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1 Logical Mthods in Computr Scinc Vol. 2 (5:4) 2006, pp Submittd Mar. 10, 2006 Publishd Nov. 8, 2006 ELGOT ALGEBRAS JIŘÍ ADÁMEK a, STEFAN MILIUS b, AND JIŘÍ VELEBIL c a,b Institut of Thortical Computr Scinc, Tchnical Univrsity, Braunschwig, Grmany -mail addrss: {adamk,milius}@iti.cs.tu-bs.d c Faculty of Elctrical Enginring, Tchnical Univrsity, Pragu, Czch Rpublic -mail addrss: vlbil@math.fld.cvut.cz Abstract. Dnotational smantics can b basd on algbras with additional structur (ordr, mtric, tc.) which maks it possibl to intrprt rcursiv spcifications. It was th ida of Elgot to bas dnotational smantics on itrativ thoris instad, i.., thoris in which abstract rcursiv spcifications ar rquird to hav uniqu solutions. Latr Bloom and Ésik studid itration thoris and itration algbras in which a spcifid solution has to oby crtain axioms. W propos so-calld Elgot algbras as a convnint structur for smantics in th prsnt papr. An Elgot algbra is an algbra with a spcifid solution for vry systm of flat rcursiv quations. That spcification satisfis two simpl and wll motivatd axioms: functoriality (stating that solutions ar stabl undr rnaming of rcursion variabls) and compositionality (stating how to prform simultanous rcursion). Ths two axioms stm canonically from Elgot s itrativ thoris: W prov that th catgory of Elgot algbras is th Eilnbrg Moor catgory of th monad givn by a fr itrativ thory. If you ar not part of th solution, you ar part of th problm. Eldridg Clavr, spch in San Francisco, Introduction W study Elgot algbras, a nw notion of algbra usful for application in th smantics of rcursiv computations. In programming, functions ar oftn spcifid by a rcursiv program schm such as ϕ(x) F(x,ϕ(Gx)) (1.1) ψ(x) F(ϕ(Gx), GGx) whr F and G ar givn functions and ϕ and ψ ar rcursivly dfind in trms of th givn ons by (1.1). W ar intrstd in th smantics of such schms. Actually, on 2000 ACM Subjct Classification: F.3.2. Ky words and phrass: Elgot algbra, rational monad, coalgbra, itrativ thoris. This papr is a full vrsion of an xtndd abstract [AMV 3] prsntd at th confrnc MFPS I.. a,c Th first and th third author acknowldg th support of th Grant Agncy of th Czch Rpublic undr th Grant No. 201/02/0148. LOGICAL METHODS Ð IN COMPUTER SCIENCE DOI: /LMCS-2 (5:4) 2006 c J. Adámk, S. Milius, and J. Vlbil CC Crativ Commons

2 2 J. ADÁMEK, S. MILIUS, AND J. VELEBIL has to distinguish btwn unintrprtd and intrprtd smantics. In th unintrprtd smantics th givns ar not functions but mrly function symbols from a signatur Σ. In th prsnt papr w prpar a basis for th intrprtd smantics in which a program schm coms togthr with a suitabl Σ-algbra A, which givs an intrprtation to all th givn function symbols. Th actual application of Elgot algbras to smantics will b dalt with in joint work of th scond author with Larry Moss [MM]. By suitabl algbra w man, of cours, on in which rcursiv program schms can b givn a smantics. For xampl, for th rcursiv program schm (1.1) w ar only intrstd in thos Σ-algbras A, whr Σ = {F,G}, in which th program schm (1.1) has a solution, i.., w can canonically obtain nw oprations ϕ A and ψ A on A so that th formal quations (1.1) bcom valid idntitis. Th qustion w addrss is: What Σ-algbras ar suitabl for smantics? (1.2) Svral answrs hav bn proposd in th litratur. On wll-known approach is to work with complt posts (CPO) in liu of sts, s.g. [GTWW]. Hr algbras hav an additional CPO structur making all oprations continuous. Anothr approach works with complt mtric spacs, s.g. [ARu]. Hr w hav an additional complt mtric making all oprations contracting. In both of ths approachs on imposs xtra structur on th algbra in a way that maks it possibl to obtain th smantics of a rcursiv computation as a join (or limit, rspctivly) of finit approximations. It was th ida of Calvin Elgot to try and work in a purly algbraic stting avoiding xtra structur lik ordr or mtric. In [El] h introducd itrativ thoris which ar algbraic thoris in which crtain systms of rcursiv quations hav uniqu solutions. Latr Evlyn Nlson [N] and Jrzy Tiuryn [T] studid itrativ algbras, which ar algbras for a signatur Σ with uniqu solutions of rcursiv quations. Whil avoiding xtra structur, ths ar still not th unifying concpt on would hop for, sinc thy do not subsum continuous algbras last fixd points ar typically not uniqu. Howvr, analyzing all th abov typs of algbras w find an intrsting common fatur which maks continuous, mtrizabl and itrativ algbras fit for us in smantics of rcursiv program schms: ths algbras allow for an intrprtation of infinit Σ-trs. Lt us mak this mor prcis. For a givn signatur Σ considr th algbra T Σ of all (finit and infinit) Σ-trs ovr, i.., rootd ordrd trs whr innr nods with n childrn ar lablld by n-ary opration symbols from Σ, and lavs ar lablld by constants or lmnts from. It is wll-known that for any continuous (or mtrizabl) algbra A thr is a uniqu continuous (or contracting, rspctivly) homomorphism from T Σ A to A which provids for any Σ-tr ovr A its rsult of computation in A. It is thn asy to giv smantics to rcursiv program schms in A. For xampl, for (1.1) on can simply tak

3 ELGOT ALGEBRAS 3 th tr unfolding which yilds th infinit trs ϕ (x) = F x F Gx F GGx ψ (x) = F F GGx Gx F GGx and thn for any argumnt x A comput ths infinit trs in A. Actually, w do not nd to b abl to comput all infinit trs: all rcursiv program schms unfold to algbraic trs, s [C] (w discuss ths brifly in Sction 6 blow). Anothr important subclass ar rational trs, which ar obtaind as all solutions of guardd finitary rcursiv quations. Thy wr charactrizd by Sussanna Ginali [G] as thos Σ- trs having up to isomorphism finitly many subtrs only. W dnot by R Σ th subalgbra of all rational trs in T Σ. With this in mind, w can rstat problm (1.2) mor formally: What Σ-algbras hav a suitabl computation of all trs? (1.3) Or all rational trs? This mans, on furthr stp mor formally: what is th largst catgory of Σ-algbras in which T Σ, or R Σ, rspctivly, act as fr algbras on? Th answr in th cas of T Σ is: complt Elgot algbras. Ths ar Σ-algbras A with an additional opration daggr assigning to vry systm of rcursiv quations in A a solution. Two (surprisingly simpl) axioms ar put on ( ) which stm from th intrnal structur of T Σ : th functor T Σ givn by T Σ is part of a monad on St, and this monad yilds th fr compltly itrativ thory on Σ, as provd in [EBT]. W will prov that th algbras for this monad (i.., th Eilnbrg Moor catgory of T Σ ) ar complt Elgot algbras. Basic xampls: continuous algbras or mtrizabl algbras ar complt Elgot algbras. Analogously, th largst catgory of Σ-algbras in which ach R Σ acts as a fr algbra is th catgory of Elgot algbras. Thy ar dfind prcisly as th complt Elgot algbras, xcpt that th systms of rcursiv quations considrd thr ar rquird to b finit. For xampl, vry itrativ algbra is an Elgot algbra. W prsnt th rsults for suitabl ndofunctors of an arbitrary catgory A satisfying vry mild conditions: for complt Elgot algbras w just nd A to hav finit coproducts, for Elgot algbras w work with locally finitly prsntabl catgoris in th sns of Ptr Gabril and Fridrich Ulmr [AR]. Rlatd Work: Solutions of rcursiv quations ar a fundamntal part of a numbr of modls of computation,. g., itrativ thoris of C. Elgot [El], itration thoris of S. Bloom and Z. Ésik [BÉ], tracd monoidal catgoris of A. Joyal, R. Strt and D. Vrity [JSV], fixd-point thoris for domains, s S. Eilnbrg [Ei] or G. Plotkin [P], tc. In som of ths modls th assignmnt of a solution to a givn typ of rcursiv quation is uniqu (. g., in itrativ thoris vry idal systm has a uniqu solution, or in domains givn by a complt mtric spac thr ar uniqu solutions of fixd-point quations, s [ARu]). Th opration thn satisfis a numbr of quational proprtis. In othr modls, lik in itration thoris, for xampl, a spcific choic of a solution is assumd, and crtain

4 4 J. ADÁMEK, S. MILIUS, AND J. VELEBIL proprtis (inspird by th modls with uniqu solutions) ar formulatd as axioms. Rcall that in a tracd monoidal catgory whos tnsor product is just th ordinary product th trac is quivalntly prsntd in form of an opration satisfying crtain axioms, s [Ha] and [H]. Th approach of th prsnt papr is mor lmntary in asking for solutions in a concrt algbra A. Hr w work with flat quations in A, which ar morphisms of th form : H + A. Howvr, flatnss is just a tchnical rstriction: in futur rsarch w will prov that mor gnral non-flat quations obtain solutions automatically. Th fact that w work with a fixd algbra A (and lt only and vary) is partly rsponsibl for th simplicity of our axioms in comparison to th work on thoris (whr A varis as wll), s. g. [BÉ] or [SP 1]. Itrativ algbras of Evlyn Nlson [N] and Jrzy Tiuryn [T], whr solutions ar rquird to b uniqu, ar a similar approach. Furthrmor, itration algbras of Zoltan Ésik [É] ar anothr on. Unfortunatly, th numbr of axioms (svn) and thir complxity mak th qustion of th rlationship of that notion to Elgot algbras a nontrivial on. W intnd to study this qustion in th futur. W work with two variations: Elgot algbras, rlatd to R Σ, whr th function ( ) assigns a solution only to finitary flat rcursiv systms, and complt Elgot algbras, rlatd to T Σ, whr th function ( ) assigns solutions to all flat rcursiv systms. This is basd on our prvious rsarch [AAMV, M, AMV 1, AMV 2 ] in which w provd that vry finitary ndofunctor H gnrats a fr itrativ monad R, and a fr compltly itrativ monad T. In th prsnt papr w study th Eilnbrg Moor catgoris of th monads R and T. Organization of th Papr: In Sction 2 w rcall (compltly) itrativ algbras and prov that th assignmnt of uniqu solutions in ths algbras fulfills th axioms of functoriality and compositionality. Elgot algbras and complt Elgot algbras ar introducd in Sction 3 as algbras quippd with a chosn assignmnt of a solution that satisfis functoriality and compositionality. In Sctions 4 and 5 w prov that (complt) Elgot algbras form Eilnbrg Moor catgory of a fr (compltly) itrativ monad. 2. Itrativ Algbras and CIAs Assumption 2.1. Throughout th papr H dnots an ndofunctor of a catgory A having binary coproducts. W dnot th corrsponding injctions by inl : A A + B and inr : B A + B. Rcall that an objct of a catgory with filtrd colimits is calld finitly prsntabl if th hom-functor A(, ) : A St is finitary, i.., if it prsrvs filtrd colimits. (In St, ths ar prcisly th finit sts. In quational classs of algbras ths ar prcisly th finitly prsntabl algbras in th usual sns.) Rcall furthr that a catgory A is calld locally finitly prsntabl if it has colimits and a st of finitly prsntabl objcts whos closur undr filtrd colimits is all of A, s [AR]. (Exampls: th catgoris of sts, posts, graphs or any finitary varity of algbras ar locally finitly prsntabl catgoris.)

5 ELGOT ALGEBRAS 5 Dfinition 2.2. By a flat quation morphism in an objct A w undrstand a morphism : H + A in A. W call finitary providd that is finitly prsntabl. 1 Suppos that A is th carrir of an H-algbra α : HA A. A solution of is a morphism : A such that th squar H + A H +A A [α,a] HA + A (2.1) commuts. If vry finitary flat quation morphism has a uniqu solution, thn A is said to b an itrativ algbra. Th algbra A is calld a compltly itrativ algbra (CIA) if vry flat quation morphism has a uniqu solution. Rmark 2.3. Itrativ algbras for polynomial ndofunctors of St wr introducd and studid by Evlyn Nlson [N]. Sh provd that th algbras R Σ of rational Σ-trs on ar fr itrativ algbras, and that th algbraic thory obtaind from thm is a fr itrativ thory of Calvin Elgot [El]. W hav rcntly studid itrativ algbras in a much mor gnral stting, working with a finitary ndofunctor of a locally finitly prsntabl catgory. Compltly itrativ algbras wr studid by Stfan Milius [M]. Exampl 2.4. Considr algbras in St with on binary opration. In that cas, th functor is H =. A flat quation morphism in an algbra A assigns to vry variabl x ithr a flat trm y z (y and z ar variabls) or an lmnt of A. A solution : A assigns to x ithr th sam lmnt as, in cas (x) A, or th rsult of (y) (z), in cas (x) = y z. For xampl, th following rcursiv quation x x x, rprsntd by th obvious morphism : {x} {x} {x} + A, has as solution an lmnt a = (x) which is idmpotnt. Consquntly, vry itrativ algbra has a uniqu idmpotnt. If A is vn compltly itrativ, thn it has, for ach squnc a 0,a 1,a 2,... of lmnts, a uniqu intrprtation of a 0 (a 1 (a 2 ))), i.., a uniqu squnc b 0,b 1,b 2,... with b 0 = a 0 b 1, b 1 = a 1 b 2, tc. In fact, w considr hr th quations x n a n x n+1 (n N). Itrativ algbras hav uniqu solutions of many non-flat quations bcaus w can flattn thm. For xampl th following rcursiv quations x 1 (x 2 a) b x 2 x 1 b ar not flat. But thy can b asily flattnd to obtain a systm x 1 z 1 z 2 x 2 x 1 z 2 z 1 x 2 z 3 z 2 b z 3 a rprsntd by a morphism : + A, whr = {x 1,x 2,z 1,z 2,z 3 }. Its solution is a map : A yilding a pair of lmnts s = (x 1 ) and t = (x 2 ) satisfying s = (t a) b and t = s b. 1 W shall only us this notion in th cas whn A is locally finitly prsntabl and H is finitary.

6 6 J. ADÁMEK, S. MILIUS, AND J. VELEBIL Exampl 2.5. Itrativ Σ-algbras. For vry finitary signatur Σ = (Σ n ) n N w can idntify Σ-algbras with algbras for th polynomial ndofunctor H Σ of St dfind on objcts by H Σ = Σ 0 + Σ 1 + Σ A Σ-trm which has th form σ(x 1,...,x k ) for som σ Σ k and for variabls x 1,...,x k from is calld flat. Thn a flat quation morphism : H Σ + A in an algbra A rprsnts a systm x t x of rcursiv quations, on for vry variabl x, whr ach t x is ithr a flat trm in, or an lmnt of A. A solution assigns to vry variabl x with t x = a, a A, th lmnt a, and if t x = σ(x 1,...,x k ) thn (x) = σ A ( (x 1 ),..., (x k )). Obsrv that vry itrativ Σ-algbra A has, for vry σ Σ k, a uniqu idmpotnt (i.., a uniqu lmnt a A with σ(a,...,a) = a). In fact, considr th flat quation x σ(x,...,x). Mor gnrally, vry Σ-trm has a uniqu idmpotnt in A. For xampl, for a trm of dpth 2, σ(τ 1,...,τ k ), whr σ Σ k and τ 1,...,τ k Σ n considr th rcursiv quations x 0 σ(x 1,x 2,...,x k ) x i τ i (x 0,x 0,...,x 0 ) (i = 1,...,k). An xampl of an itrativ Σ-algbra is th algbra T Σ of all (finit and infinit) Σ-trs. Also th subalgbra R Σ of T Σ of all rational Σ-trs is itrativ, s [N]. Exampl 2.6. In particular, for unary algbras (H = Id), an algbra α : A A is itrativ iff α k has a uniqu fixd point (k 1), s [AMV 2 ]. Th algbra A is a CIA iff, in addition to a uniqu fixd point of ach α k, thr xists no infinit squnc (a n ) n N in A with αa n+1 = a n, s [M]. Rmark 2.7. In [AMV 2 ] w hav provd that for vry finitary functor H of a locally finitly prsntabl catgory A, a fr itrativ algbra RY xists on vry objct Y. Furthrmor, w hav givn a canonical construction of RY as a colimit of all coalgbras H +Y carrid by finitly prsntabl objcts, in othr words, for vry objct Y of A, RY is a colimit of all finitary flat quations in Y. For xampl, for a polynomial functor H Σ of St th fr itrativ algbra on a st Y is th algbra R Σ Y of all rational Σ-trs ovr Y. In gnral, w call th monad R of fr itrativ algbras th rational monad gnratd by H. W hav provd in [AMV 2 ] that th rational monad R is a fr itrativ monad on H. Exampl 2.8. Compltly mtrizabl algbras. Complt mtric spacs ar wll-known to b a suitabl basis for smantics. Th first catgorical tratmnt of complt mtric spacs for smantics is du to Pirr Amrica and Jan Ruttn [ARu]. Lt CMS dnot th catgory of all complt mtric spacs (i.., such that vry Cauchy squnc has a limit) with mtrics in th intrval [0,1]. Th morphisms ar maps f : (,d ) (Y,d Y ) whr th inquality d Y (f(x),f(x )) d (x,x ) holds for all x, x in. Such maps f ar calld nonxpanding.

7 ELGOT ALGEBRAS 7 Givn complt mtric spacs and Y, th hom-st CMS(,Y ) carris th pointwis mtric d,y dfind as follows: d,y (f,g) = sup d Y (f(x),g(x)) x Amrica and Ruttn call a functor H : CMS CMS contracting if thr xists a constant ε < 1 such that for arbitrary morphisms f,g : Y w hav d H,HY (Hf,Hg) ε d,y (f,g). (2.2) Lmma 2.9. If H : CMS CMS is a contracting functor, thn vry nonmpty H-algbra is a CIA. Proof. Lt α : HA A b a nonmpty H-algbra. Rcall that th hom-st CMS(,A) is a complt mtric spac with th suprmum mtric. Dfinition 2.2 of a solution of a flat quation morphism : H + A stats that is a fixd point of th function F on CMS(, A) givn by F : (s : A) ([α,a] (Hs + A) ). This function is a contraction on CMS(,A). In fact, for two nonxpanding maps s,t : A w hav d,a (Fs,Ft) = d,a ([α,a] (Hs + A),[α,A] (Ht + A) ) (by th dfinition of F) d H,HA (Hs + A,Ht + A) (sinc composition is nonxpanding) = d H,HA (Hs,Ht) εd,a (s,t) (sinc H is contracting), whr ε < 1 is th constant of (2.2) abov. By Banach s Fixd Point Thorm, thr xists a uniqu fixd point of F: a uniqu solution of. Rmark (1) Th proof of th last thorm yilds a concrt formula for th uniqu solution of a givn flat quation morphism : H + A. This uniqu solution is givn as th limit of a Cauchy squnc in CMS(,A) as follows: = lim n n, whr 0 : A is any nonxpanding map (for xampl a constant map: us that A is nonmpty) and n+1 is dfind by th commutativity of th diagram blow: n+1 H + A H n +A A [α,a] HA + A (2.3)

8 8 J. ADÁMEK, S. MILIUS, AND J. VELEBIL (2) Many st functors H hav a lifting to contracting ndofunctors H of CMS. That is, for th forgtful functor U : CMS St th following squar CMS H CMS U St commuts. For xampl, if H = n, dfin H U St H (,d) = ( n, 1 2 d ), whr d is th maximum mtric. Thn H is a contracting functor with ε = 1 2. Sinc coproducts of 1 2 -contracting liftings ar 1 2-contracting liftings of coproducts, w conclud that vry polynomial ndofunctor has a contracting lifting to CMS. Lt us call an H-algbra α : HA A compltly mtrizabl if thr xists a complt mtric, d, on A such that α is a nonxpanding map from H (A,d) to (A,d). Corollary Evry compltly mtrizabl algbra A is a CIA. In fact, to vry quation morphism : H + A assign th uniqu solution of : (,d 0 ) H (,d 0 ) + (A,d), whr d 0 is th discrt mtric (d 0 (x,x ) = 1 iff x x ). Rmark Stfan Milius [M] provd that for any ndofunctor H of A a final coalgbra T Y for H( ) + Y is a fr CIA on Y, and convrsly. Furthrmor, assuming that th fr CIAs xist, it follows that th monad T of fr CIAs is a fr compltly itrativ monad on H. This gnralizs and xtnds th classical rsult of Elgot, Bloom and Tindll [EBT] sinc for a polynomial functor H Σ of St th fr compltly itrativ algbra on a st Y is th algbra T Σ Y of all Σ-trs ovr Y. Rmark W ar going to prov two proprtis of itrativ algbras and CIA s: functoriality and compositionality of solutions. W will us two oprations on quation morphisms. On,, is just chang of paramtr nams: givn a flat quation morphism : H +Y and a morphism h : Y Z w obtain th following quation morphism h H + Y H+h H + Z. Th othr opration combins two flat quation morphisms : H + Y and f : Y HY + A into th singl flat quation morphism f : + Y H( + Y ) + A in a canonical way: put can = [Hinl,Hinr] : H + HY H( + Y ) and dfin f +Y [,inr] H+Y H+f H+HY +A can+a H(+Y )+A, (2.4)

9 ELGOT ALGEBRAS 9 Functoriality. This stats that solutions ar invariant undr rnaming of variabls, providd, of cours, that th right-hand sids of quations ar rnamd accordingly. Formally, obsrv that vry flat quation morphism is a coalgbra for th ndofunctor H( ) + A. Givn two such coalgbras and f, a rnaming of th variabls (or morphism of quations) is a morphism h : Y which forms a coalgbra homomorphism: h Y f H + A Hh+A HY + A (2.5) Dfinition Lt A b an algbra with a choic of solutions, for all flat quation morphisms in A. W say that th choic is functorial providd that = f h (2.6) holds for all morphisms h : f of quations. In othr words: ( ) is a functor from th catgory of all flat quation morphisms in th algbra A into th comma-catgory of th objct A. Lmma In vry CIA th assignmnt ( ) is functorial. Proof. For ach morphism h of quations th diagram H + A h Hh+A f h Y f f A [α,a] HY + A Hf +A HA + A H(f h)+a commuts. Thus, f h is a solution of. Uniqunss of solutions now implis th dsird rsult. Rmark. Th sam holds for vry itrativ algbra, xcpt that thr w rstrict and Y in Dfinition 2.14 to finitly prsntabl objcts. Compositionality. This tlls us how to prform simultanous rcursion: givn an quation morphism f in A with a variabl objct Y, w can combin it with any quation morphism in Y with a variabl objct to obtain th quation morphism f in A of Rmark Compositionality dcrs that th lft-hand componnt of (f ) is just th solution of f. In othr words: in liu of solving f and simultanously w first solv f, plug in th solution in and solv th rsulting quation morphism. Dfinition Lt A b an algbra with a choic of solutions, for all flat quation morphisms in A. W say that th choic is compositional if for ach pair : H+Y and f : Y HY + A of flat quation morphisms, w hav (f ) = (f ) inl. (2.7)

10 10 J. ADÁMEK, S. MILIUS, AND J. VELEBIL Rmark Notic that th coproduct injction inr : Y + Y is a morphism of quations from f to f. Functoriality thn implis that f = (f ) inr. Thus, in th prsnc of functoriality, compositionality is quivalnt to (f ) = [(f ),f ]. (2.8) Lmma In vry CIA, th assignmnt ( ) is compositional. Proof. Dnot by r = (f ) : A th solution of f. It is sufficint to prov that th quation blow holds: (f ) = [r,f ] : + Y A. W stablish this using th uniqunss of solutions and by showing that th following diagram + Y [,inr] [r,f ] A f H + Y H+f [α,a] H + HY + A [Hr,Hf ]+A can+a H( + Y ) + A HA + A H[r,f ]+A (2.9) commuts. Commutation of th right-hand componnts (with domain Y ) of th diagram: [α,a] ([Hr,Hf ] + A) inr f = [α,a] (Hf + A) f = f bcaus f solvs f. For th lft-hand componnts (with domain ) us th commutativity of th squar dfining r = (f ) : r A f H + Y [α,a] (2.10) H+f H + A Hr+A HA + A W now only nd to show that th passags from H + Y to A in th abov squars (2.9) and (2.10) ar qual. Th lft-hand componnts ar, in both cass, α Hr : H A. For th right-hand componnts us f = [α,a] (Hf + A) f.

11 ELGOT ALGEBRAS 11 Rmark Th sam holds for vry itrativ algbra, xcpt that hr w work in a locally finitly prsntabl catgory and rstrict and Y in Dfinition 2.16 to finitly prsntabl objcts. Rmark As mntiond in th Introduction, our two axioms, functoriality and compositionality, ar not nw as idas of axiomatizing rcursion w bliv howvr, that thir concrt form is nw, and thir motivation strngthnd by th rsults blow. Functoriality rsmbls th functorial daggr implication of S. Bloom and Z. Ésik [BÉ], 5.3.3, which stats that for vry objct p of an itrativ thory th formation f f of solutions for idal morphisms f : m m + p is a functor. Compositionality rsmbls th lft pairing idntity of [BÉ], 5.3.1, which for f : n n + m + p and g : m n + m + p stats that whr [f,g] = [f [h,id p ],h ], h m g n + m + p [f,id m+p ] m + p. This idntity corrsponds also to th Bkić-Scott idntity, s. g. [Mo], Elgot Algbras Dfinition 3.1. Lt H b an ndofunctor of a catgory with finit coproducts. An Elgot algbra is an H-algbra α : HA A togthr with a function ( ) which to vry finitary flat quation morphism : H + A assigns a solution : A in such a way that functoriality (2.6) and compositionality (2.7) ar satisfid. By a complt Elgot algbra w analogously undrstand an H-algbra togthr with a function ( ) assigning to vry flat quation a solution so that functoriality and compositionality ar satisfid. Exampl 3.2. Evry join smilattic A is an Elgot algbra. Mor prcisly: considr th polynomial ndofunctor H = of St (xprssing on binary opration). Thn for vry join smilattic A thr is a canonical Elgot algbra structur on A obtaind as follows: th algbra RA of all rational binary trs on A has an intrprtation on A givn by th function α : RA A forming, for vry rational binary tr t th join of all th (finitly many!) labls of lavs of t in A. Now givn a finitary flat quation morphism : + A, it has a uniqu solution : RA in th fr itrativ algbra RA, and composd with α this yilds an Elgot algbra structur α on A. S Exampl 4.10 for a proof. Rmark 3.3. In contrast, no nontrivial join smilattic is itrativ. In fact, in an itrativ join smilattic thr must b a uniqu solution of th formal quation x x x. Exampl 3.4. Continuous algbras on cpos ar complt Elgot algbras. Lt us work hr in th catgory CPO of all ω-complt posts, which ar posts having joins of incrasing ω-chains; morphisms ar th continuous functions, i.., functions prsrving joins of ω-chains. A functor H : CPO

12 12 J. ADÁMEK, S. MILIUS, AND J. VELEBIL CPO is calld locally continuous providd that for arbitrary CPOs, and Y, th associatd function from CPO(,Y ) to CPO(H,HY ) is continuous (i.., H( f n ) = Hf n holds for all incrasing ω-chains f n : Y ). For xampl, vry polynomial ndofunctor n Σ n n of CPO (whr Σ n ar cpos) is locally continuous. Obsrv that th catgory CPO has coproducts: thy ar th disjoint unions with lmnts of diffrnt summands incompatibl. Proposition 3.5. Lt H : CPO CPO b a locally continuous functor and lt α : HA A b an H-algbra with a last lmnt A. Thn (A,α,( ) ) is a complt Elgot algbra w.r.t. th assignmnt of th last solution to vry flat quation morphism. Rmark 3.6. Notic that th last solution of : H +A rfrs to th lmntwis ordr of th hom-st CPO(,A). W can actually prov a concrt formula for as a join of th ω-chain = n ω n of approximations : 0 is th constant function to, th last lmnt of A, and givn n, thn n+1 is dfind by th commutativity of (2.3). Proof of Proposition 3.5. (1) Lt : H + A b a flat quation morphism in A. W dfin a function F on CPO(,A) by F : (s : A) ([α,a] (Hs + A) ). Sinc H is locally continuous and composition in th catgory CPO is continuous, w s that F is continuous too. By th Kln Fixd Point Thorm, thr xists a last fixd point of F and this is th last solution as dscribd in Rmark 3.6. (2) Th assignmnt is functorial. In fact, lt h Y f H + A Hh+A HY + A b a coalgbra homomorphism. It is asy to s by induction that thus, = f h. (3) W prov compositionality. Lt n = f n h (for all n 0), : H + Y b givn. W shall show that th quality and f : Y HY + A (f ) inl = (f ) holds. It suffics to prov, by induction on n, that th following two inqualitis (f ) n inl (f ) (3.1) (f ) n (f ) inl (3.2)

13 ELGOT ALGEBRAS 13 hold. First rcall that inr : (Y,f) ( +Y,f ) is a coalgbra homomorphism. Thus, th quation (f ) inr = f holds by functoriality. For th induction stp for (3.1) considr th following diagram (f ) inl (f ) n+1 + A A H + Y = f = H+f H + HY + A can+a H( + Y ) + A H(f ) n +A H+f [α,a] H + A [H(f ),Hf ]+A H(f ) +A HA + A In ordr to prov th dsird inquality in th uppr triangl, w us th fact that th outr squar commuts by dfinition of ( ). Th thr middl parts clarly bhav as indicatd (for th triangl us th induction hypothsis (3.1) and (f ) n inr (f ) inr = f ), and th lowst part commuts whn xtndd by [α, A]: In fact, for th lft-hand componnt with domain H this is trivial; for th right-hand componnt with domain Y us f = [α,a] (Hf + A) f, s (2.1). For th induction stp for (3.2) considr th following diagram (f ) n+1 inl + A (f ) A H + Y = f = H+f H + HY + A can+a H( + Y ) + A H(f ) +A H+f [α,a] H + A [H(f ) n,hf ]+A H(f ) n +A HA + A Th outr squar commuts by dfinition of (f ) n+1. Th thr middl parts bhav as indicatd (for th inquality us th induction hypothsis), and th lowst part commuts

14 14 J. ADÁMEK, S. MILIUS, AND J. VELEBIL whn xtndd by [α, A] as bfor. Thus, w obtain th dsird inquality in th uppr triangl. Rmark 3.7. Many st functors H hav a lifting to locally continuous ndofunctors H of CPO. That is, for th forgtful functor U : CPO St th following squar CPO H CPO U St H commuts. For xampl, vry polynomial functor H Σ has such a lifting. An H-algbra α : HA A is calld CPO-nrichabl if thr xists a CPO-ordring with a last lmnt on th st A such that α is a continuous function from H (A, ) to (A, ). Corollary 3.8. Evry CPO-nrichabl H-algbra A in St is a complt Elgot algbra. In fact, to vry quation morphism : H + A assign th last solution of : (, ) H (, ) + (A, ) whr is th discrt ordring of (x y iff x = y). Exampl 3.9. Unary algbras. Lt H = Id as an ndofunctor of St. Givn an H-algbra α : A A, if α has no fixd point, thn A carris no Elgot algbra structur: considr th quation x α(x). Convrsly, vry fixd point a 0 of α yilds a flat cpo structur with a last lmnt a 0 on A, i.., x y iff x = y or x = a 0. Thus, A is a complt Elgot algbra sinc it is CPO-nrichabl. Notic that for vry flat quation morphism : + A, th last solutions oprats as follows: for a variabl x w hav (x) = U St α k (a) if thr is a squnc x = x 0,x 1,...x k in that fulfils (x 0 ) = x 1,... (x k 1 ) = x k and (x k ) = a a 0 ls. Rmark For unary algbras, Exampl 3.9 dscribs all xisting Elgot algbras. In fact, lt (A,α,( ) ) b an Elgot algbra and lt a 0 A b th chosn solution of x α(x); mor prcisly, a 0 = ( ) whr = inl : A and is th uniqu lmnt of 1. Thn for vry flat quation morphism : + A th chosn solution snds a variabl x to on of th abov valus α k (a) or a 0. To prov this dnot by Y th st of all variabls for which th ls cas holds abov. Hnc no squnc x = x 0,...x k in fulfils (x i ) = x i+1, for i = 0,...,k 1, and (x k ) A. Apply functoriality to th morphism h from to 1 + : A dfind by h(y) 1 for y Y and h(x) = x ls. In fact, th chosn solution of th uniqu lmnt of 1 in 1 + must b a 0 by functoriality (considr th lft-hand coproduct injction from th flat quation morphism inl : A to 1 + ). Exampl Just as Exampl 3.4 is basd on th Kln Fixd Point Thorm, w obtain xampls of complt Elgot algbras basd on th Knastr-Tarski Fixd Point Thorm. Hr w work with th catgory Pos of all posts and ordr-prsrving functions. (In fact, vrything w say holds, much mor gnrally, in vry catgory nrichd ovr Pos with Pos-nrichd finit coproducts.) A functor H : Pos Pos is calld locally ordr-prsrving if for all ordr-prsrving functions

15 ELGOT ALGEBRAS 15 f,g : A B with f g (in th pointwis ordring of Pos(A,B), of cours), w hav Hf Hg. Proposition Lt H : Pos Pos b locally ordr-prsrving and lt α : HA A b an H-algbra which is carrid by a complt lattic A. Thn (A,α,( ) ) is a complt Elgot algbra w.r.t. th assignmnt of a last solution to vry flat quation morphism. Rmark Again, th last solution of : H + A rfrs to th lmntwis ordr of th hom-st Pos(,A). W can actually prov a concrt formula for as a join of th ordinal chain = n Ord of approximations : 0 is th constant function to, th last lmnt of A, givn n, thn n+1 is dfind by th commutativity of (2.3) and for limit ordinals n w put n = k<n k. Proof of Proposition On ssntially rpats th proof of Proposition 3.5 for as dfind in th prvious Rmark. In part (1) apply th Knastr-Tarski Fixd Point Thorm in liu of th Kln Fixd Point Thorm. For part (2) rplac vry induction argumnt by a corrsponding transfinit induction argumnt and notic that th limit stp is always trivial. Exampl Evry complt lattic A is a complt Elgot algbra for H =. Analogously to Exampl 3.2 w hav a function α : TA A assigning to vry binary tr t in TA th join of all labls of lavs of t in A. Now for vry flat quation morphism in A w hav its uniqu solution in TA and this yilds a complt Elgot algbra structur α. S Exampl 5.9 for a proof. n 4. Th Eilnbrg-Moor Catgory of th Monad R W prov now that th catgory of all Elgot algbras and solution-prsrving morphisms, dfind as xpctd, is th catgory A R of Eilnbrg-Moor algbras of th rational monad R of H, s Rmark 2.7. Assumption 4.1. Throughout this sction H dnots a finitary ndofunctor of a locally finitly prsntabl catgory A. W dnot by A fp a small full subcatgory rprsnting all finitly prsntabl objcts of A. Rcall th oprations and from Rmark Dfinition 4.2. Lt (A,α,( ) ), and (B,β,( ) ) b Elgot algbras. W say that a morphism h : A B in A prsrvs solutions providd that for vry finitary flat quation morphism : H + A w hav th following quation A h B (h ) B. (4.1) Lmma 4.3. Evry solution-prsrving morphism btwn Elgot algbras is a homomorphism of H-algbras, i.., w hav h α = β Hh.

16 16 J. ADÁMEK, S. MILIUS, AND J. VELEBIL Proof. Lt A fp /A b th comma-catgory of all arrows q : A with in A fp. Sinc A is locally finitly prsntabl, A is a filtrd colimit of th canonical diagram D A : A fp /A A givn by (q : A). Now A fp is a gnrator of A, thus, in ordr to prov th lmma it is sufficint to prov that for vry morphism p : Z HA with Z in A fp w hav h α p = β Hh p. (4.2) Sinc H is finitary, it prsrvs th abov colimit D A. This implis, sinc A(Z, ) prsrvs filtrd colimits, that p has a factorization p Z HA s H for som q : A in A fp /A and som s. For th following quation morphism w hav a commutativ squar Z + s+ H + Hinr+q H(Z + ) + A Hq Z + A s+ H + [α,a] Hinr+q H(Z + ) + A H +A HA + A Consquntly, inr = q, and this implis inl = α H( inr) s = α p. Sinc h prsrvs solutions, w hav h = (h ) and thrfor On th othr hand, considr th following diagram (h ) = [h α p,h q]. (4.3) h (h ) Z + B p+hq s+ Hq+hq H + HA + B H[αp,q]+h Hinr+q [β,b] Hh+B H(Z + ) + A H[αp,q]+B H(Z+)+h H(Z + ) + B HB + B H(h ) +B It commuts: th outr shap commuts sinc (h ) is a solution. For th lowr triangl us quation (4.3), and th rmaining triangls ar trivial. Thus, th uppr right-hand part

17 ELGOT ALGEBRAS 17 commuts: (h ) = [β Hh p,h q]. (4.4) Now th lft-hand componnts of (4.3) and (4.4) stablish th dsird quality (4.2). Exampl 4.4. Th convrs of Lmma 4.3 is tru for itrativ algbras, as provd in [AMV 2 ], but for Elgot algbras in gnral it is fals. In fact, considr th unary algbra id : A A, whr A = {0,1}. This is an Elgot algbra with th solution structur ( ) givn by th fixd point 0 A, s Exampl 3.9. Thn const 1 : A A is a homomorphism of unary algbras that dos not prsrv solutions. Indd, considr th following quation morphism : {x} {x} + A, x x. W hav (x) = 0, and thus 1 = const 1 (x) (const 1 ) (x) = (x) = 0. Notation 4.5. W dnot by Alg H th catgory of all Elgot algbras and solution-prsrving morphisms. Rmark 4.6. For th two oprations and from Rmark 2.13 w list som obvious proprtis that ths oprations hav for all : H + Y, f : Y HY + Z, s : Z Z and t : Z Z : (1) id Y =. This is trivial. (2) t (s ) = (t s). S th following diagram (3) s (f ) = (s f). S th following diagram + Y [,inr] H + Y H+s f H+s H + Y H + Y H+t s H + Y H+f H+t H + HY + Z can+z H( + Y ) + Z H(+Y )+s H + HY + Z can+z H( + Y ) + Z Proposition 4.7. A fr itrativ algbra on Y is a fr Elgot algbra on Y. Proof. (1) W first rcall th construction of th fr itrativ algbra RY on Y prsntd in [AMV 2 ]. For th functor H( ) + Y dnot by EQ Y th full subcatgory of Coalg (H( ) + Y ) givn by all coalgbras with a finitly prsntabl carrir, i.., finitary flat quation morphisms : H + Y. Th inclusion functor Eq Y : EQ Y Coalg (H( ) + Y ) is an ssntially small filtrd diagram. Put RY = colim Eq Y.

18 18 J. ADÁMEK, S. MILIUS, AND J. VELEBIL Mor prcisly, form a colimit of th abov diagram Eq Y. This is a coalgbra RY with th following coalgbra structur i : RY HRY + Y and with colimit injctions : (,) (RY,i) for all : H + Y in EQ Y. Notic that this colimit is prsrvd by th forgtful functor Coalg (H( ) + Y ) A sinc H is finitary. Th coalgbra structur i : RY HRY + Y is an isomorphism; its invrs givs an H-algbra structur ρ Y : HRY RY and a morphism η Y : Y RY. Furthrmor, w provd that th algbra (RY,ρ Y ) is a fr itrativ H-algbra on Y with th univrsal arrow η Y. Rcall furthr from [AMV 2 ] that th uniqu solution : RY for a finitary flat quation morphism : H + RY is obtaind as follows. Thr xists a factorization H + RY H+g (4.5) 0 H + Z with g : Z HZ + Y in EQ Y. Dfin inl + Z (g 0) RY This dfins ( ) from ( ). Convrsly, it is not difficult to s that th quality = (η Y ) (4.6) holds for vry : H + Y in EQ Y. Finally, th univrsal arrow η Y has for any finitly prsntabl objct Y th form η Y = inr (for inr : Y HY + Y ). (2) W ar prpard to prov th Proposition. Suppos that (A,α,( ) ) is an Elgot algbra and lt m : Y A b a morphism. W ar to prov that thr xists a uniqu solutionprsrving h : RY A with h η Y = m. In ordr to show xistnc, w dfin a morphism h : RY A by commutativity of th following triangls h RY A (m ) for all : H + Y in EQ Y. Th dfinition of h is justifid, sinc th morphisms (m ) form a cocon for Eq Y : for any coalgbra homomorphism k : (,) (Z,g) in EQ Y w hav a coalgbra homomorphism k : (,m ) (Z,m g). Thus, (m ) k =

19 ELGOT ALGEBRAS 19 (m g) holds by functoriality. For = inr : Y HY +Y, Y finitly prsntabl, w hav = η Y, thus, h η Y = (m inr) (sinc η Y = inr ) = [α,a] (H(m inr) + A) (m inr) (by (2.1)) = [α,a] (H(m inr) + A) (HY + m) inr (Dfinition of ) = m. For arbitrary objcts Y th quation h η Y = m follows asily. Lt us show that h prsrvs solutions. W hav h = h (g 0 ) inl (Dfinition of ) = (m (g 0 )) inl (Dfinition of h) = ((m g) 0 ) inl (4.6(3)) = ((m g) 0 ) (compositionality) = ((h g ) 0 ) (Dfinition of h) = (h (g 0 )) (4.6(2)) = (h ) ((4.5) and th dfinition of ) Concrning uniqunss, suppos that h with h η Y = m prsrvs solutions, thn w hav h = h (η Y ) (by (4.6)) = (h (η Y h)) (h prsrvs solutions) = ((h η Y ) ) (4.6(2)) = (m ) (sinc h η Y = m) which dtrmins h uniquly. Thorm 4.8. Th catgory Alg H of Elgot algbras is isomorphic to th Eilnbrg-Moor catgory A R of R-algbras for th rational monad R of H. Rmark 4.9. Th shortst proof w know is basd on Bck s Thorm, s blow. But this proof is not vry intuitiv. A slightly mor tchnical (and much mor illuminating) proof has th following sktch: Dnot for any objct Y by (RY,ρ Y,( ) ) a fr Elgot algbra on Y with a univrsal arrow η Y : Y RY. (1) For vry R-algbra α 0 : RA A w hav an undrlying H-algbra α HA Hη A HRA ρ A RA α 0 A, and th following formula for solving quations: givn a finitary flat quation morphism : H + A put (η A ) RA α 0 A. It is not difficult to s that this formula indd yilds a choic of solutions satisfying functoriality and compositionality. (2) Convrsly, givn an Elgot algbra α : HA A, dfin α 0 : RA A as th uniqu solution-prsrving morphism such that α 0 η A = id. It is asy to s that α 0 satisfis th two axioms of an Eilnbrg-Moor algbra. (3) It is ncssary to prov that th abov passags xtnd to th lvl of morphisms and thy form functors which ar invrs to ach othr.

20 20 J. ADÁMEK, S. MILIUS, AND J. VELEBIL Proof. (Thorm 4.8.) By Proposition 4.7 th natural forgtful functor U : Alg H A has a lft adjoint Y RY. Thus, th monad obtaind by this adjunction is R. W prov that th comparison functor K : Alg H A R is an isomorphism, using Bck s thorm (s [ML], Thorm 1 in Sction VI.7). Thus, w must prov that U crats coqualizrs of U-split pairs. Lt (A,α,( ) ) and (B,β,( ) ) b Elgot algbras, and lt f,g : A B b solution-prsrving morphisms with a splitting A f g t B in A (whr cs = id, ft = id and gt = sc). Sinc c is, thn, an absolut coqualizr of f and g, c is a coqualizr in Alg H for a uniqu H-algbra structur γ : HC C. In fact, th forgtful functor Alg H A crats vry colimit that H prsrvs. It rmains to show that C has a uniqu Elgot algbra structur such that (1) c prsrvs solutions, and (2) c is a coqualizr in Alg H. W stablish (1) and (2) in svral stps. (a) An Elgot algbra on (C,γ). For vry finitary flat quation morphism : H+C w prov that th following morphism c s C (s ) B c C is a solution of. In fact, th following diagram s (s ) B [β,b] c C H + B HB + B H+s H(s ) +B Hc+c H + C HC + C H(c (s ) )+C clarly commuts. Functoriality: any coalgbra homomorphism h Z z H + C Hh+C HZ + C is, of cours, a coalgbra homomorphism h : (, s ) (Z, s z). Thus, th quations = c (s ) = c (s z) h = z h hold by functoriality of ( ). Lt us prov compositionality: suppos w hav finitary flat quation morphisms : H + Y Thn w obtain th dsird quation as follows: and k : Y HY + C [γ,c]

21 ELGOT ALGEBRAS 21 (k ) = c (s (k )) (Dfinition of ( ) ) = c (s (c (s k) )) (Dfinition of ( ) ) = c ((s c) ((s k) )) (4.6(2)) = c ((g t) ((s k) )) (g t = s c) = c (g (t ((s k) ))) (4.6(2)) = c g (t ((s k) )) (g prsrvs solutions) = c f (t ((s k) )) (c f = c g) = c ((f t) ((s k) )) (f prsrvs solutions and 4.6(2)) = c ((s k) ) (f t = id and 4.6(1)) = c ((s k) ) inl (compositionality of ( ) ) = c (s (k )) inl (Sinc (s k) = s (k ) by 4.6(3)) = (k ) inl (Dfinition of ( ) ) (b) Th morphism c : B C is solution-prsrving. In fact, for any finitary flat quation morphism : H + B w hav th dsird quation: (c ) = c (s (c )) (Dfinition of ( ) ) = c ((s c) ) (4.6(2)) = c ((g t) ) (g t = s c) = c (g (t )) (4.6(2)) = c g (t ) (g prsrvs solutions) = c f (t ) (c f = c g) = c (f (t )) (f prsrvs solutions) = c ((f t) ) (4.6(2)) = c (id ) (f t = id) = c (4.6(1)) (c) ( ) is th uniqu Elgot algbra structur such that c is solution-prsrving: in fact, for any such Elgot algbra structur ( ) and for any finitary flat quation morphism : H + B w hav c = (c ). In particular, this is tru for any quation morphism of th form Thus, w conclud (s ) H + C H+s H + B = ((c s) ) (c s = id and 4.6(3)) = (c (s )) (4.6(2)) = c (s ) (c prsrvs solutions) (d) c is a coqualizr of f and g in Alg H. In fact, lt h : (B,β,( ) ) (D,δ,( ) + ) b a solution-prsrving morphism with h f = h g. Thr is a uniqu homomorphism

22 22 J. ADÁMEK, S. MILIUS, AND J. VELEBIL h : C D of H-algbras with h c = h (bcaus c is a coqualizr of f and g in Alg H). W prov that h is solution-prsrving. Lt : H + C b a finitary flat quation morphism. Thn w hav as dsird. This complts th proof. h = h c (s ) (Dfinition of ( ) ) = h (s ) (h = h c) = (h (s )) + (h prsrvs solutions) = ((h s) ) + (4.6(2)) = ((h c s) ) + (h = h c) = (h ) + (c s = id) Exampl Lt A b a join smilattic. Rcall from Exampl 3.2 th function α : RA A assigning to a rational binary tr t in RA th join of th labls of all lavs of t in A. Sinc joins commut with joins it follows that this is th structur of an Eilnbrg- Moor algbra on A. Thus, A is an Elgot algbra as dscribd in Exampl Complt Elgot Algbras Rcall our standing assumptions that H is an ndofunctor of a catgory A with finit coproducts. Stfan Milius [M] has stablishd that for vry objct-mapping T of A th following thr statmnts ar quivalnt: (a) for vry objct Y, TY is a final coalgbra for H( ) + Y, (b) for vry objct Y, TY is a fr compltly itrativ H-algbra on Y, and (c) T is th objct part of a fr compltly itrativ monad T on H. S also [AAMV], whr th monad T is dscribd and th implication that (a) implis (c) is provd. W ar going to add anothr quivalnt itm to th abov list, bringing complt Elgot algbras into th pictur. Th statmnts (a) to (c) ar quivalnt to (d) for vry objct Y, TY is a fr complt Elgot algbra on Y. Furthrmor, rcall from [AAMV] that H is itratabl if thr xist objcts TY such that on of th abov quivalnt statmnts holds. W will dscrib for vry itratabl ndofunctor th catgory A T of Eilnbrg Moor algbras it is isomorphic to th catgory of complt Elgot algbras for H. Exampl 5.1. For a polynomial ndofunctor H Σ of St, th abov monad T is th monad T Σ of all (finit and infinit) Σ-trs. In th following rsult th concpt of solution-prsrving morphism is dfind for complt Elgot algbras analogously to Dfinition 4.2: th quation (4.1) holds for all flat quation morphisms. W dnot by Alg c H th catgory of all complt Elgot algbras and solution-prsrving morphisms. Lmma 5.2. Evry solution-prsrving morphism btwn complt Elgot algbras is a homomorphism of H-algbras.

23 ELGOT ALGEBRAS 23 Rmark 5.3. If th bas catgory A is locally finitly prsntabl and H is finitary, thn this lmma is a spcial cas of Lmma 4.3. Howvr, th statmnt of Lmma 5.2 is mor gnral, and th proof is compltly diffrnt. Proof of Lmma 5.2. Lt (A,α,( ) ) and (B,β,( ) ) b complt Elgot algbras. Suppos that h : A B is a solution-prsrving morphism, and considr th flat quation morphism HA + A Hinr+A H(HA + A) + A. Its solution fulfils = [α,a] : HA + A A. In fact, th following diagram HA + A Hinr+A A [α,a] H(HA + A) + A HA + A H +A commuts. Thus, inr = id, and thn it follows that inl = α. Sinc h prsrvs solutions w know that h α is th lft-hand componnt of th solution of th following flat quation morphism h HA + A Hinr+A H(HA + A) + A H(HA+A)+h H(HA + A) + B ; in symbols, (h ) inl = h α. Now considr th diagram HA + A Hinr+A H(HA+A)+h H(HA + A) + A H(HA + A) + B H(inr h)+h HB + B H(HB + B) + B Hh+h h Hinr+B H(Hh+h)+B which trivially commuts. Hnc, Hh+h is a morphism of quations from h to Hinr+B. By a similar argumnt as for abov w obtain [β,b] = (Hinr + B). Thus, by functoriality w conclud that h is an H-algbra homomorphism: This complts th proof. h α = (h ) inl = [β,b] (Hh + h) inl = β Hh. Thorm 5.4. Lt Y b an objct of A. Thn th following ar quivalnt: (1) TY is a final coalgbra for H( ) + Y, and (2) TY is a fr complt Elgot algbra on Y. Bfor w prov this thorm, w nd a tchnical lmma: Construction 5.5. Lt (A,α,( ) ) b a complt Elgot algbra. For vry morphism m : Y A w construct a nw complt Elgot algbra on HA + Y as follows: (1) Th algbra structur is H(HA + Y ) H[α,m] HA inl HA + Y.

24 24 J. ADÁMEK, S. MILIUS, AND J. VELEBIL (2) Th choic ( ) of solutions is as follows: for vry flat quation morphism : H + HA + Y considr th flat quation morphism and put H + HA + Y H+[α,m] H + A, H + HA + Y [H,HA]+Y HA + Y. Notic that = [α,m]. Lmma 5.6. Th abov construction dfins a complt Elgot algbra such that [α,m] : HA + Y A is a solution-prsrving morphism into th original algbra. Proof. (1) Th morphism [α, m] is solution-prsrving: In fact, for any flat quation morphism : H + HA + Y w hav th following commutativ diagram H + HA + Y H+[α,m] H + A H +A =([α,m] ) [H,HA]+Y HA + Y HA + A [α,a] A Th lowr lft-hand part commuts sinc solvs ; th uppr part is th dfinition of ( ), th lft-hand triangl is th dfinition of, and all componnts of th innr right-hand part ar clar. (2) Th morphism is a solution of. In fact, th following diagram [α,m] H+HA+Y [H,HA]+Y HA+Y H+HA+Y H+HA+Y H+HA+Y H(H+HA+Y )+HA+Y H([H,HA]+Y )+HA+Y [inl H[α,m],HA+Y ] H(HA+Y )+HA+Y H +HA+Y commuts: th uppr and lowr part as wll as th lft-hand squar ar obvious, and so ar th middl and right-hand componnts of th right-hand squar. To s that th lft-hand componnt commuts, w rmov H and obsrv that th following diagram commuts: A [α,a] H + A HA + A [α,m] H +A H+[α,m] H + HA + Y HA + Y [H,HA]+Y

25 ELGOT ALGEBRAS 25 (3) Functoriality: Suppos w hav a morphism h : f of quations. Thn h : f is also on, and w obtain th following diagram H + HA + Y [H,HA]+Y h Z f HA + Y HZ + HA + Y [Hf,HA]+Y Hh+HA+Y f It commuts: in th triangl th componnts with domains HA and Y ar clar, for th lft-hand componnt rmov H and us functoriality of ( ), and all othr parts ar obvious. (4) Compositionality: Suppos w hav two flat quation morphisms f : H + Z and g : Z HZ + HA + Y. Obsrv that (g f) is th following morphism f g f H+Z H+g H+[Hg,HA]+Y H+HZ+HA+Y H+HA» H g f «,Hg,HA +Y HA and (g f) inl is th following morphism» H g f «,HA +Y (5.1) f g f H+Z H+g can+ha+y H+HZ+HA+Y H(+Z)+HA+Y h H (g f) i,hg,ha +Y HA+Y [H(g f ),HA]+Y (5.2) In fact, to s that th last triangl commuts considr th componnts sparatly. Th right-hand on with domain HA + Y is trivial, and for th lft-hand on with domain H + HZ it suffics to obsrv th following quations: g f = ([α,m] (g f)) (Dfinition of g f) = [ (([α,m] g) f) (4.6(3)) = (([α,m] g) f),([α,m] g) ] (by (2.8)) ] = [(g f),g (Dfinition of g) To show th dsird idntity of th morphisms in (5.1) and (5.2) it suffics to prov that th slanting arrows in thos diagrams ar qual. Th last thr componnts ar clar, and

26 26 J. ADÁMEK, S. MILIUS, AND J. VELEBIL for th first on th following quations ar sufficint: This complts th proof. g f = ([α,m] g) f (Dfinition of g) = ([α,m] g ) f ([α,m] prsrvs solutions) = [α,m] (g f) (4.6(2)) = g f (Dfinition of g f) Proof. (Thorm 5.4.) By Thorms 2.8 and 2.10 of [M], statmnt (1) is quivalnt to (1 ) TY is a fr CIA on Y, W prov now that (2) is quivalnt to (1) by showing th implications (1 ) (2) (1). W first obsrv that for a fr complt Elgot algbra on Y, (TY,τ Y,( ) ), with a univrsal arrow η Y : Y TY, th morphism [τ Y,η Y ] : HTY + Y TY is an isomorphism. In fact, by Lmma 5.6, HTY +Y carris th complt Elgot algbra structur and j = [τ Y,η Y ] is solution-prsrving and fulfils j inr = η Y. Invok th frnss of TY to obtain a uniqu solution-prsrving morphism i : TY HTY + Y such that i η Y = inr. It follows that j i = id. By Lmma 5.2, i is an H-algbra homomorphism. Thus th following squar commuts, whnc i j = id. HTY + Y Hi+Y j TY H(HTY + Y ) + Y Hj+Y HTY + Y Proof of (2) (1). Lt (TY,τ Y,( ) ) b a fr complt Elgot algbra on Y with a univrsal arrow η Y : Y TY. Thn [τ Y,η Y ] : HTY +Y TY is an isomorphism with an invrs i. W prov that (TY,i) is a final coalgbra for H( ) + Y. So lt c : H + Y b any coalgbra, and form th flat quation morphism c H + Y H+η Y H + TY. (5.3) Thn is a coalgbra homomorphism from (,c) to (TY,i); in fact, it suffics to stablish that th diagram c H + Y H+η Y H + TY H +TY i H +Y HTY + TY [τ Y,TY ] HTY +η Y TY HTY + Y [τ Y,η Y ]=i 1 commuts. Th uppr part is (5.3), th lft-hand part commuts sinc is a solution of, th right-hand on commuts trivially, and th lowr part is obvious.

27 ELGOT ALGEBRAS 27 Now suppos that s is a coalgbra homomorphism from (,c) to (TY,i). W prov that s =. Obsrv first that s is a morphism of quations from to th following flat quation morphism f TY In fact, th following diagram i HTY + Y HTY +η Y HTY + TY, (5.4) c H + Y H+η Y H + TY s TY i Hs+Y HTY + Y HTY +η Y Hs+TY HTY + TY f commuts: th lft-hand squar dos sinc s is a coalgbra homomorphism, th right-hand on commuts trivially and th uppr and lowr parts ar du to (5.3) and (5.4). By functoriality of ( ) w obtain f s =. W shall show blow that f : TY TY is a solution-prsrving map with f η Y = η Y. By th frnss of TY, w thn conclud that f = id, whnc = s as dsird. To s that f η Y = η Y considr th following diagram TY f TY i [τ Y,η Y ] f HTY +η Y HTY + Y HTY + TY [τ Y,TY ] Hf +TY HTY + TY which commuts sinc f is a solution of f. Follow th right-hand componnt of th coproduct HT Y + Y to s th dsird quation. To complt our proof w must show that th following triangl TY f (f ) TY commuts for any flat quation morphism : H + T Y. Notic first that (5.5) (f ) = (f ) inl : TY (5.6)

28 28 J. ADÁMEK, S. MILIUS, AND J. VELEBIL by compositionality. Furthrmor, w hav a morphism [,TY ] of quations from f to f. In fact, th diagram blow commuts: f + TY [,inr] H + TY H+i H + HTY + Y can+η Y H( + TY ) + TY [,TY ] H +TY HTY + TY [τ Y,TY ] [inr,i] TY HTY + Y i [H,HTY ]+Y HTY +η Y H[,TY ]+TY HTY + TY By functoriality w obtain th following quality f f [,TY ] = (f ), whos lft-hand componnt provs du to (5.6) th dsird commutativity of (5.5). (1 ) (2). W only nd to show th univrsal proprty. Suppos that (TY,τ Y,( ) ) is a fr CIA on Y with a univrsal arrow η Y : Y TY. Du to th quivalnc of (1) and (1 ), [τ Y,η Y ] has an invrs i, and (TY,i) is a final coalgbra for th functor H( )+Y. Now lt (A,α,( ) ) b a complt Elgot algbra and lt m : Y A b a morphism of A. Solv th following flat quation morphism g TY i HTY + Y HTY +m HTY + A in A to obtain a morphism h = g : TY A. W first chck that h η Y = m. In fact, th following diagram TY h A i [τ Y,η Y ] g HTY HTY + Y +m HTY + A [α,a] Hh+A HA + A commuts sinc h is a solution of g. Considr th right-hand componnt of th coproduct HTY + Y to obtain th dsird quation. Nxt lt us show that h is a solution-prsrving morphism. Mor prcisly, w show that for any quation morphism : H + T Y th triangl commuts. Sinc h = g, th quality TY h (h ) A (5.7) (h ) = (g ) inl : A (5.8)