformalimplication0
u2F)q
u2Fholdsforanystateq.Similarly,163sinceaistheonlywordsuchthat3
a2Fandonealsohas1
a2F.ThesyntacticmonoidofListhemonoidM=f1;a;b;ab;ba;aba;0gpresentedbytherelationsa2=b2=bab=0.Itssyntacticorderis1ab0,1ba0,aaba0,b0.2 1.3FreepronitemonoidsWebrie\ryrecallthedenitionofafreepronitemonoid.Moredetailscanbefoundin[1,45].AnitemonoidMseparatestwowordsuandvofAifthereisamorphism':A!Msuchthat'(u)='(v).Wesetr(u;v)=minjMjjMisanitemonoidthatseparatesuandvgandd(u;v)=2r(u;v),withtheusualconventionsmin;=+1and21=0.ThendisametriconAandthecompletionofAforthismetricisdenotedbycA.TheproductonAcanbeextendedbycontinuitytocA.Thisextendedprod-uctmakescAacompacttopologicalmonoid,calledthefreepronitemonoid.Itselementsarecalledpronitewords.Inacompactmonoid,thesmallestclosedsubsemigroupcontainingagivenelementshasauniqueidempotent,denoteds!.Thisistrueinparticularinanitemonoidandinthefreepronitemonoid.Onecanshowthateverymorphism'fromAintoa(discrete)nitemonoidMextendsuniquelytoaauniformlycontinuousmorphismb'fromcAtoM.Itfollowsthatifxisaproniteword,thenb'(x!)=b'(x)!.1.4EquationsandidentitiesLet'beamorphismfromAintoanite[ordered]monoidMandletx;ybetwopronitewordsofcA.Wesaythat'satisestheproniteequationx=y[x6y]ifb'(x)=b'(y)[b'(x)6b'(y)].AregularlanguageofAsatisesaproniteequationifitssyntacticmor-phismsatisesthisequation.Moregenerally,wesaythatasetofregularlan-guagesLisdenedasetofproniteequationsEifListhesetofallregularlanguagessatisfyingeveryequationofE.AlatticeoflanguagesisasetLoflanguagesofAcontaining;andAandclosedunderniteunionandniteintersection.Itisclosedunderquotientsif,foreachL2Landu2A,thelanguagesu1LandLu1arealsoinL.Itisprovedin[19]thatasetofregularlanguagesisalattice[Booleanalgebra]closedunderquotientifandonlyifitcanbedenedbyasetofproniteequationsoftheformu6v[u=v].Anite[ordered]monoidMsatisestheidentityx=y[x6y]ifeverymorphismfromAintoMsatisesthisequation.ThesenotionscanbeextendedtosemigroupsbyconsideringmorphismsfromthefreesemigroupA+toanitesemigroup.1.5VarietiesofmonoidsInthispaper,wewillonlyconsidervarietiesinEilenberg'ssense.Thus,forus,avarietyofsemigroupsisaclassofnitesemigroupsclosedundertakingsubsemigroups,quotientsandnitedirectproducts[18].Varietiesoforderedsemigroups,monoidsandorderedmonoidsaredenedanalogously[39].3 GivenasetEofidentities,wedenotebyJEKtheclassofallnite[ordered]monoidswhichsatisfyalltheidentitiesofE.Reiterman'stheorem[57]anditsextensiontoorderedstructures[53]statesthateveryvarietyof[ordered]monoids[semigroups]canbedenedbyasetofidentities.Forinstance,thevarietyoforderedsemigroupsJx!yx!6x!KisthevarietyoforderedsemigroupsSsuchthat,foreachidempotente2Sandforeachs2S,ese6e.Thefollowingvarietieswillbeusedinthispaper:thevarietyAofaperiodicmonoids,denedbytheidentityx!+1=x!,thevarietyR[L]ofR-trivial[L-trivial]monoids,denedbytheidentity(xy)!x=x![y(xy)!=x!]andthevarietyDA,whichconsistsoftheaperiodicmonoidswhoseregularJ-classesareidempotentsemigroups.Thisvarietyisdenedbytheidentitiesx!=x!+1and(xy)!(yx)!(xy)!=(xy)!.Wewillalsoconsidertwogroupvarieties:thevarietyGpofp-groups(foraprimep)andthevarietyGsolofsolublegroups.Finally,ifVisavarietyofmonoids,theclassofallsemigroupsSsuchthat,foreachidempotente2S,the\local"monoideSebelongstoV,formavarietyofsemigroups,denotedLV.Inparticular,thevarietyLIisthevarietyoflocallytrivialsemigroups,denedbytheidentityx!yx!=x!.1.6VarietiesoflanguagesAclassoflanguagesCassociateswitheachalphabetAasetC(A)ofregularlanguagesofA.ApositivevarietyoflanguagesisaclassoflanguagesVsuchthat,forallalphabetsAandB,(1)V(A)isalatticeoflanguagesclosedunderquotients,(2)if':A!Bisamorphism,thenL2V(B)implies'1(L)2V(A).AvarietyoflanguagesisapositivevarietyVsuchthat,foreachalphabetA,V(A)isclosedundercomplement.WecannowstateEilenberg'svarietytheorem[18]anditscounterpartfororderedmonoids[39].Theorem1.1.LetVbeavarietyofmonoids.ForeachalphabetA,letV(A)bethesetofalllanguagesofAwhosesyntacticmonoidisinV.ThenVisavarietyoflanguages.Further,thecorrespondenceV!Visabijectionbetweenvarietiesofmonoidsandvarietiesoflanguages.Theorem1.2.LetVbeavarietyoforderedmonoids.ForeachalphabetA,letV(A)bethesetofalllanguagesofAwhosesyntacticorderedmonoidisinV.ThenVisapositivevarietyoflanguages.Further,thecorrespondenceV!Visabijectionbetweenvarietiesoforderedmonoidsandpositivevarietiesoflanguages.AslightlymoregeneraldenitionwasintroducedbyStraubing[71].LetCbeaclassofmorphismsbetweenfreemonoids,closedundercompositionandcontainingalllength-preservingmorphisms.Examplesincludetheclassesofalllength-preservingmorphisms,ofalllength-multiplyingmorphisms(morphismssuchthat,forsomeintegerk,theimageofanyletterisawordoflengthk),4 allnon-erasingmorphisms(morphismsforwhichtheimageofeachletterisanonemptyword),alllength-decreasingmorphisms(morphismsforwhichtheimageofeachletteriseitheraletterortheemptyword)andallmorphisms.ApositiveC-varietyoflanguagesisaclassVofrecognisablelanguagessat-isfyingtherstconditiondeningapositivevarietyoflanguagesandasecondcondition(20)if':A!BisamorphisminC,L2V(B)implies'1(L)2V(A).AC-varietyoflanguagesisapositiveC-varietyoflanguagesclosedundercom-plement.WhenCistheclassofnon-erasingmorphisms(forwhichtheimageofaletterisanonemptyword),weusethetermne-variety.Thesene-varietiesareessentiallythesamethingasEilenberg's+-varieties(see[49,p.260{261]foradetaileddiscussion)andtheycorrespondtovarietiesofsemigroups.1.7RelationalmorphismsArelationalmorphismbetweentwomonoidsMandNisafunctionfromMintoP(N)suchthat:(1)forallM2M,(m)=;,(2)12(1),(3)forallm;n2M,(m)(n)(mn)LetVbeavarietyof[ordered]semigroups.A[relational]morphism:M!Nissaidtobea[relational]V-morphismifforevery[ordered]semigroupRofNbelongingtoV,the[ordered]semigroup1(R)alsobelongstoV.Letmepointoutanimportantsubtlety.Thedenitionofa[relational]V-morphismadoptedinthispaperistakenfrom[44]anddiersfromtheoriginaldenitiongivenforinstancein[68,42].Theoriginaldenitiononlyrequiresthat,foreachidempotente,the[ordered]semigroup1(e)alsobelongstoV.Inmanycasesthetwodenitionsareequivalent:forinstance,whenVisoneofthevarietiesA,Jx!yx!=x!K,Jx!y=x!K,Jyx!=x!KorLHwhereHisavarietyofgroups.However,thetwodenitionsarenotequivalentforthevarietyJx!yx!6x!K.2Themeandvariations:theconcatenationproductWenowcometothemaintopicofthisarticle.Justlikeapieceofclassicalmusic,theconcatenationproductincludesthemeandvariations.2.1MainthemeTheproduct(orconcatenationproduct)ofthelanguagesL0;L1;:::;LnofAisthelanguageL0L1


Ln=fu0u1


unju02L0;u12L1;


;un2Lng5 AlanguageLofAisamarkedproductofthelanguagesL0;L1;:::;LnifL=L0a1L1


anLnforsomelettersa1;:::;anofA.2.2ThreevariationsVariationsincludetheunambiguous,deterministic,bideterministicandmodularproducts,thataredenedbelow.Unambiguousproduct.AmarkedproductL=L0a1L1


anLnissaidtobeunambiguousifeverywordofLadmitsauniquedecompositionoftheformu=u0a1u1


anunwithu02L0,...,un2Ln.Forinstance,themarkedproductfa;cgaf1gbfb;cgisunambiguous.Deterministicproduct.Awordxisaprex[sux]ofaworduifthereisawordvsuchthatu=xv[u=vx].Itisaproperprex[sux]ifx=u.AsubsetCofA+isaprex[sux]codeififnoelementofCisaproperprex[sux]ofanotherelementofC.AmarkedproductL=L0a1L1


anLnofnnonemptylanguagesL0,L1,...,LnofAisleft[right]deterministicif,for16i6n,thesetL0a1L1


Li1ai[aiLi


anLn]isaprex[sux]code.ThismeansthateverywordofLhasauniqueprex[sux]inL0a1L1


Li1ai[aiLi


anLn].Itisobservedin[9,p.495]thatthemarkedproductL0a1L1


anLnisdeterministicifandonlyif,for16i6n,thelanguageLi1aiisaprexcode.Sincetheproductoftwoprexcodesisaprexcode,anyleft[right]deterministicproductofleft[right]deterministicproductsisleft[right]deterministic.Amarkedproductissaidtobebideterministicifitisbothleftandrightdeterministic.Modularproductoflanguages.LetL0;:::;LnbelanguagesofA,leta1;:::;anbelettersofAandletrandpbeintegerssuchthat06rp.WedenethemodularproductofthelanguagesL0;:::;Lnwithrespecttorandp,denoted(L0a1L1


anLn)r;p,asthesetofallwordsuinAsuchthatthenumberoffactorizationsofuintheformu=u0a1u1


anun,withui2Lifor06i6n,iscongruenttormodulop.Alanguageisap-modularproductofthelanguagesL0;:::;Lnifitisoftheform(L0a1L1


anLn)r;pforsomer.3ClassicalareaThemostimportantresultsontheconcatenationproductareduetoSchutzen-berger.TheyconcernthesmallestBooleanalgebraoflanguagesclosedundermarkedproductoroneofitsvariants.6 Recallthatthesetofstar-freelanguagesisthesmallestBooleanalgebraoflanguagesofAwhichisclosedundermarkedproduct.Theorem3.1(Schutzenberger[60]).Aregularlanguageisstar-freeifandonlyifitssyntacticmonoidisaperiodic.Thereareessentiallytwoproofsofthisresult.Schutzenberger'soriginalproof[60,35],slightlysimpliedin[30],worksbyinductionontheJ-depthofthesyn-tacticsemigroup.Schutzenberger'sproofactuallygivesastrongerresultsinceitshowsthatthestar-freelanguagesformthesmallestBooleanalgebraoflan-guagesofAwhichisclosedundermarkedproductsoftheformL!LaAandAaL.Inotherwords,markedproductswithAsucetogenerateallstar-freelanguages.Theotherproof[17,28]makesuseofaweakformoftheKrohn-Rhodestheorem:everyaperiodicsemigroupdividesawreathproductofcopiesofthemonoidU2=f1;a;bg,givenbythemultiplicationtableaa=a,ab=b,ba=bandbb=b.Theorem3.1providesanalgorithmtodecidewhetheragivenregularlan-guageisstar-free.Thecomplexityofthisalgorithmisanalysedin[16,65].Letusdeneinthesamewaythesetofunambiguous[rightdeterministic,leftdeterministic]star-freelanguagesasthesmallestBooleanalgebraoflanguagesofAcontainingthelanguagesoftheformB,forBA,whichisclosedunderunambiguous[leftdeterministic,rightdeterministic]markedproduct.Thealgebraiccharacterizationsoftheseclassesarealsoknown.Theorem3.2(Schutzenberger[61]).Aregularlanguageisunambiguousstar-freeifandonlyifitssyntacticmonoidbelongstoDA.Onecanshowthatthesetofunambiguousstar-freelanguagesofAisthesmallestsetoflanguagesofAcontainingthelanguagesoftheformB,forBA,whichisclosedundernitedisjointunionandunambiguousmarkedproduct.ThelanguagescorrespondingtoDAadmitseveralothernicecharacterizations:see[72]forasurvey.Deterministicproductswerealsostudiedin[61].Alternativedescriptionsoftheselanguagescanbefoundin[18,13].Theorem3.3([18]).Aregularlanguageisleft[right]deterministicstar-freeifandonlyifitssyntacticmonoidisR-trivial[L-trivial].Similarresultsareknownforthep-modularproduct[18,66,73,76,29,78{80].Theorem3.4.Letpbeaprime.AlanguageofAbelongstothesmallestBooleanclosedunderp-modularproductifandonlyifitssyntacticmonoidisap-group.Theorem3.5.AlanguageofAbelongstothesmallestBooleanclosedunderp-modularproductforallprimepifandonlyifitssyntacticmonoidisasolublegroup.7 Finally,onemayconsidertheproductandthep-modularproductssimulta-neously.Theorem3.6.AlanguageofAbelongstothesmallestBooleanclosedunderproductandunderp-modularproductforallprimepifandonlyifallthegroupsinitssyntacticmonoidaresoluble.Seealso[75]foranotherdescriptionofthisvarietyoflanguages.4Thegroundbass:SchutzenbergerproductsTheSchutzenbergerproductistherstalgebraictoolusedtostudytheconcate-nationproduct.ItwasrstdenedbySchutzenberger[60]andlatergeneralizedbyStraubing[67].Anintuitiveconstruction,relatedtothelinearrepresenta-tionofasuitabletransducer,wasgivenin[46,47]andisbrie\rysketchedbelow.MoreinformationontransducersandtheirlinearrepresentationscanbefoundinSakarovitch'sbook[59].4.1TransducersfortheproductTheconstructiongivenin[46,47]reliesonthefollowingobservation.LetandabethetransductionsfromAtoAAdenedby(u)=f(u1;u2)ju1u2=uga(u)=f(u1;u2)ju1au2=ugItiseasytoseethatthetwotransducerspicturedbelowrealisethesetrans-ductions.Inthesegures,cisagenericletterandthesymboljisaseparatorbetweentheinputletterandtheoutput. 1 2 1 2 cj(c;1) 1j(1;1) cj(1;c) cj(c;1) aj(1;1) cj(1;c)Thetransducerontheleft[right]realizes[a].NowL0L1=1(L0L1)andL0aL1=1a(L0L1)andthisequalityallowsonetocomputeamonoidrecognisingL0L1andL0aL1,givenmonoidsrecognisingL0andL1.Thisconstructioncanbereadilyextendedto(marked)productsofseverallanguages.Forinstance,givena1;:::;an2A,thetransductiondenedby(u)=f(u0;


;un)2(A)n+1ju0a1u1


anun=ugisrealisedbythetrans-ducer8 1 2::: n1 n cj(c;1) a1j(1;1) cj(1;c) cj(1;c) anj(1;1) cj(1;c)andthemarkedproductL0a1L1


anLnisequalto1(L0L1


Ln).Abitofalgebraisnowrequiredtomakefulluseofthistransduction.4.2LinearrepresentationsTheRbethesemiringP(AA).ThenforeachworduinA,a(u)=(u)1;2,where:A!M2(R)isdenedby(a)=(c;1)(1;1)0(1;c)and(c)=(c;1)00(1;c)ifc=aIndeed,foreachu2A,onegets(u)=(u;1)f(u0;u1)ju0au1=ug0(1;u)whichgivestheresult.Letnow0:A!M0[1:A!M1]beamonoidmorphismrecognisingthelanguageL0[L1]andletM=M0M1.Let=01.ThenisamonoidmorphismfromAAintoM,whichcanberstextendedtoasemiringmorphismfromAAtoP(M)andthentoasemiringmorphismfromM2(AA)toM2(P(M)),alsodenotedby.ItfollowsthatisamorphismfromAintoM2(P(M))anditisnotdiculttoseethatthismorphismrecognisesthelanguage1a(L0L1),thatis,L0aL1.Further,ifuisawordofA,thematrix(u)hastheform(m0;1)P0(1;m1)forsomem02M0,m12M1andPM0M1.Inparticular,L0aL1isrecog-nisedbythemonoidofmatricesofthisform.ThismonoidistheSchutzenbergerproductofthemonoidsM0andM1.AsimilarrepresentationcanbegivenforthetransducerandthisleadstothedenitionoftheSchutzenbergerproductofn+1monoidsM0;:::;Mn.Infact,onecangiveaslightlymoregeneraldenition.LetM=M0


Mn,letkbeasemiringandletk[M]bethemonoidalgebraofMoverk.TheSchutzenbergerproductoverkofthemonoidsM0;:::;Mn,isthesubmonoidofMn+1(k[M])madeupofmatricesm=(mi;j)suchthat(1)mi;j=0,fori�j,(2)mi;i=(1;:::;1;mi;1;:::;1)forsomemi2Mi,(3)mi;j2k[1


1Mi


Mj1


1],forij.9 Thismonoidisdenotedk}(M0;:::;Mn).Therstconditionmeansthatthematricesareuppertriangular,thesecondonethattheentrymi;icanbeidentiedwithanelementofMi.WhenkistheBooleansemiring,thenk[M]isisomorphictoP(M)andtheSchutzenbergerproductissimplydenoted}(M0;:::;Mn).Forinstance,amatrixof}3(M1;M2;M3)willhavetheform0@s1P1;2P1;30s2P2;300s31Awithsi2Mi,P1;2M1M2,P1;3M1M2M3andP2;3M2M3.TherstpartofthenextpropositionisduetoSchutzenberger[60]forn=1andtoStraubing[67]forthegeneralcase.Proposition4.1.LetL=L0a1L1


anLnbeamarkedproductandletMibethesyntacticmonoidofLi,for06i6n.ThentheSchutzenbergerproduct}n(M0;:::;Mn)recognisesL.Asimilarresultholdsforthep-modularproduct,foraprimep,bytakingk=Fp,theeldwithpelements[34,37,79].Proposition4.2.LetL=(L0a1L1


anLn)r;pbeap-modularproductandletMibethesyntacticmonoidofLi,for06i6n.ThentheSchutzenbergerproductFp}n(M0;:::;Mn)recognisesL.InviewofProposition4.1,anaturalquestionarises:whatarethelanguagesrecognisedbyaSchutzenbergerproduct?IntheBooleancase,theanswerwasrstgivenbyReutenauer[58]forn=2andbytheauthor[33]inthegeneralcase(seealso[80,63]).Thecasek=FpwastreatedbyWeil[79,Theorem2.2].Theorem4.3.AlanguageisrecognisedbytheSchutzenbergerproductofM0,...,MnifandonlyifitbelongstotheBooleanalgebrageneratedbythemarkedproductsoftheformLi0a1Li1


asLiswhere06i0i1


is6nandLijisrecognisedbyMijfor06j6s.Theorem4.4.AlanguageisrecognisedbythemonoidFp}(M0;:::;Mn)ifandonlyifitbelongstotheBooleanalgebrageneratedbythep-modularproductsoftheform(Li0a1Li1


asLis)r;pwhere06i0i1


is6nandLijisrecognisedbyMijfor06j6s.IntheBooleancase,itispossibletogiveanorderedversionofTheorem4.3[54,44].Indeed,the(Boolean)Schutzenbergerproductcanbeorderedbyreverseinclusion:P6P0ifandonlyiffor16i6j6n,Pi;jP0i;j.Thecorrespondingorderedmonoidisdenoted}+n(M0;:::;Mn)andiscalledtheorderedSchutzen-bergerproductofM1,...,Mn.Theorem4.5.AlanguageisrecognisedbytheorderedSchutzenbergerproductofM0,...,MnifandonlyifitbelongstothelatticegeneratedbythemarkedproductsoftheformLi0a1Li1


asLiswhere06i0i1


is6nandLijisrecognisedbyMijfor06j6s.10 4.3AlgebraicpropertiesoftheSchutzenbergerproductItfollowsfromthedenitionoftheSchutzenbergerproductthatthemapsendingamatrixtoitsdiagonalisamorphismfromk}(M0;:::;Mn)toM.ThepropertiesofthismorphismwererstanalysedbyStraubing[67]andbytheauthor[36,54,44]intheBooleancaseandbyWeil[80,Corollary3.6]whenk=Fp.Seealso[4].Proposition4.6.Themorphism:}(M0;:::;Mn)!MisaJx!yx!6x!K-morphism.Proposition4.7.Themorphism:Fp}(M0;:::;Mn)!MisaLGp-morphism.5Passacaglia:pumpingpropertiesThesecondmethodtostudytheproductistouserelationalmorphisms.ThistechniquewasinitiatedbyStraubing[68]andlaterrenedin[10,8,36,44,50,54].Werststatethemainresultundertheformofapumpinglemmabeforeturningtoamorealgebraicformulation.LetL=L0a1L1


anLnbeamarkedproductofregularlanguages.Theorem5.1.LetuandvbewordsofAsatisfyingthefollowingproperties:(1)u2Luand(2)foreachi2f0;:::;ng,u2Liuanduvu6Liu.Thenforallx;y2A,theconditionxuy2Limpliesxuvuy2L.Anotherpossibleformulationofthetheoremistosaythat,undertheassump-tions(1)and(2),Lisclosedundertherewritingsystemu!uvu.Wenowturntothealgebraicversionofthisstatement.Foreachi,letLibealanguageofA,leti:A!M(Li)beitssyntacticmorphismandlet:A!M(L0)M(L1)


M(Ln)bethemorphismdenedby(u)=(0(u);1(u);:::;n(u)).Finally,let:A!M(L)bethesyntacticmorphismofL.Theorem5.1canbereformulatedasapropertyoftherelationalmorphism(seepicturebelow)=1:M(L)!M(L0)M(L1)


M(Ln)M(L)M(L0)M(L1)


M(Ln)A  =1 11 Theorem5.2.(1)TherelationalmorphismisarelationalJx!yx!6x!K-morphism.(2)Iftheproductisunambiguous,itisarelationalJx!yx!=x!K-morphism.(3)Iftheproductisleftdeterministic,itisarelationalJx!y=x!K-morphism.(4)Iftheproductisrightdeterministic,itisarelationalJyx!=x!K-morphism.Asimilarresultholdsforthep-modularproduct.Proposition5.3.LetL=(L0a1L1


anLn)r;pbeap-modularproduct.Therelationalmorphism:M(L)!M(L0)


M(Ln)isarelationalLGp-morphism.Theorem5.2isoftenusedinthefollowingweakerform.Corollary5.4.Therelationalmorphism:M(L)!M(L0)M(L1)


M(Ln)isanaperiodicrelationalmorphism.6Chaconne:ClosurepropertiesTheresultsofSection3giveadescriptionofthesmallestBooleanalgebraclosedundermarkedproductanditsvariants.ThenextstepwouldbetocharacterizeallBooleanalgebrasclosedundermarkedproductanditsvariants.Arelatedproblemistodescribetheclassesofregularlanguagesclosedunderunionandmarkedproduct.Bothproblemshavebeensolvedinthecaseofavarietyoflanguages,butthedescriptionoftheseresultsrequiresanalgebraicdenition.LetVbeavarietyof[ordered]monoidsandletWbeavarietyoforderedsemigroups.Theclassofall[ordered]monoidsMsuchthatthereexistsaV-relationalmorphismfromMintoamonoidofVisavarietyof[ordered]monoids,denotedW1V.6.1VarietiesclosedunderproductVarietiesclosedundermarkedproductsweredescribedbyStraubing[66].Theorem6.1.LetVbeavarietyofmonoidsandletVbetheassociatedvari-etyoflanguages.ForeachalphabetA,letW(A)bethesmallestBooleanalgebracontainingV(A)andclosedunderproduct.ThenWisavarietyandtheasso-ciatedvarietyofmonoidsisA1V.ThisimportantresultcontainsTheorem3.1asaparticularcase,whenVisthetrivialvarietyofmonoids.ExamplesofvarietiesVsatisfyingtheequalityA1V=Valsoincludethevarietyofmonoidswhosegroupsbelongtoagivenvarietyofgroups.Theorem6.1hasbeenextendedtoC-varietiesin[15,Theorem4.1].12 6.2VarietiesclosedundermodularproductFinally,letusmentiontheresultsofWeil[80].AsetoflanguagesLofAisclosedunderp-modularproductif,foranylanguageL0;:::;Ln2L,foranylettera1;:::;an2Aandforanyintegerrsuchthat06rp,(L0a1L1


anLn)r;p2L.AsetoflanguagesLofAisclosedundermodularproductifitisclosedunderp-modularproduct,foreachprimep.Theorem6.2.Letpbeaprimenumber,letVbeavarietyofmonoidsandletVbetheassociatedvarietyoflanguages.ForeachalphabetA,letW(A)bethesmallestBooleanalgebracontainingV(A)andclosedunderp-modularproduct.ThenWisavarietyoflanguagesandtheassociatedvarietyofmonoidsisLG1pV.GivenavarietyofgroupsH,let HbethevarietyofallmonoidswhosegroupsbelongtoH.Theorem6.3.Letpbeaprimenumber,letVbeavarietyofmonoidsandletVbetheassociatedvarietyoflanguages.ForeachalphabetA,letW(A)bethesmallestBooleanalgebracontainingV(A)andclosedunderproductandp-modularproduct.ThenWisavarietyoflanguagesandtheassociatedvarietyofmonoidsisL G1pV.Theorem6.4.LetVbeavarietyofmonoidsandletVbetheassociatedvarietyoflanguages.ForeachalphabetA,letW(A)betheBooleanalgebracontainingV(A)andclosedundermodularproduct.ThenWisavarietyoflanguagesandtheassociatedvarietyofmonoidsisL Gsol1V.6.3PolynomialclosureLetLbealatticeoflanguages.ThepolynomialclosureofListhesetoflan-guagesthatareniteunionsofmarkedproductsoflanguagesofL.ItisdenotedPol(L).Similarly,theunambiguouspolynomialclosureofListhesetoflan-guagesthatareniteunionsofunambiguousmarkedproductsoflanguagesofL.ItisdenotedUPol(L).Theleftandrightdeterministicpolynomialclosurearedenedanalogously,byreplacing\unambiguous"by\left[right]deterministic".TheyaredenotedDlPol(V)[DrPol(V)].Analgebraiccharacterizationofthepolynomialclosureofavarietyoflan-guageswasrstgivenin[51,54].Itwasextendedtopositivevarietiesin[44].Theorem6.5.LetVbeavarietyof[ordered]monoidsandletVbetheassoci-ated[positive]varietyoflanguages.ThenPol(V)isapositivevarietyoflanguagesandtheassociatedvarietyoforderedmonoidsisJx!yx!6x!K1V.Theorem6.5hasbeenextendedtoC-varietiesin[49,Theorem7.2].Fortheunambiguousproduct,onehasthefollowingresult[32,50,4].13 Theorem6.6.LetVbeavarietyofmonoidsandletVbetheassociatedvarietyoflanguages.ThenUPol(V)isavarietyoflanguagesandtheassociatedvarietyoforderedmonoidsisJx!yx!=x!K1V.Fortheleft(resp.right)deterministicproduct,similarresultshold[32,50].Theorem6.7.LetVbeavarietyofmonoidsandletVbetheassociatedvarietyoflanguages.ThenDlPol(V)(resp.DrPol(V))isavarietyoflanguages,andtheassociatedvarietyofmonoidsisJx!y=x!K1V(resp.Jyx!=x!K1V).ItisknownthatthesmallestnontrivialvarietyofaperiodicmonoidsisthevarietyJ1=Jxy=yx;x=x2K.OnecanshowthatJx!y=x!K1J1isequaltothevarietyRofallR-trivialmonoids,whichisalsodenedbytheidentity(xy)!x=(xy)!.Thisleadstothefollowingcharacterization[18,13].Corollary6.8.ForeachalphabetA,R(A)consistsofthelanguageswhicharedisjointunionsoflanguagesoftheformA0a1A1a2


anAn,wheren�0,a1;:::an2AandtheAi'saresubsetsofAsuchthatai=2Ai1,for16i6n.AdualresultholdsforL-trivialmonoids.Finally,Jx!yx!=x!K1J1=DA,whichleadstothedescriptionofthelanguagesofDAgivenhereinabove.6.4BacktoidentitiesAgeneralresultof[52]permitstogiveidentitiesdeningthevarietiesoftheformV1W.Inparticular,wegetthefollowingresults.Theorem6.9.LetVbeavarietyofmonoids.Then(1)A1Visdenedbytheidentitiesoftheformx!+1=x!,wherexisapronitewordsuchthatVsatisestheidentityx=x2.(2)Jx!yx!=x!K1Visdenedbytheidentitiesoftheformx!yx!=x!,wherex;yarepronitewordssuchthatVsatisestheidentityx=y=x2.(3)Jx!yx!6x!K1Visdenedbytheidentitiesoftheformx!yx!6x!,wherex;yarepronitewordssuchthatVsatisestheidentityx=y=x2.7HierarchiesandbridgesTheBooleanalgebraBLgeneratedbyalatticeLiscalleditsBooleanclosure.Inparticular,BPol(L)denotestheBooleanclosureofPol(L).ConcatenationhierarchiesaredenedbyalternatingBooleanoperationsandpolynomialoperations(unionandmarkedproduct).Moreprecisely,letLbeasetofregularlanguages(ormoregenerally,aclassoflanguages).TheconcatenationhierarchybuiltonListhesequenceLndenedinductivelyasfollows1:L0=Land,foreachn�0: 1Intheliterature,concatenationhierarchiesareusuallyindexedbyhalfintegers,butitseemssimplertouseintegers.14 (1)L2n+1isthepolynomialclosureofthelevel2n,(2)L2n+2istheBooleanclosureofthelevel2n+1.Thenextresultssummarizetheresultsof[5,6,54].Proposition7.1.IfLisalatticeofregularlanguages,theneachevenlevelisalatticeofregularlanguagesandeachoddlevelisaBooleanalgebra.Further,ifLisclosedunderquotients,theneverylevelisclosedunderquotients.SincethepolynomialclosureofaC-varietyoflanguagesisapositiveC-varietyoflanguages[49,Theorem6.2],asimilarresultholdsforC-varieties.Proposition7.2.IfLisaC-varietyoflanguages,theneachevenlevelisapositiveC-varietyoflanguagesandeachoddlevelisaC-varietyoflanguages.Forinstance,theStraubing-Therien'hierarchyVn[74,67,69]isbuiltonthetrivialBooleanalgebraV0=f;;Ag.ThestartingpointofBrzozowski's\dot-depth"hierarchyBn[12]wasoriginallydenedastheBooleanalgebraofniteandconitelanguagesbutitwaslatersuggestedtostartwiththeBooleanalgebraB0(A)=fFAG[HjF,G,HarenitelanguagesgThissuggestionwasmotivatedbyTheorem7.4below.Anotherseriesofconcatenationhierarchiesisobtainedasfollows.LetHbeavarietyofgroupsandletHbetheassociatedvarietyoflanguages.TheconcatenationhierarchybuiltonHisdenotedbyHnandthesehierarchiesarecalledgrouphierarchies.Itisnotimmediatetoseethatallthesehierarchiesdonotcollapse.ThiswasrstprovedbyBrzozowskiandKnast[14]forthedot-depthhierarchy,buttheresultalsoholdsfortheotherhierarchies[26].Theorem7.3.TheStraubing-Therien'hierarchy,thedot-depthhierarchyandthegrouphierarchiesareinnite.LetVnbethevarietyofmonoidscorrespondingtoVnandletBnbethevarietyofsemigroupscorrespondingtoBn.ThereisanicealgebraicconnectionbetweenVnandBn,discoveredbyStraubing[69].Givenavarietyof[ordered]monoidsVandavarietyofmonoids[semigroups]W,letVWbethevarietyof[ordered]monoidsgeneratedbythesemidirectproductsMNwithM2VandN2W.Theorem7.4.TheequalityBn=VnLIholdsforeachn�0.ThereisasimilarbridgebetweenVnandHnforeachvarietyofgroupsH[43,44].Theorem7.5.TheequalityHn=VnHholdsforeachn�0.Itisstillanoutstandingopenproblemtoknowwhetherthereisanalgorithmtocomputetheconcatenationlevelofagivenregularlanguage.Hereisabriefsummaryoftheknownresults.Letusstartwiththelevel1[26,39,41,56].LetGbethevarietyofallgroups.15 Theorem7.6.Thefollowingrelationshold:V1=Jx61K,B1=Jx!yx!6x!KandG1=Jx!61K.Inparticular,thesevarietiesaredecidable.ThelanguagesofG1arealsoknowntobetheopenregularsetsfortheprogrouptopology[26].ExtensionsofthisresulttothevarietiesH1whereHisavarietyofgroupsisthetopicofintensiveresearch.SeeinparticularSteinberg'article[64].Therstdecidabilityresultforthelevel2wasobtainedbySimon[62].Theorem7.7.AlanguagebelongstoV2ifandonlyifitssyntacticmonoidisJ-trivial.ThecorrespondingresultforB2isduetoKnast[24,25]Theorem7.8.AlanguagebelongstoB2ifandonlyifitssyntacticsemigroupsatisestheidentity(x!py!qx!)!x!py!sx!(x!ry!sx!)!=(x!py!qx!)!(x!ry!sx!)!:ThecorrespondingresultforG2hasalongstory,relatedindetailin[38],whereseveralothercharacterizationscanbefound.Theorem7.9.AlanguagebelongstoG2ifandonlyifinitssyntacticmonoid,thesubmonoidgeneratedbytheidempotentsisJ-trivial.Theorem7.9showsthatG2isdecidable.Again,thereisalotofongoingworktotrytoextendthisresulttovarietiesoftheformH2.Seeinparticular[7].Sincelevel3isthepolynomialclosureoflevel2,Theorem6.5canbeap-plied.Onegetsinparticularthefollowingdecidabilityresult[54].Recallthatthecontentofawordisthesetoflettersoccurringinthisword.Theorem7.10.AlanguagebelongstoV3ifandonlyifitssyntacticorderedmonoidsatisestheidentitiesx!yx!6x!forallpronitewordsx;ywiththesamecontent.ThecorrespondingproblemforB3isstudiedin[20,22,56].Infact,Theorem7.4canbeusedtoprovethefollowingmoregeneraldecidabilityresult[56,69].Theorem7.11.Foreveryintegern,thevarietyBnisdecidableifandonlyifVnisdecidable.ItisstillanopenproblemtoknowwhetherasimilarreductionexistsforthehierarchyGn.Forthelevel4,severalpartialresultsareknown[48,70]andseveralconjec-tureshavebeenformulatedandthendisproved[54,64,55].Duetothelackofspace,wewillnotdetailtheseresultshere.Somepartialresultsarealsoknownforthelevel5[21].16 8HarmonywithlogicOneofthereasonswhythedecidabilityproblemisparticularlyappealingisitscloseconnectionwithnitemodeltheory,rstexploredbyBuchiintheearlysixties.Buchi'slogiccomprisesarelationsymboland,foreachlettera2A,aunarypredicatesymbola.ThesetFO[]ofrstorderformulasisbuiltintheusualwaybyusingthesesymbols,theequalitysymbol,rstordervariables,Booleanconnectivesandquantiers.Aworduisrepresentedasastructure(Dom(u);(a)a2A;)whereDom(u)=f1;:::;jujganda=fi2Dom(u)ju(i)=ag.Thebinaryrelationsymbolisinterpretedastheusualorder.Thus,ifu=abbaab,Dom(u)=f1;:::;6g,a=f1;4;5gandb=f2;4;6g.Formulascannowbeinterpretedonwords.Forinstance,thesentence'=9x9y(xy)^(ax)^(by)
means\thereexisttwointegersxysuchthat,inu,theletterinpositionxisanaandtheletterinpositionyisab".Therefore,thesetofwordssatisfying'isAaAbA.Moregenerally,thelanguagedenedbyasentence'isthesetofwordsusuchthat'satisesu.Theconnectionwithstar-freelanguageswasestablishedbyMcNaughtonandPapert[27].Theorem8.1.AlanguageisFO[]-denableifandonlyifitisstar-free.Thomas[77](seealso[31])renedthisresultbyshowingthattheconcate-nationhierarchyofstar-freelanguagescorresponds,levelbylevel,tothen-hierarchy,denedinductivelyasfollows:(1)0consistsofthequantier-freeformulas.(2)nconsistsoftheformulasoftheform989


'withnalternatingblocksofquantiersand'quantier-free.(3)BndenotestheclassofformulasthatareBooleancombinationsofn-formulas.Forinstance,9x19x28x38x48x59x6',where'isquantierfree,isin3.ThenexttheoremisduetoThomas[77](seealso[31,40]).Theorem8.2.(1)Alanguageisn[]-denableifandonlyifitbelongstoV2n1.(2)AlanguageisBn[]-denableifandonlyifitbelongstoV2n.Aslightlyexpandedlogicisrequiredforthedot-depthhierarchy.Letmin[max]beapredicatesymbolinterpretedastheminimum[maximum]ofthedo-mainandletP[S]bearelationsymbolinterpretedasthepredecessor[successor]relation.LetLocbethesignaturefmin;max;S;Pg[f(a)a2Ag.Theorem8.3.(1)Alanguageisn[Loc]-denableifandonlyifitbelongstoB2n1.(2)AlanguageisBn[Loc]-denableifandonlyifitbelongstoB2n.Thusdecidingwhetheralanguagehaslevelnisequivalenttoaverynaturalprobleminnitemodeltheory.17 9Othervariations,recentadvancesSomespecializedtopicsrequireevenmoresophisticatedalgebraictools,likethekernelcategoryofamorphism.Thisisthecaseforinstanceforthebidetermin-isticproduct[9{11]orforthemarkedproductoftwolanguages[4].Anothertopicthatwedidnotmentionatall,butwhichishighlyinteresting,istheextensionoftheseresultstoinnitewordsoreventowordsoverordinalsorlinearorders.Iwouldliketoconcludewitharecentresult,whichopensanewresearchdirection.WehavegiveninSection6variousclosurepropertiesforvarietiesorevenforC-varieties.ThenextresultofBrancoandtheauthor[8]ismuchmoregeneral.Theorem9.1.IfLisalatticeoflanguagesclosedunderquotients,thenPol(L)isdenedbythesetofequationsoftheformx!yx!6x!,wherex;yarepronitewordssuchthattheequationsx=x2andy6xaresatisedbyL.WorkisinprogresstoextendtheotherresultsofSection6tothismoregeneralsetting.ThedicultystemsfromthefactthatdenitionslikeV1Warenolongeravailableinthiscontextandonehastoworkdirectlyonproniteidentities.References1.J.Almeida,Finitesemigroupsanduniversalalgebra,WorldScienticPublishingCo.Inc.,RiverEdge,NJ,1994.Translatedfromthe1992Portugueseoriginalandrevisedbytheauthor.2.J.Almeida,Finitesemigroups:anintroductiontoauniedtheoryofpseudovari-eties,inSemigroups,Algorithms,AutomataandLanguages,G.M.S.Gomes,J.-E.PinandP.Silva(eds.),pp.3{64,WorldScientic,Singapore,2002.3.J.Almeida,Pronitesemigroupsandapplications,inStructuraltheoryofau-tomata,semigroups,anduniversalalgebra,pp.1{45,NATOSci.Ser.IIMath.Phys.Chem.vol.207,Lect.NotesComp.Sci.,Dordrecht,2005.4.J.Almeida,S.Margolis,B.SteinbergandM.Volkov,Representationthe-oryofnitesemigroups,semigroupradicalsandformallanguagetheory,Trans.Amer.Math.Soc.361,3(2009),1429{1461.5.M.Arfi,Polynomialoperationsonrationallanguages,inSTACS87(Passau,1987),pp.198{206,Lect.NotesComp.Sci.vol.247,Lect.NotesComp.Sci.,Berlin,1987.6.M.Arfi,Operationspolynomialesethierarchiesdeconcatenation,Theoret.Com-put.Sci.91,1(1991),71{84.7.K.AuingerandB.Steinberg,VarietiesofnitesupersolvablegroupswiththeM.Hallproperty,Math.Ann.335,4(2006),853{877.8.M.J.BrancoandJ.-E.Pin,Equationsforthepolynomialclosure,inICALP2009,PartII,S.Albers,A.Marchetti-Spaccamela,Y.Matias,S.NikoletseasandW.Thomas(eds.),Berlin,2009,pp.115{126,Lect.NotesComp.Sci.vol.5556,Springer.18 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