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