Monoid-labeled transition systemsH.PeterGumm,TobiasSchroderFachbereich Mathematik und InformatikPhilipps-Universitat MarburgMarburg, Germany{gumm,tschroed}@mathematik.uni-marburg.deAbstractGivena -complete(semi)lattice L, weconsider L-labeledtransitionsystemsascoalgebras of a functor L(), associating with a set X the set LXof all L-fuzzy sub-sets. We describe simulations and bisimulations of L-coalgebras to show that L()weakly preserves nonempty kernel pairs i it weakly preserves nonempty pullbacksi L is join innitely distributive (JID).Exchanging L for a commutative monoid M, we consider the functor M()whichassociates with a set X all nite multisets containing elements of X with multiplici-ties m M. The corresponding functor weakly preserves nonempty pullbacks alonginjectives i 0 is the only invertible element of M, and it preserves nonempty kernelpairs i M is renable, in the sense that two sum representations of the same value,r1 + . . . + rm = c1 + . . . + cn, have a common renement matrix (mi,j) whosek-throw sums to rk and whose l-th column sums to cl for any 1 k m and 1 l n.Key words: Coalgebra, transition system, fuzzy transition,multiset, weak pullback preservation, bisimulation, renablemonoid, distributive lattice.1 IntroductionIt is well known that transition systems can be described as coalgebras of thecovariant powerset functor P. The general theory of coalgebras automaticallysuppliesthefundamental notionsof homomorphismandbisimulation. Thefactthatthefunctor Piswell behaved, i.e. thatitpreserves(generalized)weakpullbacks, guarantees acollectionof useful properties. Inparticular,therelational product of bisimulations is abisimulation, thelargest bisim-ulationisanequivalencerelationandkernelsof homomorphismsarealwaysbisimulations. Thesubclassofimage-nitetransitionsystemscorrespondstocoalgebras of thenitepowerset functor P. This functor, inaddition, isPreprintsubmittedtoElsevierPreprint 26February2001bounded,soaterminal P-coalgebraexists,providingsemanticsandaproofprinciple(coinduction)forimage-nitetransitionsystems.Introducinglabels does not complicatethesituation. Givenaset Loflabels, anL-labeledtransitionsystemona state set S is a ternaryrela-tionT S L S. Again, the equivalent coalgebraic viewas a map: S P(S)L, i.e. as a P()L-coalgebra, better captures the dynamicaspects of labeled transition systems. What has been said above for P-,resp.P-coalgebrascarriesoverto P()L-, resp. P()L-coalgebras. InfactanL-labeledtransitionsystemisnothingbutacollectionof (plain)transitionsystems,oneforeachlabel.Thesituationbecomesmoreinteresting,whenthelabelscarrysomealge-braic structure. This structure is relevant, when arc labels denote, for instance,owcapacitiesordurations.In general, we shall assume the labels to dene a commutativemonoid, i.e.acommutative, associativeoperationwithaneutral element0. Thisallowsusacoalgebraicinterpretationof labeledtransitionsystemsasamapfromstatestogradedsetsofsuccessorstates. Thisviewwillbeseentoprovideaninterestingintertwiningof coalgebraicstructurewithalgebraicpropertiesofthemonoid.Another motivation for this work is the study of Set-functors. In previouswork([Gum98],[GSb],[GS00])wehavestudiedthecoalgebraicsignicanceofvariouspreservationpropertiesof Set-endofunctors. Hereweconstructsuchfunctors, that are parameterized by a commutative monoid, so we can custombuildfunctorsbyselectingcommutativemonoidswithappropriatealgebraicproperties.Intherstsections,westartwithanarbitrary

-completesemilattice L,andweconsider L-multisets. AmultisetScanbethoughtof asaset, eachof whose elements sis containedinSwithsome multiplicity(probability,certainty)l L. Thereareplentyof natural choicesfor L, suchas {0, 1},N{}, or the real interval [0, 1], giving rise to the standard notions of (plainold)set,bag(multiset),orfuzzyset.The semilattice L gives rise to a functor L()which generalizes the powersetfunctor Pfor L = {0, 1}. Thus L()-coalgebrasgeneralizetransitionsystemstomultisettransitionsystemsandfuzzytransitionsystems.Weshowthat L()alwayspreservesnonemptypullbacksalonginjectivemaps, and that it preserves arbitrary nonempty weak pullbacks just in case Lis joininnitelydistributive, that is, nite meets distribute over innite joins.Insemilattices, thepossibilityof deninginnitesumsiscloselytiedtotheidempotentlaw. Inarbitrarycommutativemonoids,wecanforminnitesums only in cases where all but nitely many summands are zero. This leadstoaslightlydierent functor, M(), for anarbitrarycommutativemonoidM. Westudyit inthesecondhalf of this paper. This functor preserves2nonempty pullbacks along injective maps i the monoid is positiveand it pre-serves nonempty weak pullbacks of arbitrary maps i additionally the monoidisrenableinasensetobedenedlater.2 BasicnotionsLetR A BandS B Cbebinaryrelations. Wedenoteby(R; S)theircompositionorrelational product:(R; S) := {(a, c) | b B.(a, b) R, (b, c) S}.With Rwe denote the converse of R, i.e. R:= {(b, a) | (a, b) R}. We useaRbasashorthandfor(a, b) R.If : A BisamapthenwedenotebyG()itsgraph,thatisG() := {(a, (a)) | a A}.Withthesedenitionsweget: G( ) = G() ; G().2.1 F-coalgebrasandhomomorphismsLet F: Set Set be a functor from the category of sets to itself. A coalgebraof type Fis a pair A = (A, ), consisting of a set A and a map : A F(A).Aiscalledthecarrierset andiscalledthestructuremapof A.If A = (A, )and B= (B, )areF-coalgebras,thenamap : A Biscalledahomomorphism,if = F() ,thatis,ifthefollowingdiagramcommutes:A

B

F(A)F()

F(B)F-coalgebrasandtheirhomomorphismsformacategory SetF. Itiswellknownthatallcolimitsin SetFexist,andtheyareformedjustasin Set. Inparticular,thesum

iIAiofafamilyofF-coalgebras Ai=(Ai, i)hasascarrierthedisjointunion

iI Aiandthecoalgebrastructureistheuniquemap :

iI Ai F(

iI Ai)with Ai= F(Ai) ifor all i I, where each Aiis the canonical embedding of Aiinto the disjointunion

iI Ai.32.2 SubcoalgebrasAsubsetU Aiscalledasubcoalgebraof A = (A, ),providedthereexistsacoalgebrastructure: U F(U)sothattheinclusionmap AU: U Aisahomomorphismfrom U= (U, )to A.Onereadsothedenitionof homomorphism, thatUisasubcoalgebraof A = (A, )iu U. v F(U).(u) = F(AU)(v).U= isalwaysasubcoalgebra. IfU = ,thentheinclusionmap AUhasaleftinverse. Consequently, F(AU)hasaleftinversetoo, inparticularitisinjective. Thus, if astructuremapasaboveexists, itisunique. Forthatreasonweusethetermsubcoalgebrainterchangeablyforthecoalgebra UaswellasforitscarriersetU.2.3 BisimulationsA binary relation R ABis called a bisimulation between A and Bif it ispossibletodeneacoalgebrastructure: R F(R),sothattheprojectionmaps 1: R A and 2: R B are homomorphisms. This can be expressedbythefollowingcommutativediagram:A

R

12

B

F(A) F(R)F(1)F(2)

F(B)If R is a bisimulation between A and B, then its converse R is a bisimula-tion between B and A. The union of a family of bisimulations is a bisimulationagain, so there is always a largest bisimulation A,B between coalgebras A andB.If A = B, then we shall call Ra bisimulation on A. The diagonal relationA= {(a, a) | a A} is always a bisimulation, hence the largest bisimulationon A,denotedby A,isalwaysreexiveandsymmetric. Ingeneral,though,itneednotbetransitive.Amap: A Bisahomomorphismbetweencoalgebras Aand B, iitsgraphG()isabisimulation. If Risacoalgebra,and : R Aand:R Care homomorphisms, then {((r), (r)) | r R} = (G(); G()) isa bisimulation between A and B. As a consequence, a bisimulation R betweenA and Bgives rise to a bisimulationR := {(A(a), B(b)) | (a, b) R} on theirsum A+B.43 Thefunctor L()Let L be a complete

-semilattice. L can be made into a complete lattice bydeningforanysubsetS:

S=

{l L | s S.l s}.ForanysetA,let LAdenotethesetofallmaps: A L. Eachsuchcanbethoughtofasthecharacteristicfunctionofan L-multiset:x l(x) = l.Givenamapf: A B,wedeneamap Lf: LA LBbyLf()(b) :=

{(a) | f(a) = b},thenitiseasytocheck:Lemma3.1 L()isa(covariant) Set-endofunctor.Proof. Given sets A, B, and Cand maps f: A B, g: B C, we calculateforarbitrary: A L,a A,andc C:LidA()(a) =

{(x) | idA(x) = a}=(a)=idLA()(a).(Lg Lf)()(c) =

{Lf()(b) | g(b) = c}=

{

{(a) | f(a) = b} | g(b) = c}=

{(a) | (g f)(a) = c}=Lgf()(c).Obviously, when Listhetwo-elementlattice {0, 1}, thisfunctorisnatu-rallyisomorphictothepowersetfunctor Pvia: L() P,denedforanysetXbyX() := {x X | (x) = 1}.It is also straightforward to check that each

-preserving map : L L

inducesanaturaltransformationbetween L()and L

().3.1 L-Coalgebras.Accordingtothegeneraldenitionofcoalgebras,an L()-coalgebraisapair(A, ) consisting of a set A and a map : A LA. Given two L()-coalgebras5(A, )and(B, ), amap:A Bisahomomorphismif L = ,thatis,ifforalla Aandb Bwehave((a))(b) = L((a))(b) =

{(a)(a

) | a

A, (a

) = b}.In the sequel we shall speak of L-coalgebras rather than of L()-coalgebras.It is convenient tointroduce anarrownotationfamiliar fromtransitionsystemsasashorthand. Wewriteala

i (a)(a

) = l.Obviously,the L-coalgebrastructure : A LAmayalsobeinterpretedasan L-gradedrelation : A A Lbysetting (a, a

) := (a)(a

).When Listhetwo-elementlatticethenan L-coalgebraisjustaKripke-frame. Inthegeneral case, Lprovidesasetof measuresforindicatinghowstrongapair(a, a

)shouldbeinthe L-gradedrelation ,or,alternatively,howcondentwemightbethat(a, a

)isin . If Listheunitinterval,then isjustafuzzyrelation.3.2 SubcoalgebrasThefunctor L()isnotstandardinthesensedenedin[Mos99],thatisL(AU)= LALU.Rather,wealwayshave:LAU()(a) =

(a), ifa U0, otherwise.Fromthatoneobtainsstraightforwardly:Lemma3.2U Aisasubcoalgebraof A = (A, )iu U, a A. m = 0.uma =a U.Proof.Uisasubcoalgebraof A = (A, ) u U. : U L.(u) = LAU() u U. a (A U).(u)(a) = 0 u U, a A m = 0.uma =a U.For coalgebras of arbitrarytype F it is knownthat subcoalgebras arealways closed under nite intersections (see [GSa]). For the functor F= L(),weevengetfromtheabove:6Corollary3.3The intersection of an arbitrary family (Ui)iIof subcoalgebrasofan L-coalgebra Aisagainasubcoalgebraof A.3.3 L-SimulationsDenition3.4Wedeneasimulationbetween Aand BasarelationS A Bwhereforall (a, b) Sandall a

Aama

=

{l | y B.bly, a

Sy} m.Thus, if a is simulated by b and if a

is an m-successor of a, then there mustbesucientlymanyli-successorsbiofb,eachonesimulatinga

andtogetheryielding

iI li m.We write AS B, if Sis a simulation between A and B. Then we concludedirectlyfromthedenition:Lemma3.5If AS Band BT C, then AS ;T C. Thustheclassof all L-coalgebraswithsimulationsasarrowsformsacategory.3.4 L-HomomorphismsHomomorphismsturnouttobemapsbetweencoalgebraswhichstrengthenthegradedrelationandwhoseconverseisasimulation,thatis:Lemma3.6Amap: A Bbetweencoalgebras A=(A, ) and B=(B, ) is a homomorphismif and only if the following two conditions aresatisedforall a, a

A,all b

Bandforall m L:ama

=(a)m

(a

)forsomem

m (1)(a)mb

=m

{l | x A.alx, (x) = b

}. (2)Proof. The homomorphism condition requires that for every a A and everyb

Bwehavetheequation((a))(b

) = L((a))(b

).Translatingthis,usingthearrownotation,weget((a), b

) =

{l | x A.alx, (x) = b

}.The two homomorphism conditions are equivalent to the inequalities which areobtainedbyreplacing=abovebyand. Fromthersthomomor-phism condition we obtain((a), b

) l whenever alx and (x) = b

, thus((a))(b

)

{l | x A.alx, (x) = b

}. Fromthisinequality,inturn,wegetthersthomomorphismconditionbackbyconsideringonlyx=a

in7the right hand side, to obtain((a), (a

)) l whenever ala

. The secondhomomorphism condition is obviously equivalent with the inequality obtainedfromreplacing=with.Usingthedescriptionofhomomorphismsandofsubcoalgebras, itisnowstraightforwardtocheck:Corollary3.7If : A B is a homomorphism and V Bis a subcoalgebraof B,then[V ]isasubcoalgebraof A.Observethattherst,resp. second,homomorphismconditionjuststatesthatG(),resp. G(),isasimulation,thusweget:Corollary3.8Amap:A Bisahomomorphismbetween L-coalgebrasAand BifandonlyifbothG()andG()aresimulations.3.5 L-BisimulationsWehavealreadyremarkedthatamapbetweencoalgebras Aand Bisaho-momorphismiits graphis abisimulation. However, arelationRis notnecessarilyabisimulationevenifRandRaresimulations. Toseethecon-nectionswehavetocharacterizebisimulations:Proposition3.9A relation R between coalgebras A = (A, ) and B = (B, )isabisimulationif andonlyif forall (a, b) Randall a

A, b

Bwehave:ama

=

{m l | y B.bly, a

Ry} = m, and (3)bmb

=

{m l | x A.alx, xRb

} = m. (4)Proof. Let Rbeabisimulation, and(a, b) R, thenthereis some :=(a, b) : R Lwith(a) =L1(), and (5)(b) =L2(). (6)Fora

Awehavethereforeby(5) (a, a

) =L1()(a

)=

{(u) | 1(u) = a

}=

{(a

, y) | a

Ry}.By(6)wehaveforeveryywitha

Ry:8(b, y) =L2()(y)=

{(v) | 2(v) = y}=

{(x, y) | xRy}(a

, y).Combiningtheabove,weget (a, a

) =

{(a

, y) | a

Ry}=

{ (a, a

) (a

, y) | a

Ry}

{ (a, a

) (b, y) | a

Ry} (a, a

).Hence, (a, a

) =

{ (a, a

)(b, y) | a

Ry}, which is the rst bisimulationcondition. Thesecondoneisobtainedsymmetrically.Toshowtheconverse, letthebisimulationconditions(3)and(4)besat-ised. WeneedtoshowthatRisabisimulation. Deneastructuremap: R LRby(a, b)(x, y) := (a, x) (b, y).Foranarbitrarya

AwehaveL1((a, b))(a

) =

{(a, b)(u) | 1(u) = a}=

{(a, b)(a

, y) | a

Ry}= (a, a

)=(a)(a

).Hence L1((a, b)) = (a) = ( 1)(a, b),thus L1 = 1,and,analo-gously, L2 = 2. Therefore,Risabisimulation.Noticethat lemma3.6couldbeinferredfromtheabove, sincefor gen-eral coalgebras, homomorphismsarepreciselythosemapswhosegraphisabisimulation. Wealsogetthefollowingcorollary:Corollary3.10IfRisabisimulationthenbothRandRaresimulations.4 TheroleofdistributivityIntheapplicationsofcoalgebrasoneoftenhastypefunctorswhichsatisfyanimportanttechnical property: Theypreserveweakpullbacks. Thisistosaythat a (weak) pullback diagram is transformed into a weak pullback diagram.Manyresultsincoalgebratheory(e.g. [Rut00])hingeonthisproperty. In[GSa] itwasshownthatafunctorFpreservesnonemptyweakpullbacksifand only if bisimulations between F-coalgebras are closed under composition.9Nowconsider corollary3.10. If its converseis truethenbylemma3.5bisimulationswill beclosedundercomposition, thus L()will preserveweakpullbacks. In this section we shall show that this and related properties hingeonacertaindistributivityconditiononthelattice L.Denition4.1([Gra98]) Alattice is called join innite distributive (inshortJID),ifitsatisesthelawx

{xi | i I} =

{x xi | i I}.If Lisacompletesemilattice, weshall saythat LisJIDithisisthecaseforthelatticeinducedon L.Whenever LisJID, thebisimilarityconditionsfor L-coalgebrassimplifyto:Lemma4.2If LisJIDthenarelationR ABisabisimulationbetweencoalgebras Aand Bifandonlyifama

=

{l | bly, a

Ry} m, and (7)bmb

=

{l | alx, xRb

} m. (8)Thefollowingtheorem,amongstotherthings,providesaconversetothislemmaandtocorollary3.10:Theorem4.3Let Lbea

-semilattice,thenthefollowingareequivalent:(i) LisJID.(ii) R A BisabisimulationRandRaresimulations.(iii) Bisimulationsareclosedundercompositions.(iv) L()preservesnonemptyweakpullbacks.(v) L()weaklypreservesnonemptykernel pairs.(vi) Thelargestbisimulation Aonevery L-coalgebraistransitive.Proof. (i) (ii)followsfromlemma4.2. (ii) (iii)isaconsequenceoflemma3.5. (iii) (iv)isshownforarbitraryfunctorsin[GS00]. (iv) (v) (vi)canbefoundin[GS00],sowemayconcentrateonproving(vi) (i).Letafamily(li)iIandafurtherelementmof Lbegiven, itsucestoshowm

iIli

iI(m li),sincethereverseinclusionholdsinanylattice. Onthesets10A:={ai | i I} {a, a

}B:={bi | i I} {b}C :={c, c

}denecoalgebrastructures,,andgiveninarrow-notationasfollows:aea

, wheree := m

lialiai, foralli Iblibi, foralli Ic

lic

.Usinglemma3.6itiseasytoseethat: A C, denedas(a)=cand(a

) =(ai) =c

for all i I, and: B Cgivenby(b) =cand(bi)=c

foralli Iarehomomorphisms. Consequently, G()andG()arebisimulations.aezzzzzzzzzzzzzzzzzzli

lj

2222222222222c

li

bli

lj

2222222222222

a

ai. . . ajc

bi. . .bjConsidernowthesumofthethreecoalgebras A+ B + C. WestillhavethatG()andG()arebisimulations, inparticular, theyarecontainedinthelargestbisimulation =A+B+C. Thusa candc b. Byassumption,nowa b.Proposition3.9tellsusthatthereisasubcollection(bj)jJIwithe

jJ{e lj | bljbj}

iI(e li).Hencem

iIli= e

iI(e li) =

iI(m li),nishingtheproof.Theimplication(i) (iv)isduetoS. Pfeier[Pfe99]. Anitelatticeis distributive iit does not containone of the characteristic ve-elementnondistributivesublatticesM3orN5,see([Gra98]). Byseparatelyexcludingthesecases,shealsoobtainedtheconverse(iv) (i)incasethat Lisniteanddistributive.Observe that nonempty weak pullbacks along injective maps, resp. nonemptypullbacksof anarbitrarycollectionof injectivemaps, arealwayspreserved,11without assumingJID. In[GS00], theseconditions havebeenshowntobeequivalent tohomomorphicpreimages of subcoalgebras, resp. arbitraryin-tersectionof subcoalgebras, beingsubcoalgebrasagain. Thus, theseresultsfollowfromcorollaries3.7and3.3.Onemightwonder, whetheringeneral, simulationscouldnothavebeendenedbyjustoneclauseof 3.9sothatcondition(ii)wouldautomaticallybesatisedforarbitrary

-semilattices L. Notice, however, thatwithsuchadenitionwewouldnothavebeenabletoshowthatsimulationsareclosedundercomposition.5 LabelingwithacommutativemonoidIn the denition of the functor L()it is essential that L has arbitrary suprema,i.e. that Lis

-complete. Whentryingtoreplace Lbyanarbitrarycommu-tative monoid M = (M, +, 0), we do not have innite sums available anymore,unlesswhenalmostallsummandsare0. Hence,wemustredenethefunctorbyonlyconsideringmaps: X Mwithnitesupport:Denition5.1Let M = (M, +, 0)beacommutativemonoid. WheneverXis aset andall but nitelymanyelements of afamily(g(x))xXare0, wedenoteitssumby

(g(x) | x X).GivenanysetXandamap: X M,wecallsupp() := {x X | (x) = 0}thesupportof. LetMX:= {: X M | |supp()| < }betheset of all maps fromXtoMwithnitesupport, andfor anymapf: X Y let Mfbethemapdenedonany MXby:y Y. Mf()(y) :=

((x) | x X, f(x) = y).Oneeasilychecksthat Mf()isamapfromY toMwithnitesupport,sousingassociativityandcommutativityof+,oneveriesasbefore:Lemma5.2 M()isa Set-endofunctor.When Misthetwo-elementBooleanalgebra({0, 1}, , 0)then M()isjustthenitepowersetfunctor P().Coalgebrasoftype M(), inthesequelcalled M-coalgebras, mayagainbe viewed as graphs with arcs labeled by elements of M, so we continue usingthe arrow-notation as in the case of L-coalgebras. In particular, if (A, ) is anM-coalgebra, a, a

Aandm M, wecanchoosebetweentheequivalent12notations(a)(a

) = m, or (a, a

) = m, or ama

.For M-coalgebras, thebasiccoalgebraicconstructionscanbeeasilyde-scribed:Lemma5.3Let A = (A, )and B = (B, )be M-coalgebras,then(i) U Aisasubcoalgebraof A,iforall u U,andall a A:uma, m = 0 =a U.(ii) : A Bisahomomorphismiforall a A, b

B:(a)mb

m =

(m

| a

A.am

a

, (a

) = b

).Corollary5.4The intersection of an arbitrary family (U)iIof subcoalgebrasofan M-coalgebra Aisagainasubcoalgebraof A.If: A BisahomomorphismandU Basubcoalgebraof B, then[U] need not be a subcoalgebra of A. This stands in contrast to the situationforlatticelabeledcoalgebras(corollary3.7). Thus,thefunctor M()doesingeneral notpreservenonemptypullbacksalonginjectivemaps([GS00]). Inthe next section, we shall study algebraic conditions on the monoid M whichareresponsibleforsuchpropertiesofthefunctor.We conclude this sectionwithacharacterizationof bisimulations R A Bbetween M-coalgebras Aand B. Forthisweconsidertheelementsof MAas vectors with |A| manycomponents andtheelements of MRas|A| |B|-matriceswithentriesfromM. Fromthedenitionofbisimulationinsection2.3,weobtain:Lemma5.5Let A=(A, )and B=(B, )be M-coalgebras. ArelationR ABisabisimulationiforevery(a, b) Rthereexistsan |A| |B|-matrix(mx,y)withentriesfromMsuchthat:all butnitelymanymx,yare0,mx,y = 0implies(x, y) R,(a)isthevectorofall row-sumsof(mx,y),i.e.x A. (a, x) =

(mx,y | y B),(b)isthevectorofall column-sumsof(mx,y),i.e.y B.(b, y) =

(mx,y | x A).135.1 Positivemonoids.Anycommutativesemigroupcanbeturnedintoacommutativemonoidbysimplyadjoininganewelement 0. Theobtainedmonoidis rather specialthough, it can be internally characterized by the fact that no nonzero elementisinvertible:Denition5.6Amonoidelementm Miscalledinvertibleifthereexistssomem Mwithm+m= 0. Amonoid M = (M, +, 0)iscalledpositiveif0istheonlyinvertibleelement.Ifacommutativemonoidisnot positive, wecanobtainonebyfactoringout the invertible elements, or by deleting all invertible ones, except for 0, foritiseasytocheck:Lemma5.7TheinvertibleelementsofacommutativemonoidformagroupI(M). Factoring Mby the inducedcongruence relationyields the largestpositive factor of M. At the same time, any commutative monoid is theunionofthesubgroupI(M)ofinvertibleelementswithapositivesubmonoidM+.Example5.8Thefollowingmonoidsarepositive:(i) (N, +, 0)(ii) (N \ {0}, , 1)(iii) (L, , 0)forany(semi)latticewith0.5.2 Renablemonoids.We shall needtoconsider afurther monoidconditionwhichwe shall callrenable. For this, let us consider anm n-matrix(ai,j) of monoidele-ments. Consider their row-sums ri=

1jmai,j, andtheir columnsumscj=

1inai,j, thenbyassociativityandcommutativityoneobviouslyhasr1 +. . . +rn= c1 +. . . +cm. Renability is just the inverse condition, that is:Denition5.9Givenm, n N, amonoid Miscalled(m, n)-renable, ifforanyr1, . . . , rm, c1, . . . , cn Mwithr1 + . . . + rm=c1 + . . . + cnonecanndanmn-matrix(ai,j)ofelementsofM,whoserowsumsarer1, . . . , rmandwhosecolumnsumsarec1, . . . , cn.a1,1 a1,nr1... ... am,1 am,nrmc1 cn14Obviously,whenm= 1orn= 1,theconditionisvacuous. Whenm> 1then M is (m, 0)-renable i it is positive. For the remaining cases we prove:Proposition5.10For anym, n>1we have: Acommutative monoidis(m, n)-renable,iitis(2, 2)-renable.Proof. In an (m, n+1)-renement of r1+. . .+rm= c1+. . .+cn+0, we can addcorresponding elements from the last two rows to obtain an (m, n)-renementofr1 +. . . +rm= c1 +. . . +cn,soonedirectionisclear.Theother directionis provedbyaneasyinductionover thenumber ofcolumns,followedbyasimilarinductionoverthenumberofrows. Asahint,weshowhowtogetfrom(2, 2)to(3, 2):Givenr1 + r2 + r3=c1 + c2, usethe(2, 2)renementpropertytonda2 2-matrix(ai,j)withcolumnsumsc1,c2androwsumsr1,(r2 + r3). Nowa2,1 +a2,2= r2 +r3, so there is another 2 2 matrix with row sums r2, r3andcolumnsumsa2,1,a2,1:a1,1a1,2r1a2,1a2,2r2 +r3c1c2andb1,1b1,2r2b2,1b2,2r3a2,1a2,2obviouslynow,thefollowingmatrixsolvestheoriginalproblem:a1,1a1,2r1b1,1b1,2r2b2,1b2,2r3c1c2Asaconsequenceofthisproposition, wemaysimplycall acommutativemonoid renableif it is (2, 2)-renable. Renability does not imply positivity,sincenontrivialabeliangroups,forinstance,arerenable,butnotpositive.Forthenextsection,weshallneedthefollowingobservation,referringtoinnitematrices:Lemma5.11Supposethat Misrenable. GivenX,Y nonemptysets, MXand MYwith

((x) | x X) =

((y) | y Y ). Thereexistsan|X| |Y |-matrix(mx,y)withrowsums

(mx,y |y Y )=(x)andcolumnsums

(mx,y | x X) = (y),whereall butnitelymanymx,yare0.Proposition5.10makesiteasytocheckthatthersttwoinstancesofex-ample 5.8 are renable. In fact, any commutative monoid which is cancellative15andwhichsatisesc, r M.x M.c = x +rorr = x +ciseasilyseentoberenable. Thisalsocoversthecaseof(N, +, 0).Inthecaseof(N \ {0}, , 1), renabilityisaconsequenceofthefactthateveryelementhasauniqueprimefactordecomposition.Renabilitydoes not carryover tosubmonoids. Consider, for instance,thesubmonoid(N \ {0, 2}, , 1)of thepreviousexample, andtherenementproblem 5 6 = 3 10. Any renement would require the prime factor 2, whichisunavailable.Inthecaseofalattice L,weobtainafamiliarproperty:Lemma5.12If Lisalatticewithsmallestelement0,then(L, , 0)isren-ableifandonlyif Lisdistributive.Proof. Given a distributive lattice L and a, b, c, d L with ab = cd, thenwehavearenementa ca dab cb dbc dConversely,if Lisnotdistributive, thenoneofthefollowinglattices,knownasN5,resp. M3,mustbeasublatticeof L(seee.g. [Gra98]):p

PPPPPP p

???????cN5= b M3= abcaq???????nnnnnnq???????

Inbothcases,a b = b c. Supposewehadarenementxyazubb cwithx, y, u, v L. Fromthe table, it follows that u b andu c, sou b c = q a. Also,y a,henceu y= c a. Butc a,bothinM3andinN5.165.3 WeakpullbackpreservationWe nowstudyconditions under whichthe functor M()weaklypreservesnonemptykernelpairs,pullbacksalonginjectives,orarbitrarypullbacks.AfunctorFissaidto(weakly)preservepullbacksalonginjectivemaps,providedfor anyf : XZandg : Y Zwithg injective, a(weak)pullbackoffwithgistransformedbyFintoaweakpullbackofF(f)withF(g). In[GS00], wehaveshownthata Set-endofunctorFweaklypreservesnonemptypullbacksalonginjectivemapsif andonlythepreimage[V ] ofanyF-subcoalgebra V Bunderahomomorphism: A Bisagainasubcoalgebraof A.For L-coalgebras, thisconditionisalwayssatisedaccordingtocorollary3.7. We shall see, however, that this is not necessarilythe case for M-coalgebras. Infactweshall algebraicallycharacterizethosemonoids Mforwhich M()preservesweakpullbacksalonginjectivemaps.Finally,weconsiderpreservationofarbitraryweakpullbacks.Theorem5.13Let M = (M, +, 0)beacommutativemonoid.(i) M()(weakly)preservesnonemptypullbacksalonginjectivemapsi Mispositive.(ii) M()weaklypreservesnonemptykernel pairsi Misrenable.(iii) M()weakly preserves nonempty pullbacks i M is positive and renable.Proof. (i): Assumethat Mispositive, andlet: A Bbeahomomor-phismof M-coalgebrasand Vasubcoalgebraof B. Weneedtoshowthat1[V ] is a subcoalgebra of A. Given a 1[V ], a

/ 1[V ] and ama

, weneedtoshowthatm = 0. Weknowthat(a) V and(a

)/ V ,sobypart(i) of lemma 5.3 we conclude (a)0(a

). Part (ii) of the same lemma thenyields

(n | anx, (x) = (a

)) = 0. Positivity forces each summand to be0,inparticularm = 0.Toprove the converse, let m1, m2Mbe givenwithm1+m2=0.Consider the coalgebra A,given by a point p and two transitions to points q1andq2,labeledwithm1andm2. Let Bconsistoftwopointsrandswithnotransitions(all transitionslabeledwith0). Wegetahomomorphismwith(p)=rand(q1)=(q2)=s. Now {r}isasubcoalgebraof B, andtheassumptionforces1{r}= {p}tobeasubcoalgebraof A, butthisimpliesm1= m2= 0.pm1

m2

????????r0

A =

= Bq1q2 sAslightmodicationof thisconstructionalsogivesusthebackwarddi-17rectionof(ii): AssumethatFweaklypreservesnonemptykernelpairs,thenkernelsofhomomorphismsarebisimulations. Supposem1 + m2=s1 + s2inM. We take a copy A

of A as above, but we label the arcs of A

with s1ands2. If B

is obtainedfrom Bby changing theedgelabel to m1 +m2= s1 +s2,thereisanobvioushomomorphism: A + A

B

. Itskernel mustbeabisimulation, so lemma 5.5, provides us with a renement of m1+m2= s1+s2.We combine the if-directions of (ii) and (iii): Assume that Mis renable.Givenhomomorphisms : A C,and: B C,weneedtoshowthatpb(, ) := {(a, b) | (a) = (b)}isabisimulationbetween Aand B.Let(a, b) pb(, )begiven. Weshalldenean |A| |B|-matrix(mx,y)withentriesfromM,satisfyingtheconditionsoflemma5.5.For any c [A] [B], put X:= 1({c}) and rx:= (a, x), i.e. arxx,foranyx X. Similarly, Y :=1({c})andcy=(b, y)foreveryy Y .With lemma 5.11 we obtain an |X| |Y | matrix (mcx,y) with row sums (rx)xXandcolumnsums(cy)yY . Observethatwecanachievethatforallbutnitelymanycis(mcx,y)the0-matrixallbutnitelymanyentriesineach(mcx,y)are0.The nal |A| |B|-matrix (mx,y) is obtained by putting all (mcx,y) togetherandllingupwithzeroes: 0 0 (mcx,y) 0 0 mx,y:=

mcx,yif(x) = c = (y),0 otherwise.Byconstruction, mx,y =0implies (x, y) pb(, ). Moreover, all butnitelymanyentriesof(mx,y)arezero. Supposenowthatara

. Weneedtoshowthatthea

-throwsumisr,i.e.

(ma

,b | b B) = r.Letc:=(a

). If 1({c}) = then

(ma

,b | b B)=

(mca

,b | b B)=r. If 1({c})= (thiscasecannothappenin(ii)), weshall invokepositivitytoshowr = 0. Specically,forswith(a)scwehave

(m | ama

, (a

) = c) = s =

(m | bmb

, (b

) = c) = 0.Hencer +u = 0forsomeu M,whencer = 0.18A more elegant way to see (iii) is to directly conclude it from (i) and (ii),byinvokingthefollowinglemmafromthesecondauthorsthesis:Lemma5.14[Sch01] A Set-endofunctor weakly preserves pullbacks i it weaklypreserveskernel pairsandpullbacksalonginjectivemaps.6 DiscussionOne motivation for this study was to provide a repository of examples of Set-endofunctors with particular combinations of preservation properties. This weachieve by parameterizing a certain class of functors with algebraic structuresandtranslatingthefunctorial properties intocorrespondingalgebraiclaws.For instance, theorem5.13canbeusedtoobtainanexampleof afunctorweakly preserving nonempty kernel pairs, but not weakly preserving nonemptypullbacks: Simplychoosefor Manynontrivialabeliangroup.Of course, L-coalgebras and M()-coalgebras as L-, resp. M-labeled tran-sitionsystems areof independent interest. L-valuedsets andrelations areconsideredbyGoguenin[Gog67]. Inthebook[FS90], FreydandScedrovconsider the following operations on L-valued relations R : AB L andS: B C L:(R S)(a, c) :=

{R(a, b) S(b, c) | b B}.WhenL = {0, 1},thisagreeswiththefamiliarcompositionofrelations. Theauthorsremarkthatthisoperationisassociativei Lisjoininnitelydis-tributive(JID),alsocalledalocalein[Bor94].L.Moss,in[Mos99],considersthefollowingsubfunctorof R()+:Q(X) := {f: X R | supp(f)nite,

xXf(x) = 1}.Coalgebras of this functor are stochastic transition systems([Mos99],[dVR99]).MossinvokestheRow/Column-theoremfor R+, whichistosaythat Ris(m, n)-renable for each m, n > 1, to show that Q weakly preserves pullbacks.(HeattributestheproofoftheRow/ColumntheoremtoSaleyAliyari).We have borrowedthe termrenable fromaclassical line of algebraicinvestigation, askingfor theexistenceof uniqueproduct decompositions ofnitealgebras. If A1 . . . Am = B1 . . . Bnaretworepresentationsofthesamenitealgebraasaproductof indecomposables, onewouldliketoconcludem = nandBi = A(i),forsomepermutation.Itiseasytocomeupwithexamplesof nitealgebrasthatdonothaveuniquedecompositions. Insuchcasesitmaystill bepossibletoproveare-nement property: Given that A1. . . Am = B1. . . Bn, then each factor19canbefurtherdecomposedintoaproductofsmalleralgebrasQi,j, untilonehasthesamecollectionoffactorsQi,jontheleftandontherightside.J.D.H. Smithhas remindedus of aresult of B. JonssonandA. Tarski[JT47]whichstatesthataclassofalgebras,amongstwhoseoperationsareabinaryoperation+andaconstant0,whichisneutralwithrespectto+andidempotent for all fundamental operations, has the (m, n)-renement property.This means that the class of all nite Jonsson-Tarski algebras, with the monoidstructure given by the direct product () and with {0} as neutralelement,isarenablemonoid.JonssonandTarski neededthe operations +and0torepresent directproductsasinnerproducts. Withoutsomesuchassumptions, renementis not possible ingeneral, for renement implies cancellability: A B=A C = B = C. When Ahas a1-element subalgebra, cancellabilityholds, accordingtoL. Lovasz([Lov67]), butotherwise, oneneedstoreplaceisomorphybytheweakernotionof isotopy, see[Gum77]. Arenementtheoremuptoisotopyforalgebrasincongruencemodularvarietieshasbeenprovedin[GH79].References[Bor94] F. Borceux, Handbook of categorical algebra 1: basic category theory,Cambridge University Press, 1994.[dVR99] E.P. de Vink and J.J.M.M. Rutten, Bisimulation for probabilistic transitionsystems: acoalgebraic approach, Theoretical Computer Science (1999),no. 211, 271293.[FS90] P. Freyd and A. Scedrov, Categories, allegories, Elsevier, 1990.[GH79] H.P. Gummand C. Herrmann, Algebras in modular varieties: Baerrenements, cancellationandisotopy, HoustonJournal of Mathematics5 (1979), no. 4, 503523.[Gog67] J.A. Goguen, L-fuzzy sets, Journal of Mathematical Analysis andApplications (1967), no. 18, 145174.[Gra98] G. Gratzer, General lattice theory, Birkhauser Verlag, 1998.[GSa] H.P. Gumm and T. Schroder, Coalgebras of bounded type, Submitted.[GSb] H.P. Gumm and T. Schroder, Products of coalgebras, Algebra Universalis,to appear.[GS00] H.P. Gummand T. Schroder, Coalgebraic structure fromweak limitpreserving functors, CMCS (2000), no. 33, 113133.20[Gum77] H.P.Gumm,Acancellationtheoremfornitealgebras,Coll.Math.Soc.Janos Bolyai (1977), 341344.[Gum98] H.P. Gumm, Functors for coalgebras, Algebra Universalis, to appear, 1998.[JT47] B. Jonsson and A. Tarski, Direct decompositions of nite algebraic systems,Notre Dame Mathematical Lectures (1947), no. 5.[Lov67] L. Lovasz, Operations with structures, Acta. Math. Acad. Sci. Hungar. 18(1967), 321328.[Mos99] L.S. Moss, Coalgebraic logic, Annals of Pure and Applied Logic 96 (1999),277317.[Pfe99] S. Pfeier, Funktoren f ur Coalgebren, Master Thesis, Universitat Marburg,1999.[Rut00] J.J.M.M. Rutten, Universal coalgebra: atheoryof systems, TheoreticalComputer Science (2000), no. 249, 380.[Sch01] T. Schroder, Coalgebren und Funktoren, PhD-Thesis, Philipps-UniversitatMarburg, 2001.21