Completions of ordered magmas

Transcription

1 ANNALES SOCIETATIS ATHEATICÆ POLONÆ Seres VI : Fundamenta Informatcæ III1 (1980), Completons of ordered magmas BRUNO COURCELLE Unversty of Bordeaux-I JEAN-CLAUDE RAOULT IRIA, Parsq XI Receved arch 29, 1978 AS Categores : 06A75, 18A40, 68A05 AbstractWe gve a completon theorem for ordered magmas (e ordered algebras wth monotone operatons) n a general form Partcular nstances of ths theorem are already known, and new results follow The semantcs of programmng languages s the motvaton of such nvestgatons Key wordscomplete partal orders, semantcs of programmng languages Introducton When defnng recursve functons by systems of equatons (Kleene [5]), one ntroduces an order relaton whch means that a partal result approxmates another one Ths partal order s complete (e every ascendng chan admts a least upper bound), thus allowng a mnmal soluton to be defned for the system Ths matter has been rebult by Scott, and many authors after hm, wthn the framework of complete lattces ; that last theory has been developed for ts own sake by several authors, among whch Brkhoff [1] Frequently, the lattce structure does not seem necessary and creates nstead addtonal troubles (Plotkn [9], lner [8] for nstance) The noton of complete partal order s good enough, and fts better to the most common nstances Ths algebrac framework s sutable for studyng program schemes([2], [3]) We then need dstngush between the base functons and the program-defned functons, wth the help of base functons and varous control structures (recursve call, teraton, etc ) Thus, our domans wll be ordered magmas, e partal orders equpped wth monotone operators (no nformaton s lost durng a computaton) And we shall be concerned wth completeness (the operators beng supposed contnuous) ore precsely, we shall study the possble embeddngs of an ordered magma nto a complete ordered magma Some of the ascendng chans may keep ther lub, or may be added a new one ; ths gves dfferent completons, each characterzed by a unversal property We shall thus defne the Γ-completon as the completon whch preserves the lub whch already exst n a set Γ of subsetes of the magma From ths general theorem s defved the deal completon of [9], [1], [2], [4], the chan-completon of [8], and the exstence of factor objects n the category of ordered magmas The above mentonned authors woule use nether operators not magmas, but only partal orders (except [4]) But eht chan-completon n the category of partal orders need not be a complete ordered magma (cf Corollary 2) Defntons and the man theorem Let F be a set of operators wth arty An F-magma s a doman D together wth a functon f : D k D for each f F wth arty k The homomorphsms of F-magmas, or F-morphsms, shall be compatble wth the operators : ϕ : s a F-morphsm when ϕ(f (a 1,, a k )) = f (ϕ(a 1 ),, ϕ(a k )) for all f F wth arty k, and all a 1,, a k D In ths paper, we shall only consder ordered magmas (therefore magma wll mean ordered magma ), wth a partal order denoted by, a least element Ω (assocated wth the symbol Ω of arty 0 whch s always supposed to be an element of F), and monotone operators f An F-morphsm between ordered magmas must be monotone

2 2 Completons of ordered magmas Fnally, f s a magma, a sub-f-magma N of s a F-magma such that D N D, and that the ncluson : n s a (monotone) F-morphsm It s full when d d mples d N d for every d, d D N Let Γ denote a set of non-empty subsets of D The F-magma s sad to be Γ-complete when every A Γ admts a lub n D, denoted by sup A or x A x ; f m (A 1,, A k ) = {f (a 1,, a k ); a 1 A 1, a k A k } Γ and sup f (A 1,, A k ) = f (sup A 1,, sup A k ) for all f wth arty k and all A 1,, A k Γ A monotone functon ϕ : k N where s Γ-complete and N s an arbtrary magma s Γ-contnuous f for all A 1,, A k Γ, ϕ(a 1,, A k ) admts a lub n N and sup N ϕ(a 1,, A k ) = ϕ(sup A 1,, sup A k ) Wth each F-magma s naturally assocated Ξ() = the set of non-empty subsets of D, Φ() = the set of non-empty fnte subsets of D, () = the set of non-empty drected subsets of D (A () d, d A, d A d d and d d, α () = the set of non-empty drected subsets of cardnalty at most α, Λ() = the set of non-empty drected subsets of D whch admt a lub n D, Θ α () = the set of monotone morphsms of an ordnal α nto (the α-chans), etc Hence Λ Ξ Thus, a partal order s a complete lattce (cf Brkhoff [1]) f and only f t s Ξ-complete, a jon-sem-lattce f and only f t s Φ-complete, a lattce f and only f t s Φ -complete wth Φ () = {{d D ; d d 1 & & d d k }; k N, d 1,, d k D } A F-magma s pre-complete f and only f t s Λ-complete Notce that ths condton only affects the f m s If F only contans symbols of arty 0, then every F-magma s precomplete (cf corollares 1 and 2, and theorem 3) To each of these famles Γ s attached a category the objects of whch are the Γ-complete F-magmas, and the arrows of whch are the monotone F-morphsms f : N that send a subset A Γ() on a subset f (A) n Γ(N) and furthermore such that f (sup A) = sup f (A) : the Γ-contnuous morphsms These restrctons can be automatcally satsfed by the monotone morphsms for some famles, but t s not always the case, as notced for Λ In the sequel, we shall restrct ourselves to those functoral famles and related morphsms The man theorem of ths paper reads as follows THÉORÈE 1 Let Γ Γ two famles of subsets, and a Γ-complete F-magma There exsts a Γ -complete F-magma Γ Γ and a Γ-contnuous njectve F-morphsm : Γ γ such that for all Γ-contnuous morphsms j : N where n s Γ -complete, there exsts a unque Γ -contnuous morphsm h : Γ Γ N such that j = h Γ Γ j h N Proof : The constructon of Γ Γ wll be carred out n the partcular case when Γ = Ξ A non-empty subset A of D s Γ-closed when (1) d d and d A d A (hence Ω A), (2) B A and B Γ sup B A The ntersecton of a famly of closed subsets s closed ; therefore for all non-empty subset A of D, there exsts a smallest closed subset contanng A, ts closure, denoted by C Γ (A) or C(A) when Γ s clear from the context ; the set ˆD of non-empty sebsets of D ordered by set ncluson s a complete lattce : for all famly (A ) I (whch contans Ω D ) A A = I I A = C ( ) A I I

3 3 The lattce ˆD s gven a structure of an F-magma by settng for all closed subsets C ˆD f ˆD (C 1,, C k ) = C ( {f (d 1,, d k ; d C } ) One checks that, for all subsets A 1,, A k of D ( ) C ( f (C(A 1 ),, C(A k )) ) = C ( f (A 1,, A k ) ) As a result, ˆD s a complete F-magma ; let us check t for the frst argument : f ˆD ( ( ) A, B 2,, B k ) = f ˆD C( A ), B 2,, B k = C ( f ( A, B 2,, B k ) ) from ( ) = C ( f (A, B 2,, B k ) ) = f ˆD (A, B 2,, B k ) because for all famly of subsets P, C ( P ) = C ( C(P ) ) holds snce C s a closure operaton (Brkhoff [1], p 111) The correspondng F-magma s denoted by Ξ Γ DEFINITION OF Let : D ˆD be defned by (d) = C({d}) = {x D; x d} Clearly d d (d) (d ) ; furthermore, s Γ-contnuous : f d = sup A wth A Γ, one shoud have (d) = {x D; x d} = x A (x) Now that last element s C(A) and snce A Γ, sup A = d C(A) Fnally, ( ) shows that s an F-morphsm CONSTRUCTION OF Γ Γ Set E 0 = {(d); d D} and let E denote the smallest subset of ˆD contanng E 0 whch s Γ -complete (e such that A E for all famly (A ) I n Γ (E)) It can be descrbed more explctly by E = α E α (α s an ordnal), E α+1 = E α {sup X; X Γ (E α )}, ˆD E β = α<β E α f β s a lmt ordnal In fact, E = E γ for some ordnal γ because ˆD s a set Notce that E 1 {C(A); A Γ (D)} Snce the subsets of Γ are sent by f onto subsets of Γ, E s nvarant under f ˆD Hence (E,, (f E) f F ) s a sub-f-magma of Ξ Γ (f E s the restrcton of f ˆD to E) whch s Γ -complete for the nduced order We shall denote t by Γ Γ UNIVERSAL PROPERTY OF (, Γ Γ ) Let j : N be a Γ-contnuous morphsm nto a Γ -complete F-magma We shal defne h : Γ Γ N by transfnte nducton : h s defned over (D) = E 0 by the condton h = j : h(c({d}) = j(d) Suppose that h s defned over E α for all α < β Then, f β s a lmt ordnal, then h s defned over E β = α<β E α Else h s defned by the Γ -contnuty : f e E β E α then e = sup X where X Γ (E α )

4 4 Completons of ordered magmas and {h(x); x X} Γ (N) admts a lub n n N We set h(e) = n, so that h s now defned over E β By nducton, t s defned over E, and s clearly Γ -contnuous by constructon Fnally, one can check that h s ndeed a morphsm, e that t s compatble wth the operators Ths constructon generalzes the completon of arkowsk-rosen [7] and arkowsk [6] wth the ntroducton of operators (cf theorem 3 and corollary 4) and that of Blecher-Schneder [2] by usng a parameterzed noton of lmts and contnuty That last possblty s also consdered n [10] Snce the magma Γ Γ s completely specfed, t s the object functon of a functor from the category of Γ-complete magmas to that of Γ -complete magmas, and s a left adjont to the obvous forgetful functor From ths fact, t can be deduced easly that f Γ Γ Γ, then ( Γ Γ ) Γ Γ = Γ Γ II Consequences of theorem 1 An mportant pont n the proof of the theorem s that all morphsms should belong to the approprate category, ncludng the functons f It goes wthout sayng when Γ s Ξ of Φ or even, that any monotone functon wll send a subset of Γ() onto another subset of Γ() (or Γ(N) f t s a morphsm N) It s not so obvous when Γ s Λ : ths was the purpose of the remark followng the defnton of a precomplete magma If s a lattce, e a Φ -complete ordered set, Ξ Φ s exactly the deal completon as defned by Brkhoff ([1] theorem 5 p 113) If s also a F-magma, the correspondng object n the category of all -complete F-magmas s One checks easly that ts doman s the set of non-empty deals In partcular, f = (F) s the free ordered F- magma (dentfed wth the set of fnte terms over alphabet F), the correspondng objetct s the free -complete F-magma (from theorem 1), whch can also be dentfed wth the set of nfnte terms or trees, as defned n [3], [4] (and [6] wth a somewhat dfferent termnology) Completon by cuts Let be a precomplete F-magma, e a Λ-complete, magma where Λ s the set of drected subsets of whch admt a lub In ths case, Λ s a completon of whch preserves the already exstng lub s of drected sets, and corresponds to the completon by cuts n lattces as defned by Brkhoff ([1], theorem 22, p 126) Ths constructon can also be found n arkowsk [6] These results are regrouped n the followng corollary COROLLAIRE 2 Let be an oredered (resp precomplete) F-magma There exsts a -complete F-magma (resp Λ ) such that s a full sub-magma (and furthermore the ncluson : Λ s Λ-contnuous), and for all -complete F-magma N, and all morphsms j : : N (resp Λ-contnuous morphsms), there exsts a unque -contnuous morphsm k : N (resp k : Λ N) extendng j Remark : It s necessary for to be precomplete f t s to be embedded n a complete ordered magma Λ, so that COROLLAIRE 3 A magma s precomplete f and only f t s a full -sub-magma of a -complete magma Notce that f s not precomplete, Λ s an F-magma wth doman -complete but the f are not -contnous Ths pont wll be dealt wth n corollary 4 We shall use as an other notaton for Λ Remarks : cnelle s constructon gven n Brkhoff [1] conssts n takng the set of all subsets of the lattce to be completed, then ther upper bounds, and fnally the sets of lower bounds of these upper bounds, as elements of the completed doman But t does not gve the desred result when appled to a precomplete magma Anyway, cnelle s completon does not satsfy the expected unversal property An example s shown on fgure 1 Let + denote cnelle s completon of lattce The ncluson : N s a lattce-morphsm whch preserves all lub s (and all glb s) exstng n It does not extend nto a morphsm of complete lattces + N III Quotents of complete F-magmas Let be a Γ-complete F-magma, for some Γ, and π an F-preorder over, e a preorder over D such that

5 5 ω α β b n b n b n a n a n a n b 1 b 1 b 1 a 1 a 1 a 1 b 0 b 0 b 0 a 0 a 0 a 0 Lattce Lattce + Lattce N Fg 1 1) d d entals d π d for all d, d D, 2) d π d for = 1,, k entals f (d 1,, d k ) π f m (d 1,, d k ), and furthermore, whch s Γ-contnuous, e 3) for all A Γ and all d D such that d π d for all d A, then (sup A) π d Let /π be the ordered F-magma quotent of by the equvalence assocated wth the preorder π (e d π d and d π d), h π : /π the natural epmorphsm, Γ-contnuous from 3) and Γ(/π) = {h π (A); A Γ} Then we have the followng theorem THEORE 2 Let be a Γ-complete F-magma for Γ Γ and π a Γ-contnuous preorder on Then there exsts a Γ-contnuous morphsm wth kernel π : (/π) Γ Γ n a Γ -complete magma, such that for all Γ-contnuous morphsm j : N whose kernel contans π, there exsts a unque Γ -contnuous morphsm k : (/π) Γ Γ N such that j = k The kernel of a morphsm : s the preorder κ such that d κ d ff (d) (d ) Proof : Clearly the magmaˆ/π s Γ(/π)-complete and h π : /π s Γ-contnuous We apply theorem 1 and get (/π) Γ Γ whch s Γ -complete and a Γ-contnuous morphsm : /π (/π)γ Γ the kernel of whch s the order on /π The morphsm = h π : (/π) Γ Γ s thus Γ-contnuous wth kernel π All morphsms j : N wth kernel contanng π can be factored nto the composton hπ /π j N, and j s Γ(/ p)-contnuous From theorem 1, there exsts a unque Γ -contnuous k : (/π) Γ Γ N such that j = k, hence j = j h π = k h π = k Ths proof can be llustrated by the followng commutatve dagram whch determnes completely j and k : hπ /π j j k N (/π) Γ Γ COROLLARY 3 Let be a -complete F-magma The -contnuous F-preorders on are exactly the kernels of -contnuous morphsms nto -complete F-magmas One derves also the followng generalzaton of corollary 1 for magmas whch are not precomplete THEORE 3 Let be an ordered F-magma There exsts a -complete F-magma and a -contnuous morphsm : such that for all -complete F-magma N and -contnuous morphsm j : N, there exsts a unque -contnuous morphsm k : N such that j = k

6 6 Completons of ordered magmas Proof : Let (D, ) be a preorder An element d of D s a lub of A D when 1) d d for all d A, 2) d s smaller than all upper bounds of A, e d d for all d A mples that d d Notce that A may well have several lub s Let be an ordered F-magma There exsts a least F-preorder π such that ( ) f A 1, A k are π-drected subsets of D wth lub s δ 1,, δ k respectvely, and f the π-drected subset f (A 1,, A k ) admts an upper bound δ for π, then f (δ 1,, δ k ) π δ We clam that /π s precomplete and satsfes the followng unversal property : for all -contnuous F-morphsm j : N n a -contnuous F-magma N, there exsts a unque -contnuous F-morphsm k : /π N such that j = kh π (h π beng the natural epmorphsm /π) The theorem wll be deduced from corollary 1 wth = /π Let A D, A = h π (A) D /π, and h π (d) the lub of A It s easy to see that A s drected for π and D s one of the lub s of A for π If A 1,, A k /π admt lub s h π (d 1 ),, h π (d k ) and d = f (d 1,, d k ), we must show that h π (d) s the lub of f /π (A 1,, A k ) Now, snce π s an F-preorder, f /π (h π (a 1 ),, h π (a k )) = hπ(f m (a 1,, a k )) h π (f (d 1,, d k )) = h π (d) On the other hand, f h π (d ) f /π (A 1,, A k ), then d s an upper bound of f (A 1,, A k ) wth respect to π ; hence f (d 1,, d k ) π d, e h π (d) h π (d ), QED Therefore the followng dagram n whch exsts even f s not precomplete s commutatve : 1 2 COROLLARYN 4 The morphsm h s onto It s one-to-one f and only f for all A 1,, A k such that f (A 1,, A k ) then f ( sup A 1,, sup A k ) = sup f (A 1,, A k ) Proof : On one hand, f the condton holds, then tself satsfes ( ), hence π = Therefore = /π = On the other hand, suppose that h s one-to-one, and that A 1,, A k, f (A 1,, A k ) wth respectve lub s δ 1,, δ k, δ, and δ = f (δ 1,, δ k ) Obvously δ δ But δ π δ entals that 2 (δ) = 2 (δ ) Snce h s one-to-one, and the dagram commutes, 1 (δ) = 1 (δ ) Snce 1 s th ncluson, the same holds n : δ = δ, QED EXAPLE Let denote the F-magma N {α, β} ordered by j α β for j n N The only element of N s of arty one defned by { f () = f N, f (α) = f (β) = β, whch s not -contnuous Snce the doman of s -complete, as an ordered set, = But n order to have a contnuous f, one must dentfy α and β, thus obtanng wth doman N γ and h :, as shown by fgure 2 β α γ h 0 0 =

7 7 Fg 2 References [1] Brkhoff, G, Lattce theory, Amer ath Soc, Provdence 1967 [2] Blecher, N, et al, Permanence of denttes on algebras, Algebra Unversals 3 (1973), [3] Courcelle, B, Nvat,, Algebrac famles of nterpretatons, 17 th Symposum on FOCS, Houston 1976 [4] Goguen, J, et al, Intal algebra semantcs and contnuous algebras, J Assoc Comput ach 24 (1977), [5] Kleene, SC, Introducton to metamathematcs, Van Nostran, Prnceton 1952 [6] arkowsk G, Chan-complete posets and drected sets wth applcatons, Algebra Unv 6 (1976), [7] arkowsk G, Rosen, B, Bases for chan-complete posets, IB J Res Develop 20 (1976), [8] lner, R, Fullty abstract models of types lambda-calcul, Theoretcal Computer Scence 4 (1977), 1 22 [9] Plotkn, G, T ω as a unversal doman, JCSS 17 (1978), [10] Thatcher, J, Wagner, E, Wrght, J, A unform approach to nductve posets and nductble closure, Theoretcal Computer Scence 7 (1978), [11] Vullemn, J Syntaxe, sémantque et axomatque d un langage de programmaton smple, Brkhaüser, Basel 1975 References added n proof (June 1980) and not cted n the text : [12] Bshop, A, A unversal characterzaton of the completon by cuts, Algebra Unversals 8 (1978), [13] Courcelle, B, Nvat,, The algebrac semantcs of recursve program schemes, athematcal Foundatons of Computer Scence 78, Lec Notes Comput Sc 64 (1978), [14] eseguer, J, Completons, factorzatons and colmts for ω-posets, Proc of Coll Logc n Programmng, Salgotarjan, Hungary (1978) ; also Semantcs and computaton report 13, U C L A (revsed verson, June 1979)