Completions of ordered magmas

We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations. Key wordscomplete partial orders, semantics of programming languages. Introduction. When defining recursive functions by systems of equations (Kleene [5]), one introduces an order relation which means that a partial result approximates another one. This partial order is complete (i.e. every ascending chain admits a least upper bound), thus allowing a minimal solution to be defined for the system. This matter has been rebuilt by Scott, and many authors after him, within the framework of complete lattices ; that last theory has been developed for its own sake by several authors , among which Birkhoff [1]. Frequently, the lattice structure does not seem necessary and creates instead additional troubles (Plotkin [9], Milner [8] for instance). The notion of complete partial order is good enough, and fits better to the most common instances. This algebraic framework is suitable for studying program schemes([2], [3]). We then need distinguish between the base functions and the program-defined functions, with the help of base functions and various control structures (recursive call, iteration, etc...). Thus, our domains will be ordered magmas, i.e. partial orders equipped with monotone operators (no information is lost during a computation). And we shall be concerned with completeness (the operators being supposed continuous). More precisely, we shall study the possible embeddings of an ordered magma into a complete ordered magma. Some of the ascending chains may keep their l.u.b., or may be added a new one ; this gives different completions, each characterized by a universal property. We shall thus define the Γ-completion as the completion which preserves the l.u.b. which already exist in a set Γ of subsetes of the magma. From this general theorem is defived the “ideal completion” of [9], [1], [2], [4], the “chain-completion” of [8], and the existence of factor objects in the category of ordered magmas. The above mentionned authors woule use neither operators not magmas, but only partial orders (except [4]). But eht “chain-completion” in the category of partial orders need not be a complete ordered magma (cf. Corollary 2). Definitions and the main theorem. Let F be a set of operators with arity. An F-magma M is a domain DM together with a function fM : DM → DM for each f ∈ F with arity k. The homomorphisms of F-magmas, or F-morphisms, shall be compatible with the operators : φ : M → M′ is a F-morphism when φ(fM(a1, ... , ak)) = fM′(φ(a1), ... , φ(ak)) for all f ∈ F with arity k, and all a1, ... , ak ∈ DM. In this paper, we shall only consider ordered magmas (therefore “magma” will mean “ordered magma”), with a partial order denoted by ≤M, a least element ΩM (associated with the symbol Ω of arity 0 whichi is always supposed to be an element of F), and monotone operators fM. An F-morphism between ordered magmas must be monotone. 2 Completions of ordered magmas Finally, if M is a magma, a sub-F-magma N of M is a F-magma such that DN ⊆ DM, and that the inclusion i : n → M is a (monotone) F-morphism. It is full when d ≤M d′ implies d ≤N d′ for every d, d′ ∈ DN. Let Γ denote a set of non-empty subsets of DM. The F-magma M is said to be Γ-complete when — every A ∈ Γ admits a l.u.b. in DM, denoted by sup A or ∨ x∈A x ; — fm(A1, ... , Ak) = {f (a1, ... , ak); a1 ∈ A1, ... ak ∈ Ak} ∈ Γ and sup fM(A1, ... , Ak) = fM(sup A1, ... , sup Ak) for all f with arity k and all A1, ... , Ak ∈ Γ. A monotone function φ : Mk → N where M is Γ-complete and N is an arbitrary magma is Γ-continuous if for all A1, ... , Ak ∈ Γ, φ(A1, ... , Ak) admits a l.u.b. in N and sup N φ(A1, ... , Ak) = φ(sup M A1, ... , sup M Ak). With each F-magma M is naturally associated Ξ(M) = the set of non-empty subsets of DM, Φ(M) = the set of non-empty finite subsets of DM, ∆(M) = the set of non-empty directed subsets of DM (A ∈ ∆(M) ⇐⇒ ∀d, d′ ∈ A, ∃d′′ ∈ A d ≤ d′′ and d′ ≤ d′′, ∆α(M) = the set of non-empty directed subsets of cardinality at most α, Λ(M) = the set of non-empty directed subsets of DM which admit a l.u.b. in DM, Θα(M) = the set of monotone morphisms of an ordinal α into M (the α-chains), etc. Hence −Λ ⊆ ∆ ⊆ Ξ. Thus, a partial order is a complete lattice (cf. Birkhoff [1]) if and only if it is Ξ-complete, a join-semi-lattice if and only if it is Φ-complete, a lattice if and only if it is Φ′-complete with Φ′(M) = {{d′ ∈ DM; d′ ≤ d1 & ... & d′ ≤ dk}; k ∈ N, d1, ... , dk ∈ DM}. A F-magma is pre-complete if and only if it is Λ-complete. Notice that this condition only affects the fm′s. If F only contains symbols of arity 0, then every F-magma is precomplete (cf. corollaries 1 and 2, and theorem 3). To each of these families Γ is attached a category the objects of which are the Γ-complete F-magmas, and the arrows of which are the monotone F-morphisms f : M → N that send a subset A ∈ Γ(M) on a subset f (A) in Γ(N) and furthermore such that f (sup A) = sup f (A) : the Γ-continuous morphisms. These restrictions can be automatically satisfied by the monotone morphisms for some families, but it is not always the case, as noticed for Λ. In the sequel, we shall restrict ourselves to those “functorial” families and related morphisms. The main theorem of this paper reads as follows. THÉORÈME 1. — Let Γ ⊆ Γ′ two families of subsets, and M a Γ-complete F-magma. There exists a Γ′-complete F-magma MΓ′ Γ and a Γ-continuous injective F-morphism i : M → MΓ′ γ such that for all Γ-continuous morphisms j : M → N where n is Γ′-complete, there exists a unique Γ′-continuous morphism h : MΓ′ Γ → N such that j = hi. M i → MΓ′ Γ j ↘ ↓ h N Proof : The construction of MΓ′ Γ will be carried out in the particular case when Γ′ = Ξ. A non-empty subset A of D is Γ-closed when (1) d ≤ d′ and d′ ∈ A ⇒ d ∈ A (hence ΩM ∈ A), (2) B ⊆ A and B ∈ Γ ⇒ sup B ∈ A. The intersection of a family of closed subsets is closed ; therefore — for all non-empty subset A of D, there exists a smallest closed subset containing A, its closure, denoted by CΓ(A) or C(A) when Γ is clear from the context ; — the set D̂ of non-empty sebsets of D ordered by set inclusion is a complete lattice : for all family (Ai)i∈I (which contains ΩD) ∧