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Operads, quasiorders and regular languages S. Giraudo ∗ J.-G. Luque† L. Mignot ‡ F. Nicart§ 4 1 0 2 Abstract n Wegeneralizetheconstructionofmultitildesintheaimtoprovidemultitildeoperatorsforregular a languages. Weshowthattheunderliyingalgebraicstructureinvolvestheactionofsomeoperads. J An operad is an algebraic structure that mimics the composition of the functions. The involved 9 operads are described in terms of combinatorial objects. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counter-parts are investigated. One ] of these operads acts faithfully on languages in the sense that two different operators act in two L differentways. F . s c 1 Introduction [ 1 FollowingtheChomsky-Schützenbergerhierarchy[5],regularlanguagesaredefinedtobetheformal v languagesthataregeneratedbyType-3grammars(alsocalledregulargrammars). Theseparticular 0 languageshavebeenstudiedfromseveralyearssincetheyhavemanyapplicationstopatternmach- 1 ing,compilation,verification,bioinformatics,etc. Theirgeneralizationasrationalserieslinksthem 0 tovariousalgebraicorcombinatorialtopics: enumeration(manipulationsofgeneratingfunctions), 2 rationalapproximation(forinstance Pade approximation),representationtheory(moduleviewed 1. asautomaton),combinatorialoptimization((max,+)-automata),etc. 0 Oneoftheirmaininterestisthattheycanberepresentedbyvarioustools: regulargrammars,au- 4 tomata,regularexpressions,etc. Whilstregularlanguagescanberepresentedbybothautomataand 1 regularexpressions[8],thesetoolsarenotequivalent. Indeed,EhrenfeuchtandZeiger[6]showed v: a one parameter family of automata whose shortest equivalent regular expressionshave a width i exponentiallygrowingwiththenumbersofstates. Notethat,itispossibletocomputeanautomaton X fromaregularexpressionEsuchthat the number ofits states isalinear functionof the alphabet r width(i.e. thenumberofoccurrencesofalphabetsymbols)ofE[1,4,7,13]. a Intheaimtoincreaseexpressivenessofexpressionsforaboundedlength,Caronetal.[3]introduced the so-called multi-tilde operators and applied it to representfinite languages. Investigating the equivalenceoftwomulti-tildeexpressions,theydefineanaturalnotionofcompositionwhichen- dowsthesetofmulti-tildeoperatorswithastructureofoperad.Thisstructurehasbeeninvestigated in[10]. Originatingfromthealgebraictopology[2,12],operadtheoryhasbeendevelopedasafieldof abstractalgebraconcernedbyprototypicalalgebrasthatmodelclassicalpropertiessuchascommu- tativityandassociativity[9]. Generallydefinedintermsofcategories,thisnotioncanbenaturally appliedtocomputerscience. Indeed,anoperadisjustasetofoperations,eachonehavingexactly one outputand afixedfinite number of inputs, endowedwith the compositionoperation. So an operadcanmodelthe compositionsoffunctions occurringduringtheexecutionofaprogram. In ∗[email protected];IGMLABINFOUMR8049,Laboratoired’informatiqueGaspardMonge,UniversitéParis- Est,CitéDescartes,BâtCopernic5,bdDescartesChampssurMarne77454Marne-la-ValléeCedex2FRANCE. †[email protected];LaboratoireLITIS-EA4108UniversitédeRouen. Avenuedel’Université-BP876801 Saint-Étienne-du-RouvrayCedex ‡[email protected]; LaboratoireLITIS-EA4108UniversitédeRouen. Avenuedel’Université-BP876801 Saint-Étienne-du-RouvrayCedex §fl[email protected]; Laboratoire LITIS - EA 4108 Université de Rouen. Avenue de l’Université - BP 8 76801 Saint-Étienne-du-RouvrayCedex 1 termsof theoreticalcomputerscience, this canbe representedbytreeswith branching rules. The wholepointoftheoperadsinthecontextofthecomputerscienceisthatthisallowstousedifferent toolsandconceptsfromalgebra(forinstance: morphisms,quotients,modulesetc.). Intheaimtoillustratethispointofview,letusrecallthemainresultsofourpreviouspaper[10]. Inthispaper,wefirstshowedthatthesetofmulti-tildeoperatorshasastructureofoperad.Weused theconceptofmorphismintheaimtochoosetheoperadallowingustodescribeinthesimplestway agivenoperationoraproperty.Forinstance,theoriginaldefinitionoftheactionofthemulti-tildes onlanguagesisrathercomplicated. But,viaanintermediateoperadbasedonsetofbooleanvectors, theactionwasdescribedinamorenaturalway. Inthesameway,theequivalenceproblemisclearer whenaskedinaoperadbasedonantisymmetricandreflexiverelationswhichisisomorphictothe operad of multi-tildes: two operators are equivalent if and only if they have the same transitive closure.Thetransitiveclosurebeingcompatiblewiththecomposition,wedefinedanoperadbased onpartialorderedsetsasaquotientofthepreviousoperadandweshowedthatthisrepresentation isoptimalinthesensethattwodifferentoperatorsactintwodifferentwaysonlanguages. Thisnot onlyhelpstoclarifyconstructionsbutalsotoasknewquestions. Forinstance,howmanydifferent ways do k-ary multi-tildes act on languages? Precisely, the answer is the number of posets on {1,...,k+1}thatarecompatiblewiththenaturalorderonintegers. The aim of this paper is to generalize the construction to regular languages. We investigate several operads (based on double multi-tildes, antireflexive relations or quasiorders)allowing to representaregularlanguageasak-aryoperatorOactingonak-upletofsymbols(α ,...,α )where 1 k the α aresymbolsor∅. The operatorsgeneralizethe multi-tildesand the investigatedproperties i involvetheoperads. Thepaperisorganizedasfollows. FirstwerecallinSection2severalnotionsconcerningoperad theoryandmulti-tildeoperations.InSection3,weremarkthatmanyoftheoperadsinvolvedin[10] andinthispaperhavesomecommonproperties. Moreprecisely,theycanbedescribedcompletely bymeansof“shifting”operations. Thisleadstothedefinitionofthecategoryofprecompositions together with a functor to the category of operads. Also we define and investigate the notion of quotientofprecompositions.Thesestructuresserveasmodelfortheoperadsdefinedinthesequel. Intheaimtoillustratehowtousethesetools,werevisit,inSection4,theoperadsdefinedin[10]and describethemintermsofprecompositions. InSection5,wedefinethedoublemulti-tildeoperad DT asthegradedtensorsquareofthemulti-tildeoperad. Weconstructalsoanisomorphicoperad ARefbasedonantireflexiverelationsandaquotientbasedonquasiordersQOSet. InSection6,we describetheactionoftheoperadsonthelanguages.Inparticular,weshowthatanyregularlanguage canbewrittenasO (α ,...,α )wheretheα arelettersor∅andO isak-aryoperationbelongingto k 1 k i k ARef,DT orQOSet. Finally,weprovethattheactionofQOSetonregularlanguagesisfaithful,that istwodifferentoperatorsactintwodifferentways. 2 Some Combinatorial Operators in Language Theory Werecallheresomebasicnotionsaboutthetheoryofoperadsandsetournotationsforthesequel ofthepaper. Inparticular,werecallwhatareoperads,freeoperads,andmodulesoveranoperad. Weconcludethissectionbypresentingtheoperadofmulti-tildesintroducedin[10]. 2.1 Whatis anoperad? Operadsarealgebraicgradedstructureswhich mimicthe compositionof n-aryoperators. Letus recallthemaindefinitionsandproperties. LetP = n∈N\{0}Pn beagradedset( meansthatthe setsaredisjoint);theelementsofPnarecalledn-aryoFperators.ThesetPisendowFedwithfunctions (calledcompositions) ◦:Pn×Pk1×···×Pkn →Pk1+···+kn. Thepair(P,◦)isanoperadifthecompositionssatisfy: 1. Associativity: p◦(p ◦(p ,...,p ),...,p ◦(p ,...,p ))=(p◦(p ,...,p ))◦(p ,...,p ,...,p ,...,p ). 1 1,1 1,k1 n n,1 n,kn 1 n 1,1 1,k1 n,1 n,kn 2 2. Identity: Thereexistsaspecialelement1∈P suchthat 1 p◦(1,...,1)=1◦p=p. Forconvenience,manyauthorsuseanalternativedefinitionofoperadsinvolvingpartialcomposi- tions. Apartialcomposition◦ isamap(seee.g.[9]) i ◦i :Pm×Pn →Pm+n−1, definedby ×i−1 × m− i p ◦ p :=p ◦(1,...,1,p ,1,...,1) 1 i 2 1 2 z}|{ z}|{ for1≤i≤n. Letp ∈P ,p ∈P andp ∈P . Whencestatedintermsofpartialcompositions,theassociativity 1 m 2 n 3 q conditionsplitsintotworules: 1. Associativity1: If1≤ j<i≤nthen (p1◦ip2)◦jp3=(p1◦jp3)◦i+q−1p2. 2. Associativity2: If j≤nthen p1◦i(p2◦jp3)=(p1◦ip2)◦i+j−1p3. Notethatthecompositionsarerecoveredfromthepartialcompositionsbytheformula: p◦(p ,...,p )=(...(p◦ p )◦ p )◦ ...p )◦ p . 1 n n n n−1 n−1 2 2 1 1 Thereaderscouldreferto[9,11]foramorecompletedescriptionofthestructures. Consider two operads (P,◦) and (P′,◦′). A morphism is a graded map φ : P → P′ satisfying φ(p ◦ p ) = φ(p )◦′ φ(p ) for each p ∈ P , p ∈ P and 1 ≤ i ≤ m. Let (P,◦) be an operad, 1 i 2 1 i 2 1 m 2 n P′= P′ beagradedset.SupposethatP′isendowedwithbinaryoperators◦′:P′ ×P′ →P′ n n i m n n+m−1 andthSereexistsasurjectivegradedmapη : P → P′ satisfyingη(p1◦ip2) = η(p1)◦′i η(p2). Theset P′isautomaticallyendowedwithastructureofoperad(P′,◦′). Indeed,itsufficestoshowthatthe associativityrulesaresatisfied:Letp′ ∈P′,p′ ∈P′andp′ ∈P′. Sincetheηissurjective,thereexist 1 m 2 n 3 q p ∈P ,p ∈P ,p ∈P suchthatη(p)=p′fori=1...3. Hence, 1 m 2 n 3 q i i p′ ◦′(p′ ◦′p′) = η(p )◦′(η(p )◦′η(p )) 1 i 2 j 3 1 i 2 j 3 = η(p ◦ (p ◦ p )) 1 i 2 j 3 = η((p1◦jp3)◦i+q−1p2) = (η(p )◦′η(p ))◦′ η(p ) 1 j 3 i+q−1 2 = (p′1◦jp′3)◦i+q−1p′2. Thisprovesthefirstruleofassociativity. Thesecondrulescanbeprovedinthesameway. Further- morethe imageη(1) is the identity inP′. So (P′,◦′) is an operad. Remark that ifη is a bijection then p′ ◦′p′ =η(η−1(p )◦ η−1(p )). (1) 1 i 2 1 i 2 IfQ ⊂ P,thesuboperadofPgeneratedbyQisthesmallestsubsetofPcontainingQand1which is stable by composition. Let G = (G ) be a collection of sets. The set Free(G) is the set of k k n planar rootedtrees with n leaves with labeled nodes where nodes with k childrenare labeled by the elements of G . The freeoperad on G is obtained by endowing the set Free(G) = Free(G) k n n withthecompositionp1◦ip2 whichconsistsingraftingtheithleafofp1 withtherootSofp2. Note thatFree(G)containsacopyofGwhichisthesetofthetreeswithonlyoneinnernode(theroot) labeled by elements of G; for simplicitywe will identifyit with G. Clearly, Free(G) is generated byG. Theuniversalitymeansthatforanymapϕ : G → Pitexistsauniqueoperadicmorphism φ:Free(G)→Psuchthatφ(g)=ϕ(g)foreachg∈G. 3 Let ≡ be a graded equivalence relation on P. The relation ≡ is a congruence, if for any p ,p ,p′,p′ ∈ P we have p ≡ p′ and p ≡ p′ implies p ◦ p ≡ p′ ◦ p′. Hence, this natu- 1 2 1 2 1 1 2 2 1 i 2 1 i 2 rallyendowsthequotientP/ withastructureofoperad. Notethatifφ : P → P′ isasurjective ≡ morphism of operads then the equivalence defined by p ≡ p if and only if φ(p ) = φ(p ) is a 1 2 1 2 congruence. Let(P,◦)and(P′,◦′)betwooperads.ThegradedsetT(P,P′):= n∈NTn(P,P′),withTn(P,P′):= Pn ×P′n, is naturally endowed with a structure of operad whereSthe composition is defined by (p ,p′)◦(p ,p′):=(p ◦p ,p′◦′p′)withp ∈P ,p ∈P ,p′ ∈P′ ,p′ ∈P and1≤i≤k .Consider 1 1 i 2 2 1 i 2 1 i 2 1 k1 2 k2 1 k1 2 k2 1 asetStogetherwithanactionofanoperadP. Thatis,foreachp∈P wedefineamapp:Sn →S. n WesaythatSisaP-moduleiftheactionofPiscompatiblewiththecompositioninthefollowing sense: foreachp1 ∈Pm,p2∈Pn,1≤i≤m,s1,...,sm+n−1 ∈Sonehas: p1(s1,...,si−1,p2(si,...,si+n−1),si+n,...,sm+n−1)=(p1◦ip2)(s1,...,sm+n−1). Furthermore, if for each k > 0 and p , p′ ∈ P there exists a ,...,a ∈ S such that p(a ,...,a ) , k 1 k 1 k p′(a ,...,a )thenwesaythatthemoduleSisfaithful. 1 k 2.2 Multi-tildesand relatedoperads In[10],wehavedefinedseveraloperads.Letusrecallbrieflythemainconstructions. Firstwedefined the operad T = T of multi-tildes. A multi-tilde of T is a subset of {(x,y) : 1 ≤ x ≤ y ≤ n}. n n n Notethat nmeaFnsthatthesamesetbelongingintwodifferentgradedcomponentsTnandTmare consideredFasdifferentoperators. Foranypair(x,y)wedefine 1. ≫k (x,y)=(x+k,y+k) (x,y) ify<k, 2. ‘n,k(x,y)= (x,y+n−1) ifx≤k≤ y,  ≫n−1(x,y) otherwise. Theactionsofthetwooperatorsareextendedtothesetofpairsby 1. ≫k (E)={≫k (x,y):(x,y)∈E}, n,k n,k 2. ‘(E)={‘(x,y):(x,y)∈E}. Weshownthefollowingresult: Theorem1([10]). ThesetT endowedwiththepartialcompositions T ×T → T m n n+m−1 ◦i : T1◦iT2 = ‘n,i (T1)∪≫i−1(T2), isanoperad.  Wealsodefinetheoperators n,i x ifx≤i, „(x)= ( x+n−1 otherwise n,i n,i n,i n,i n,i „(x,y)=(„(x),„(y))and„(E)={„(x,y):(x,y)∈E}. Theoperad(T,◦)isisomorphictoanotheroperad(RAS,^)whoseunderlyingsetisthesetRAS= RAS where RAS denotesthe setof Reflexive and AntisymmetricSubrelations of the natural n n n oFrder≤on{1,...,n+1}. ThepartialcompositionsofRASaredefinedby R ^R =„n,i (R )∪≫i−1(R ), 1 i 2 1 2 ifR ∈ RAS andR ∈ RAS . TheisomorphismbetweenT andRASsendsT ∈ T to{(x,y+1) : 1 m 2 n n (x,y)∈T}∪{(x,x):x∈{1,...,n+1}}. See[10]formoredetails. 4 3 Breaking operads Theobjectiveofthissectionistointroducenewalgebraicobjects, namelythe precompositions. We presenthereafunctorfromthecategoryofprecompositionstothecategoryofoperads.Weshalluse thisfunctorinthesequeltoreconstructsomealreadyknownoperadsandtoconstructnewones. 3.1 Precompositions Weconsiderthemonoid definedbygenerators{î,k :i∈Z,k∈N\{0}}andrelations: e î,k =ˆ0,k foranyi<0. (2) î,1 =ˆ0,1 =1 foranyi. (3) e î,k ˆj,k′ = j+ˆk−1,k′î,k ifi≤ jori,j≤0, (4) î+j,kî,k′ =i,kˆ+k′−1if0≤ j<k′. (5) Let(S,⊕)beacommutativemonoidendowedwithafiltrationS= n∈N\{0}SnwithS1 ⊂···⊂Sn ⊂ ··· andaunity1S ∈S1. S Aprecompositionisamonoidmorphism◦: →Hom(S,S)satisfying: e ◦(î,k ):Sn→Sn+k−1 (6) i,k ◦ ˆ | =Id ifn<i (7) Sn Sn (cid:18) (cid:19) i,k i,k where| denotestherestrictiontoS .Forsimplicitywedenote‘:=◦(ˆ).Let◦: →Hom(S,S) Sn n e and⊲: →Hom(S′,S′)betwoprecompositions.Amapφ:S→S′isaprecompositionmorphism e from◦to⊲ifandonlyifitisamonoidmorphismsatisfying φ:S →S′ (8) n n ⊲k→,n(φ(x))=φ ‘k,n(x) . (9) (cid:18) (cid:19) WedenotebyHom(◦,⊲)thesetofprecompositionmorphismfrom◦to⊲. Let◦ : → Hom(S,S),⊲ : → Hom(S′,S′)and^ : → Hom(S′′,S′′)bethreeprecompositions e e e together with φ ∈ Hom(◦,⊲) and ϕ ∈ Hom(⊲,^). Remark that the compositionϕφ : S → S′′ is a morphismsendingS toS′′andsatisfying n n ϕφ ‘i,k (x) =ϕ ⊲i→,k φ(x) =„i,k ϕφ(x) (cid:18) (cid:19) (cid:18) (cid:16) (cid:17)(cid:19) (cid:16) (cid:17) foreachx∈Sandeachi∈Zandk∈N\{0}. Hence,ϕφ∈Hom(◦,^). Foreachprecomposition◦ : → Hom(S,S)wedefineId := Id . Clearly,Id ∈ Hom(◦,◦)andfor e ◦ S ◦ eachφ∈Hom(◦,◦)wehaveφId =Id φ=φ. ◦ ◦ Now,ifφ∈Hom(◦,⊲),ϕ∈Hom(⊲,^)andψ∈Hom(^,(cid:3))thenwehave,straightforwardly,(ψϕ)φ= ψ(ϕφ). Hence: Proposition 1. The family PreComp of precompositions endowed with the arrows Hom(◦,⊲) for each ◦,⊲∈PreCompisacategory. 3.2 From precompositions tooperads We considera precomposition◦ : → Hom(S,S). For simplicitywe denote ≫k := 0‘,k+1 (≫k for e ◦ shortwhenthereisnoambiguity). FromSwedefineS := {a(k) : s ∈ S }andS := S . Foreach k s k k k as(k)∈Skweset S ‘i,k′ (a(k)):= ‘i,k′ (s) ifi≤k s   s otherwise,  5 and≫k′ (a(k))=0‘,k′+1(a(k)). s s Nowforeach1 ≤ i ≤ kwedefinethebinaryoperator◦i : Sk×Sk′ → Sk+k′−1 byas(k)◦ias(k′′) := a(sk′′+k′−1) wheres′′=‘i,k′ (as(k))⊕≫i−1(as(k′′))∈Sk+k′−1. Proposition2. ThesetSendowedwiththepartialcompositions◦ isanoperad. i Proof. Firstremarkthattheidentityofthestructureis1S:=a(1). Indeed: 1S 1. Wehave1S◦1as(k)=as(k′)withs′=‘1,k (a(11S))⊕≫0 (as(k)). But,‘i,k (a(11S))=‘i,k (1S)=1Sand≫0 (as(k))=s (because≫0 =◦ 1e ). Hence,s′=sand1S◦1as(k)=as(k). (cid:16) (cid:17) 2. Let1≤ i≤ k. WehaveaT(k)◦i1S =aT(k′) withs′ =‘i,1 (as(k))⊕≫k−1(a(11S)). But,‘i,1 (as(k))= ‘0,1(as(k))= s (becauseˆ0,1 =1e)and≫k−1(a(11S))=1S. Hence,s′=sandas(k)◦i1S=as(k). Now,letusprovethetwoassociativityrules: 1. Let k,k′,k′′,i,j be five integers such that 1 ≤ i < j ≤ k. Consider also s ∈ Sk, s′ ∈ Sk′ and s′′ ∈Sk′′. Applyingthedefinitionofthecomposition◦i,wefind: (as(k)◦jas(k′′))◦iaks′′′′ =as(k(3+)k′+k′′−2) where s(3) =‘i,k′′(a(k+k′−1))⊕≫i−1(a(k′′)), s(4) s′′ ands(4)=‘j,k′(a(k))⊕≫j−1(a)(k′). s s′ Sincei≤k,wehave‘i,k′′(a(k+k′−1))=‘i,k′′ s(4) . Furthermores(4) =‘j,k′(s)⊕≫j−1(s′)since j≤kand s(4) 0≤k′. Hence, (cid:16) (cid:17) ‘i,k′′(a(k+k′−1)) = ‘i,k′′(‘j,k′(s)⊕≫j−1(s′))=‘i,k′′ ‘j,k′(s) ⊕‘i,k′′ ≫j−1(s′) s(4) ! ! = ◦ î,k′′ˆj,k′ (s)⊕◦ î,k′′ˆ0,j (s′) ! ! From(4),wehaveî,k′′ˆj,k′ = j+kˆ′′−1,k′î,k′′. Inthesameway,(4)givesî,k′′ˆ0,j =ˆ0,j i−ˆj+1,k′′andsince i−j+1≤0,therule(2)givesî,k′′ˆ0,j =ˆ0,k′′ˆ0,j =0,jˆ+k′′−1= j+ˆk′′−2from(5). Onededuces s(3)=◦ j+kˆ′′−1,k′î,k′′ (s)⊕ j+≫k′′−2 (T′)⊕ ≫i−1 (T′′). (10) ! ! (cid:18) (cid:19) Nowexamine(as(k)◦ias(k′′′′))◦j+k′′−1aks′′ =aks˜(+3)k′+k′′−2with s˜(3)= j+k‘′′−1,k′(a(k+k′′−1))⊕j+≫k′′−2(a(k′)) s˜(4) s′ and s˜(4) = ‘i,k′′(a(k)) ⊕≫i−1(a(k′′)). Since i ≤ k and 0 ≤ k′′ we deduce s˜(4) = ‘i,k′′(s) ⊕≫i−1(s′). s s′′ Furthermore,since j≤kand0≤k′′,wehave s˜(3) = j+k‘′′−1,k′(s˜(4))⊕j+≫k′′−2(T′)=◦ j+kˆ′′−1,k′î,k′′ (s)⊕◦ j+kˆ′′−1,k′ˆ0,i (s′′)⊕j+≫k′′−2s′. ! ! j+k′′−1,k′ 0,i 0,i j−i+k′′,k′ j−i+k′′,k′ But ˆ ˆ=ˆ ˆ and ‘ (s′′)=s′′from(7)since j>i. Hence,weobtain s˜(3)=◦ j+kˆ′′−1,k′î,k′′ (s)⊕≫i−1(s′′)⊕j+≫k′′−2(s′)=s(3). ! Hence, (as(k)◦jas(k′′))◦iaks′′′′ =(as(k)◦ias(k′′′′))◦j+k′′−1aks′′. 6 2. Letk,k′,k′′,i,jbefiveintegerssuchthat1≤i≤k′,1≤ j≤kand1≤k,k′,k′′. Considers∈S , k s′ ∈Sk′ ands′′∈Sk′′. Applyingthedefinitionof◦i,onehas a(k)◦ (a(k′)◦ a(k′′))=a(k+k′+k′′−2). s j s′ i s′′ s(3) wheres(3) = j,k′‘+k′′−1(a(k))⊕≫j−1(a(k′+k′′−1))ands(4) = ‘i,k′′(a(k′))⊕≫i−1(a(k′′)). Sincei ≤k′ and0≤ k′′, s s(4) s′ s′′ weobtains(4)=‘i,k′′(s′)⊕≫i−1(s′′). Furthermore,j≤kand0≤k+k′−1imply s(3) = j,k′‘+k′′−1(s)⊕≫j−1(s(4)) = j,k′‘+k′′−1(s)⊕◦ ˆ0,j î,k′′ (s′)⊕◦ ˆ0,j ˆ0,i (s′′) ! ! = j,k′‘+k′′−1(s)⊕◦ i+ˆj−1,k′′ˆ0,j (s′)⊕i≫+j−2(s′′). ! Now, letusexamine: (as(k)◦jas(k′′))◦i+j−1 as(k′′′′) = aks˜(+3)k′+k′′−2 withs˜(3) = i+‘j−1,k′′(as(˜k(4+)k′−1))⊕i+≫j−2(as(k′′′′)) and s˜(4) = ‘j,k′(a(k))⊕≫j−1(a(k′)). Since j ≤ k and 0 ≤ k′ we have s˜(4) = ‘j,k′(s)⊕≫j−1(s′). Since, s s′ i+j−1≤k+k′−1and0≤k′′,weobtain s˜(3) = ◦ i+ˆj−1,k′′ˆj,k′ (s)⊕◦ i+ˆj−1,k′′ˆj−1 (s′)⊕i≫+j−2(s′′) ! ! i+j−1,k′′ j,k′ j,k′+k′′−1 Buti−1<k′implies ˆ ˆ= ˆ (eq. (5). Hence, s˜(3)= j,k′‘+k′′−1(s)⊕◦ i+ˆj−1,k′′ˆ0,j (s′)⊕i+≫j−2(s′′)=s(3) ! Hence, as(k)◦j(as(k′′)◦ias(k′′′′))=(as(k)◦jas(k′′))◦i+j−1as(k′′′′). Thecompositions◦ satisfythetwoassertionsrulesandadmitaunity. ThesetShasastructureof i operad. (cid:3) WedefineOP(◦):=(S,◦)asdefinedintheconstruction. Letφ∈Hom(◦,⊲),wedefine i φOP:OP(◦)→OP(⊲) by φOP(a(k))=a(k) . s φ(s) Theorem2. ThearrowOP:PreComp→Operadwhichassociateswitheachprecomposition◦theoperad OP(◦)andtoeachhomomorphismφ∈Hom(◦,⊲)theoperadicmorphismφOPisafunctor. Proof. Wehavetoprovethreeproperties 1. OPsatisfiestheequality: IdOP=Id . ◦ OP(◦) Thisisstraightforwardfromthedefinition. 2. EachφOPisamorphismofoperad. Indeed,letφ∈Hom(◦,⊲),itsufficestocomputeφOP(a(k1)◦ s1 i a(k2))fors ∈S ,s ∈S and1≤i≤k . Wehave s2 1 k1 2 k2 1 φOP(a(k1)◦ a(k2))=a(k1+k2−1) s1 i s2 s3 where s = φ ‘i,k1(T )⊕‘0,i−1(T ) 3 1 2 ! = ⊲i→,k1 φ(s ) ⊕0⊲,→i−1 φ(s ) . 1 2 (cid:16) (cid:17) (cid:16) (cid:17) 7 Hence φOP(a(k)◦ a(k′)) = a(k) ⊲ a(k′) s1 i T2 = φφO(sP1()a(ki))φ⊲(s2φ)OP(a(k′) ). s1 i φ(s2) WededucethatφOPisanoperadicmorphism. 3. OP is compatible with the composition of homomorphisms. Indeed, let φ ∈ Hom(◦,⊲) and ϕ∈Hom(⊲,^). Foranys∈S ,wehave k ϕOPφOP(a(k))=ϕOP a(k) =a(k) =(ϕφ)OP a(k) . s φ(s) ϕ(φ(s)) s (cid:16) (cid:17) (cid:16) (cid:17) Wehavethenshownthat(ϕφ)OP=ϕOPφOP. Hence,thearrowOPsatisfiesthethreerequiredpropertiestobeafunctor. (cid:3) 3.3 Quotientsof precompositions Let ◦ : → Hom(S,S) be a precompositionand γ : S → S be an idempotent (γ2 = γ) monoid e i,k i,k morphismsendingS toS andsatisfying: ‘γ=γ‘. k k Wedefineγ:S→Sbyγa(k) =a(k). s γs Proposition3. Thetwofollowingconditionshold: 1. Foreachs∈Sk,s′∈Sk′ and1≤i≤k: γ γ(a(k))◦ γ(a(k′)) =γ a(k)◦ a(k′) s i s′ s i s′ (cid:16) (cid:17) (cid:16) (cid:17) 2. γ(s )=γ(s′)andγ(s )=γ(s′)impliesγ(a(k)◦ a(k′))=γ(a(k)◦ ak′) 1 2 2 s1 i s2 s′1 i s′2 Proof. 1. Wehaveγ(a(k))◦ γ(a(k′))=a(k)◦ a(k′)=a(k+k′−1)with s i s′ γs i γs′ s′′ s′′=‘i,k′ (a(k))⊕‘0,i (a(k′))=‘i,k′ (γs)⊕‘0,i (γs′)=γ ‘i,k′ (s)⊕‘0,i (s′) . γs γs′ ! Hence γ γ(a(k))◦ γ(a(k′)) = γ γ ‘i,k′ (s)⊕‘0,i (s′) =γ ‘i,k′ (s)⊕‘0,i (s′) s i s′ !! ! (cid:16) (cid:17) = γ ‘i,k′ (a(k))⊕‘0,i (a(k′)) =γ a(k)◦ ak′ . s s′ ! s i s′ (cid:16) (cid:17) 2. Supposeγ(s )=γ(s′)andγ(s )=γ(s′)thenwehave 1 2 2 γ(a(k)◦ ak′)=γγ(a(k)◦ ak′)=γ(a(k) ◦ ak′ )=γ(a(k) ◦ ak′ )=γ(a(k)◦ ak′). s1 i s2 s1 i s2 γs1 i γs2 γs′1 i γs′2 s′1 i s′2 (cid:3) Considernowtheequivalencerelation∼ onSdefinedforanys,s′ ∈Sbys∼ s′ ifandonlyif γ γ γ(s) = γ(s′). By definition of γ, ∼ is a monoid congruence of S and hence, S/ is a monoid. γ ∼γ Consideralsotheequivalencerelation≡ onOP(◦)definedforanya(k),a(k) ∈OP(◦)bya(k) ≡ a(k) if γ s s′ s γ s′ andonlyifs ∼ s′.Proposition3showsthat≡ isactuallyanoperadiccongruenceandhence,that γ γ OP(◦)/ isanoperad. ≡γ Lettheprecomposition ⊙:m→Hom S/∼γ,S/∼γ (11) (cid:16) (cid:17) i,k i,k definedforany∼ -equivalenceclass[s] by“ [s] :=[‘(s)] . Wethenhave γ ∼γ ∼γ ∼γ (cid:16) (cid:17) Corollary1. TheoperadsOP(◦)/ andOP(⊙)areisomorphic. ≡γ 8 Proof. Letusdenoteby◦γthecompositionmapofOP(◦)/ . Letthemap i ≡γ φ:OP(◦)/ →OP(⊙) (12) ≡γ definedforany≡ -equivalenceclass[a(k)] by γ s ≡γ φ([a(k)] ):=a(k) . (13) s ≡γ [s]∼γ Letusshowthatφisanoperadmorphism. Forthat,let[a(k)] and[a(k′)] betwo≡ equivalence s ≡γ s′ ≡γ γ classes. Onehas φ([a(k)] ◦γ[a(k′)] ) = φ([a(k)◦ a(k′)] ) = φ([a(k+k′−1)] ) = a(k+k′−1), (14) s ≡γ i s′ ≡γ s i s′ ≡γ s′′ ≡γ [s′′]∼γ i,k′ 0,i wheres′′:=‘(s)⊕‘(s′). Wemoreoverhave φ([a(k)] )⊙ φ([a(k′)] ) = a(k) ⊙ a(k′) = a(k+k′−1), (15) s ≡γ i s′ ≡γ [s]∼γ i [s′]∼γ [s′′′]∼γ i,k′ 0,i where[s′′′] :=“([s] )⊕“([s′] ). Now,byusingthefactthat∼ isamonoidcongruence,one ∼γ ∼γ ∼γ γ has i,k′ 0,i [s′′′] =“([s] )⊕“([s′] ) ∼γ ∼γ ∼γ i,k′ 0,i =[‘(s)] ⊕[‘(s′)] ∼γ ∼γ (16) i,k′ 0,i =[‘(s)⊕‘(s′)] ∼γ =[s′′] . ∼γ Thisshowsthat(14)and(15)areequalandhence,thatφisanoperadmorphism. Furthermore,thedefinitionsof∼ and≡ implythatφisabijection. Therefore,φisanoperad γ γ isomorphism. (cid:3) 4 Multi-tildes and precompositions In[10],weinvestigatedseveraloperadsallowingtodescribethebehaviourofthemulti-tildeopera- tors. Inthissection,weshowthatsomeofthemadmitanalternativedefinitionusingthenotionof precomposition. 4.1 The operadT revisited We consider the sets ST = 2{(x,y):1≤x≤y≤n} for each n > 0. Noting that ST ⊂ ST we define n n n+1 ST := n∈N\{0}STn. Considering the binary operation ∪ as a product, the pair (ST,∪) defines a commuStativemonoidwhoseunityis1ST =∅∈ST1. Thisisacommutativemonoidgeneratedbythe set{{(x,y)} }. 1≤x≤y Nowdefine◦: → Hom(ST,ST)by e i,k i,k ◦(ˆ):=‘ i,k whereeachhomomorphism‘isdefinedbyitsvaluesonthegenerators: {(x,y)} ify<i, i,k ‘({(x,y)})= {(x,y+k−1)} ifx≤i≤ y,  {(x+k−1,y+k−1} otherwise. Remarkthat◦isamonoidmorphism. Indeed, i,k 1. The setofthe homomorphisms‘generatesasubmonoidofHom(ST,ST)(which unityis Id ) ST 9 i,k i,k 2. Byconstruction,‘: STn → STn+k−1and‘|STn =IdSTn ifn<i. i,k 3. Theoperators‘satisfy(see[10]) i,k 0,k • ‘=‘foreachi<0, i,1 0,1 • ‘=‘=Id foreachi ST i,k j,k′ j+k−1,k′ i,k • ‘‘= ‘ ‘ifi≤ jori,j≤0 i+j,k i,k′ i,k+k′−1 • ‘‘= ‘ if0≤ j<k′. Hence◦isaprecomposition. Moreprecisely,theoperadT canbeseenastheoperadconstructed fromtheprecomposition◦: Proposition4. TheoperadsT andOP(◦)areisomorphic. Proof. TheisomorphismisgivenbythemapfromT toS sendinganyelementTtoa(k). (cid:3) k k T 4.2 The operadRASrevisited In[10],weconsideredanoperadRASonreflexiveandantisymmetricrelationsthatarecompatible withthenaturalorderonintegers(i.e. (x,y)∈RASimpliesx≤ y). Sincetheelements(x,x)donot playanyroleintheconstruction,weproposehereanalternativeconstructionbasedonantireflexive andantisymmetricrelations. ConsiderthesetsS^ =2{(x,y):1≤x<y≤n+1}foreachn>0. ByconstructionwehaveS^ ⊂S^ . Endowed n n n+1 with the binary operation ∪ the set S^ := n∈N\{0}S^n is a commutative monoid generated by {{(x,y)}1≤x<y}. S Letusdefine⋄: → Hom(S^,S^)by⋄(î,k ):=„i,k with e {(x,y)} ify≤i, i,k „({(x,y)})= {(x,y+k−1)} ifx≤i< y, (17)  {(x+k−1,y+k−1} otherwise. SimilarlytoSection4.1,weconsiderthesubmonoidofHom(S^,S^)generatedbytheelements „i,k . Wehave„i,k : S^n → S^n+k−1 and„i,k |S^n = IdS^n whenn < i. Furthermore,theelements„i,k satisfytheproperties i,k 0,k • „=„foreachi<0, i,1 0,1 • „=„=IdS^ foreachi i,k j,k′ j+k−1,k′ i,k • „„= „ „ifi≤ jori,j≤0 i+j,k i,k′ i,k+k′−1 • „„= „ if0≤ j<k′. Themap⋄isamonoidmorphismandsoaprecomposition. WesetARAS:= OP(⋄) = (S^,⋄). The operadARASisanalternativeclosedconstructionfortheoperadRASasshownby: Proposition5. TheoperadsRASandARASareisomorphic. Proof. TheisomorphismisgivenbythemapfromRAS toS^sendinganyelementRtoa(k) ,where k k R\∆ ∆={(x,x):x∈N}. (cid:3) 10