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Duality Between Modal Algebras and Neighbourhood Frames

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Duality Between Modal Algebras and Neighbourhood Frames KOsA Duality Between Modal Algebras OoEN and Neighbourhood Frames Abstract. This paper presents duality results between categories of neighbourhood frames for modal logic and categories of modal algebras (i.e. Boolean algebras with an additional unary operation). These results extend results of Goldblatt and Thomason about categories of relational frames for modal logic. The representation theory for Boolean algebras with additional unary operations started with the work of J6nsson and Tarski [6]. In the seventies, modal logicians became aware of this algebraic theory, and they incorporated it in the theory of so-called Kripke relational frames for modal logic. Two standard results from this period are to be found in [4] and I1 i], where it is shown that certain categories of relational frames are dual to categories of Boolean algebras with operations called normal modal algebras (see Sections 8-10,16 below); when we say that two categories are dual, we mean that they are equivalent by two contravariant functors. Some notions which appear in these duality results seem to have been formulated independently by modal logicians and algebraists, as witness the notion of frame homomorphism from [4] (p. 53; see Section l0 below), whose analogues we can find in [8] (p. 37) and [12] (p. 2). In general, one might say that on the logical side a greater stress is made on the set-theoretical aspects of the underlying Stone duality, whereas on the algebraic side topological aspects are more prominent (see, for example, [5]; however, the related duality theory for Heyting algebras, worked out by the logician Esakia in [2] and [3], stresses also topological aspects). In this paper we shall present in a rather self-contained manner what, starting from a logical background, can be said concerning duality for Boolean algebras with an arbitrary additional unary operation. These algebras, which we shall call modal algebras, form a wider class than normal modal algebras. For this duality we shall consider a more general class of frames which occur in modal logic, viz. neighbourhood frames. The underlying Stone duality will be manifested by its set-theoretical aspects. In the first part of the paper we show what category of neighbourhood frames is dual to the category defined by modal algebras and their homomor- phisms. These neighbourhood frames, and the required frame morphisms, are a generalization of the descriptive relational frames and frame homomorphisms of Goldblatt [4]. In general, the duality results we present are an extension of ideas in [4]. We shall compare our results with Goldblatt's, and show how they are related. In the second, shorter part of the paper, we consider what category 220 K. Dogen of modal algebras is dual to the category of full neighbourhood frames and appropriate frame morphisms. The results of this part build on corresponding results of Thomason [11] about categories of full relational frames. At the end, we shall compare our results with Thomason's. The general background of this paper is provided by [4]. Neighbourhood frames for modal logic (sometimes called Scott-Montagueframes) are treated in [10] and [1] (Part IIl; there these frames are called minimal). Basic notions which we presuppose about Boolean algebras and categories may be found in [9] and [7]. A number of proofs, which we consider straightforward, will be omitted. The remaining proofs are mostly sketches, but the reader should have no difficulty in reconstructing complete proofs. General neighbourhood frames 1. The category MA. A modal algebra is an algebra A = (D, ^, v, ~, 1,0, l--I),where (D, A, V, ~, 1, 0) is a non-degenerate Boolean algebra, and the domain D is closed under the unary operation []. For elements of D we shall use b, b I , b 2 ..... and for elements of the power set .~"D of D we shall use X, Xl, X 2 .... The category MA of modal algebras is defined by the following: objects: modal algebras, morphisms: homomorphisms. We shall use h, h 1, h2, ... for homomorphisms between modal algebras, and Id a will be the identity homomorphism of A. 2. Descriptive neighbourhood flames. A general neighbourhood frame, or general frame for short, is F = (C, N, Dr), where: (l) C # O (C is the carrier of F), (2) D F _ ~C is closed under ~ (finite set intersection), - (set complemen- tation with respect to C), and a unary operation L, and Dr# 0, (3) N:C--,~D r (if xeC, then N(x) is the set of neighbourhoods of x), (4) (V B e D v) (x e LB r B e S (x)). For elements of C we shall use x, x 1, x2, ..., and for elements of ~C we shall use B, B 1, B 2.... For every general frame F let JF = (D r, n, w, -, C, 0, L). It is clear that the following holds: THEOREM 1A. ~F is a modal algebra. Intuitively, a general frame is a set modal algebra spread over a carrier C, and by (4) we can take that either L is defined in terms of N, or N in terms of L. (To define a valuation from a modal propositional language with a single modal operator into a modal algebra, we proceed in the usual obvious way. In Duality between modal algebras ... 221 virtue of the connexion between L and N, a valuation on .~r serves as a valuation on F.) Had we allowed modal algebras to be based also on degenerate Boolean algebras (with a single element), then in general frames we should take C to be any set, including O, and nothing would change essentially in the results which follow. (However, we do not allow degenerate algebras and frames with an empty carrier, because in them we could validate the inconsistent logic made of all formulae.) As over a general frame F, i.e. over its carrier, we have spread a modal algebra dF, so over a modal algebra A we can spread a general frame ,~A = (C A, N A, D~A>, where: (1) C A = {X _= D: X is an ultrafilter of A}, (2) q:D~(~D) is defined by q(b)={XeCA: beX}, and D~A={q(b): beD}, (3) NA:C'I~D '~A is defined by NA(X)= {q(b): DbeX}, (4) (VbED) LA(q(b)) = {XeC'~: q(b)~NA(X)}. (By an ultrafilter we understand as usual a maximal proper filter.) To confirm that .~-A is indeed a general frame we must verify that D*A is closed under ~, - and LA. The mapping q:D--*D ~A is a mapping from A to ~r It is not difficult to prove the following for q: THEOREM 2A. The mapping q is an isomorphism from A to zC(,~A). Analogously to the mapping q we define a mapping p from F to J~(.zCF), i.e. from the carrier of F to the carrier of ff(~CF). Let p:C--,g2(.~ be defined by p(x) = {B e Dr: x e B}. It is easy to check that p(x) is an ultrafilter of ~r hence, p:C~C ~r. Now we can define descriptive neighbourhood frames, or descriptive frames for short, as general frames F which satisfy: (I) p(x~) = p(x2)='xl = x2, (II) (VXeC~F)(qxeC) X = p(x). In other words, (I) p is one-one, and (II) p is onto. 3. Frame morphisms. The objects of the category DF of descriptive frames will be descriptive frames. Now we shall define the morphisms of this category. Let F t = (C1, N t, DF~> and F 2 = (C2,N2,D F2> be general frames. Let f:Cl ~C2, and for every BzeD r2, let (.~r {xl: f(xl)cB2}. Then f is a frame morphism from F 1 to F 2 iff for every B2eD ~'2 and every xleCl: (i) (~r D F', (ii) (s~cf)(B2)e Nl(Xt)c:,B2~ N2(f (xl)). We shall use f, ]'1, f2,-.- for frame morphisms. 6 - Studia Logica 222 K. Dogen It is a straightforward matter to prove the following theorem, where d.f is as above: THEOREM 3A. The mapping fi C 1 --. C z is a frame morphism from Ft to F a iff sl f is a homomorphism from dF 2 to dF r So, if in the definition of general frames we were to take L as primitive, and N as defined, it would be natural to define a frame morphism from F 1 to F z as an f:C 1-*C 2 such that sCf is a homomorphism from ~r 2 to ~r r A frame morphism f from F t to F 2 is a frame isomorphism ifff is one-one and onto, andf -1 is a frame morphism from F 2 to F 1. (In general, there is no guarantee that if f is one-one and onto, then f-J is a flame morphism: for example, if DF# ~C, the identity map f from (C, N, .~C> to (C, N, Dr> is a frame morphism, but f-1 does not satisfy (i); this example is from [4], p. 53.) An alternative definition of frame isomorphisms is provided by the following. Let f be a frame morphism from F1 to F 2 and for every BleD r' let fiB1] = {f(xt): xteBt}. Then f is an embedding ifff is one-one and: (iii) (VB t eDrl)(3B2eD r2) fiB1] =f[Cl] nB 2. We have the following lemma: LEMMA 1. Suppose the frame morphism f from F t to F z is one-one and onto. Then f -t is a frame morphism from F 2 to F t iff f is an embedding.
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