View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 318 (2004) 355–372 www.elsevier.com/locate/tcs Continuous monoids and semirings Georg Karner∗ Alcatel Austria, Scheydgasse 41, A-1211 Vienna, Austria Received 30 October 2002; received in revised form 11 December 2003; accepted 19 January 2004 Communicated by Z. Esik Abstract Di1erent kinds of monoids and semirings have been deÿned in the literature, all of them named “continuous”. We show their relations. The main technical tools are suitable topologies, among others a variant of the well-known Scott topology for complete partial orders. c 2004 Elsevier B.V. All rights reserved. MSC: 68Q70; 16Y60; 06F05 Keywords: Distributive -algebras; Distributive multioperator monoids; Complete semirings; Continuous semirings; Complete partial orders; Scott topology 1. Introduction Continuous semirings were deÿned by the “French school” (cf. e.g. [16,22]) using a purely algebraic deÿnition. In a recent paper, Esik and Kuich [6] deÿne continuous semirings via the well-known concept of a CPO (complete partial order) and claim that this is a generalisation of the established notion. However, they do not give any proof to substantiate this claim. In a similar vein, Kuich deÿned continuous distributive multioperator monoids (DM-monoids) ÿrst algebraically [17,18] and then, again with Esik, via CPOs. (In the latter paper, DM-monoids are called distributive -algebras.) Also here the relation between the two notions of continuity remains unclear. The present paper clariÿes these issues and discusses also more general concepts. A lot of examples are included. For the sake of clarity and conciseness, the presentation is restricted to monoids and semirings. ∗ Corresponding author. E-mail address: [email protected] (G. Karner). 0304-3975/$ - see front matter c 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2004.01.020 356 G.Karner / Theoretical Computer Science 318 (2004) 355–372 The main technical tool are some topological concepts presented ÿrst by the author in [13]. At that time the topological approach was used mainly to justify the algebraic deÿnition. In the present paper, however, the topologies used serve as tools in the proofs. The paper is organised as follows. Section 2 collects the necessary deÿnitions from the papers mentioned above. In Section 3, we discuss the transition from the CPO- based deÿnition to the purely algebraic one. Section 4 presents results converse to the previous one. Section 5 summarises our results in the framework of category theory. Finally, we consider yet another notion of continuity, this time from the realm of CPOs, and relate it to our results. 2. Basic concepts The presentation in this section follows [5] but covers also related work. For basic topological notions, we refer the reader to standard textbooks on the topic, e.g. [14]. We consider monoids (A; ·; 1), which are commutative in most cases and will then be written additively, i.e. as (A; +; 0). Non-commutative monoids will occur as the multiplicative structure of semirings. A function f : An → A, n¿1, on a commutative monoid (A; +; 0) is distributive,if for all j6n and all a1;:::;an;b1;b2 ∈ A, f(a1;:::;aj−1; 0;aj+1;:::;an)=0; (1) f(a1;:::;aj−1;b1 + b2;aj+1;:::;an)=f(a1;:::;aj−1;b1;aj+1;:::;an) + f(a1;:::;aj−1;b2;aj+1;:::;an): (2) A commutative monoid (A; +; 0) together with a number of constants and distributive functions on A is a distributive algebra (or DM-monoid [17,18]). For the sake of conciseness and clarity, we will consider only one such class of algebras explicitly, viz. the semirings. A semiring (A; +; 0; ·; 1) is a distributive algebra such that (A; ·; 1) is a monoid. 2.1. Complete monoids A complete monoid (A; +; 0;) is a commutative monoid together with sums for all families (ai|i ∈ I) of elements of A, where I is an arbitrary index set, such that the following conditions are satisÿed: ai =0; ai = aj; (3) i∈∅ i∈{j} ai = aj + ak for j = k; (4) i∈{j;k} ai = ai if Ij = I and Ij ∩ Ij = ∅ for j = j : (5) j∈J i∈Ij i∈I j∈J G.Karner / Theoretical Computer Science 318 (2004) 355–372 357 Complete monoids have been studied in detail by Krob [15]. Given a family (ai|i ∈ I), we write F ⊆ÿn I for ÿnite subsets F ⊆ I, and aF for i∈F ai, and (aF |F ⊆ÿn I) for the net of all aF ’s over the partially ordered index set ({F|F ⊆ÿn I}; ⊆). A function f : An → A, n¿1 on a complete monoid (A; +; 0;)iscomplete or -preserving, if for all j6n, all a1;:::;an ∈ A, and families (bi|i ∈ I)onA, f a1;:::;aj−1; bi;aj+1;:::;ak = f(a1;:::;aj−1;bi;aj+1;:::;ak ): (6) i∈I i∈I Note that every -preserving function is distributive. A complete semiring (A; +; ·; 0; 1;) is a complete monoid (A; +; 0;) which is at the same time a semiring (A; +; ·; 0; 1) such that · is -preserving. If (A; +; 0;) is complete, the summation topology on A is the collection of all sets O ⊆ A satisfying the following condition. If i∈I ai ∈ O then there is F0 ⊆ÿn I such that aF ∈ O for each F with F0 ⊆ F ⊆ÿn I. The following characterization follows directly from the deÿnitions. Lemma 1. Let (A; +; 0;) be a complete monoid. A subset X ⊆ A is closed in the summation topology i0, for every net (aF |F ⊆ÿn I) which is frequently in X we have i∈I ai ∈ X . The summation topology is not Hausdor1, in general, see [13]. In the case of semir- ings, the topology has been named Karner topology in [9]. We now give an example for a complete monoid and semiring. A semitopological monoid (A; ·; 1;) is a monoid (A; ·; 1) with a Hausdor1 topology such that · is continuous in each argument. (Note: Golan [9] uses the term with a di1erent meaning.) A commutative semitopological monoid (A; +; 0;)isst-complete, if all nets (aF |F ⊆ÿn I) converge. Theorem 2 (Karner [13, Property 3]). Every st-complete monoid (A; +; 0;) becomes a complete monoid (A; +; 0;) under ai = lim(aF |F ⊆ÿn I); (7) i∈I where lim is taken with respect to . Moreover, every distributive, argumentwise continuous function on (A; +; 0;) is -preserving. Theorem 2 generalises easily, cf. again [13, Property 3]. A semitopological semiring (A; +; ·; 0; 1;) is a semiring (A; +; ·; 0; 1) with a topology such that both (A; +; 0;) and (A; ·; 1;) are semitopological monoids. A semitopological semiring is st-complete if the underlying semitopological monoid is. Corollary 3. Every st-complete semiring (A; +; ·; 0; 1;) is a complete semiring (A; +; ·; 0; 1;), where the summation is given by (7). 358 G.Karner / Theoretical Computer Science 318 (2004) 355–372 In fact, Corollary 3 only uses the fact that · is distributive and argumentwise contin- uous. This allows us to use Corollary 3 for arbitrary st-complete distributive algebras. Such generalisations will be possible for other results as well but will be stated for semirings only. ∞ Example 4. Let Q =(Q+ ; +; ·; 0; 1; 6;) be the one-point compactiÿcation of the dis- crete space Q+ of non-negative rational numbers and let 6 denote the usual order. ∞ Let the semiring operations be extended for x ∈ Q+ as follows: 0ifx =0; x + ∞ = ∞ + x = ∞;x·∞= ∞·x = (8) ∞ otherwise: Then Q is an st-complete semiring, hence complete by Corollary 3. The summation is given as follows: aF if F := {ai =0} is ÿnite; ai = (9) i∈I ∞ otherwise: Proof. All points x ∈ Q+ are isolated in . Furthermore, the neighbourhoods of ∞ are ∞ ∞ the sets with ÿnite complements, i.e. all sets Y ⊆ Q+ with ∞∈Y and Q+ \Y ÿnite. ∞ Let (ai|i ∈ I) be a family of elements of Q+ .Ifai = 0 for all but ÿnitely many i ∈ I or ai = ∞ for at least one i ∈ I, the net (aF |F ⊆ÿn I) converges trivially. So let us assume that ai¿0 for inÿnitely many i ∈ I and that ai¡∞ for all i ∈ I. Set ∞ s = sup(aF |F ⊆ÿn I) where sup is taken in R+ . By our assumptions, the sup cannot be attained, i.e. aF ¡s for all F ⊆ÿn I. Let now b ∈ Q.Ifb¡s then there is Fb ⊆ÿn I with a ¿b and thus a ¿b for all F ⊆ F ⊆ I.Ifb¿s, a = b for all F ⊆ I.Weset Fb F b ÿn F ÿn ∞ Fb = ∅ in this case. Let now Y ⊆ Q+ be a neighbourhood of ∞ and set FY = b=∈Y Fb. ∞ Since the complement of Y in Q+ is ÿnite, FY is a ÿnite set. Then for FY ⊆ F ⊆ÿn I, aF ∈ Y . This shows that lim(aF |F ⊆ÿn I)=∞ as required. 2.2. Finitary and continuous monoids A partially ordered ( p.o.) monoid (A; +; 0; 6) is a commutative monoid (A; +; 0) together with a partial order 6 on A such that for all a; b and c ∈ A,06a, and a6b implies a+c6b+c. A is called sum-ordered if a6b holds i1 a+c = b for some c ∈ A. (This is also called natural order or di1erence order.) A p.o. semiring (A; +; ·; 0; 1; 6) is a semiring where (A; +; 0; 6) is a p.o. monoid and · is monotone in each argument. 1 According to Goldstern [10], a ÿnitary monoid (A; +; 0; 6;) is a p.o. monoid (A; +; 0; 6) which is a complete monoid (A; +; 0;) under the deÿnition ai := sup(aF |F ⊆ÿn I): (10) i∈I More precisely, A is ÿnitary if all the sups on the right-hand side of (10) exist and the thus deÿned satisÿes (3)–(5).