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. PROOF. (=:-) Let f-~ be a frame morphism from F 2 to F t. Then, by Theorem 3A, ~r t): D r, __, Dr2 is a homomorphism. It is not difficult to check that f[B1] = f[Cl] ~(~r (r Let (iii). hold for f To check (i) for f-t we show that (zg(f-a))(Bt) = B z, It is not difficult to check (ii) for f-i q.e.d.
4. The category DF. It is easy to verify that the identity mapping on C is a frame morphism from F to F; we call this mapping Id r. To verify that the composition of frame morphisms is a frame morphism we just check that ,~r = (zaCf0o(zaCf2), and use Theorem 3A. So, we may define the category DF of descriptive frames by the following: objects: descriptive neighbourhood frames, morphisms: frame morphisms.
5. The funetor ,d. In Sections 2 and 3 we have shown how to associate with each general frame F a modal algebra ~r and with each frame morphismffrom F t to F z a homomorphism ~r from ~r 2 to sCF r It is easy to check the following theorem: THEOREM 4A. (a) ~ld F = Ida, F; (b) ..~r (~f,)~ So, .~g defines a contravariant functor from DF to MA. Duality between modal algebras ... 223
We can also easily check the following theorem for frame morphisms f: THEOREM 5A. (a) lf f is an embedding, then ~f is onto. (b) lf f is onto, then ~f is one-one.
6. The functor ~-J. In Section 2 we have shown how to associate with each modal algebra A a general frame ~-~A. For this general frame we can prove the following: THEOREM I F. ~ A is a descriptive frame. PROOF. (I) Suppose p(X1)= p(X2), i.e. q(b)ep(Xt)c:,q(b)ep(X2). Since beX,~q(b)ep(X), it follows that X t = X 2. (II) IfX' is an ultrafilter of ~r then let X = {b: q(b)eX'}. It is easy to check that X is an ultrafilter of A, and that X'= p(X). q.e.d. If h is a homomorphism from A t to A2, and X 2 is an ultrafilter of A 2, let (~h)(X2) = {bl: h(bl)~X2}. For ~-h we can prove the following theorem: THEOREM 3F. The mapping h: D 1 ~ D 2 is a homomorphism from A t to A 2 iff ~ h is a frame morphism from ~A 2 to ~A r PROOF. (=>) Suppose h: D t --*D2 is a homomorphism from A t to A 2, and X 2 is an ultrafilter of A 2. It is easy to check that (~-h)(X2) is an ultrafilter of At; so, ~:h:Ca2--*C al. It is also easy to check that ~r s~al ~D~a2. Let ql be the isomorphism from A m to d(~Al) and q2 the isomorphism from A z to sC(~-A2). Then we can prove that ~r q2ohoq{ t, and so, ~r is a homomorphism. It follows, by Theorem 3A, that ,~-h is a frame morphism from ~-A 2 to ~-A 1. (~=) If~-h is a frame morphism from ~A z to ~A 1, it follows, by Theorem 3A, that ~(~h) is a homomorphism from ~r to ,~r It remains to show that h= q~lod(~h)oql. So, h is a homomorphism from A t to A 2. q.e.d. So, with each modal algebra A we can associate a descriptive frame ~-A, and with each homomorphism h from A1 to A 2 we can associate a frame morphism ~-h from ~-A 2 to ~A r It remains to check the following theorem: THEOREM 4F. (a) ~Id A = Ida:A; (b) ~-(h2ohl) = (~ht)o(~-h2). So, .~- defines a contravariant functor from MA to DF. We can prove the following analogue of Theorem 5A: THEOREM 5F. (a) If h is onto, then ~h is an embedding. (b) If h is one-one, then ~h is onto. PROOF. (a) Suppose h:DI--,D 2 is onto. It is easy to check that ~-h is one-one. To check (iii) for ~h we proceed as follows. If B 2 e D ~a~, then for some 224 K. Do.qet~ b2ED 2 we have B 2 = q(b2), and since h. is onto, for some b~D 1 we have bE = h(bl). It is not difficult to prove that (,~h)[B2] =(,~h)[CA:]c~q(bt). (b) Suppose h:DI-~D 2 is one-one, and XIeC A'. Let X 2 ={b2~D2: (:lb~Xl) h(b) <~ b2}. We can check that X z is a proper filter of A 2. Hence, X 2 can be extended to an ultrafilter X~ of A 2. Using the fact that h is one-one and that (VbI~XI) (h(bi)~X 2 or -h(bi)~X2) , we can show that Xt--h-~[XE]=(~h)(X~), where h-t[X2] is {b~eDt:h(b~)eX2}. q.e.d. (The proof of 5F (b) is not exactly parallel to the analogous proof of Theorem 10.9 (2) in [4], p. 71. l was unable to prove 10.9 (2) the way Goldblatt did. Instead, one can take y' = {a: (3ai ~x)d/(al) <<, a}, prove that y' is a proper filter, and extend it to an ultrafilter y, for which we have ~h~(y)= x.)
7. Duality between MA and DF. First we prove the following analogue of Theorem 2A: THEOREM 2F. If F is descriptive, then p is a frame isomorphism from F to :~ (~/ F). PROOF. It is trivial that p:C--.C ~F is one-one and onto. I~ remains to prove that p is a frame morphism and that p is an embedding. To prove that p is a frame morphism, we prove first that if q is the isomorphism from .~F to m/(~( ql By Theorem 2A we have that ql and q2 are isomorphisms, and by Theorem 2F we have that p~ and Pz are frame isomorphisms, q.e.d. Duality between modal algebras ... 225 8. Normal modal algebras and filter frames. A modal algebra A is normal iff for every bl, b2eD we have R(b 1 ^ b2)= [-]b t ^ []b 2 and []1 = 1. The category NMA of normal modal algebras is defined by the following: objects: normal modal algebras, morphisms: homomorphisms. A general frame F is a filter frame iff for every xeC the set N(x) is a filter (not necessarily proper) of ~/F. The category DFF of descriptive filter frames is defined by the following: objects: descriptive filter frames, morphisms: frame morphisms. It is not difficult to prove the following theorems: THEOREM 7A. F is a filter frame iff ~F is normal. THEOREM 7F. ~ A is a flter frame iff A is normal. These two theorems, together with the results of the preceding sections, enable us to prove that the categories NMA and DFF are dual by the functors ~t/and 9. Reducible frames and hyperfiiter frames. A general relational frame is Fr = (C, R, Dr), where: (1) c~o, (2) D r~_/~C is closed under r - and a unary operation L, and Dr# f~, (3) R ~_ C 2, and S(x)= df{Xz: xRxl}, (4) (VB~Dr)(x~LBoS(x) ~ B). (General relational frames are sometimes called first-order frames.) In the definition above S (the set S(x) is the set of successors of x) is defined in terms of R; but, alternatively, we could take S as primitive and define R in terms of S by x I Rx2ox 2 E S(xl). Clause (4) enables us to take L as defined in terms of R (or S), but, in contradistinction to what we had with general neighbourhood frames, it does not give a definition of R in terms of L. We shall say that a general relational frame Fr is a reducible frame iff for every x 1, x2~C we have: (5) (V B e D r) (x i e LB =~ x 2 e B) => x I Rx 2. This clause (which is called Axiom II in [4], p. 64) enables us to take R as defined in terms of L in reducible frames, since the converse of (5) holds for all general relational frames. In terms of S, the clause corresponding to (5) would read: (5') N{B~Or: x~LB} ~_S(x). The converse inclusion of (5') holds for all general relational frames. Reducible frames are "intertranslatable" with a special kind of general neighbourhood frames which we proceed to define. Let us call a general neighbourhood frame F = (C, N, DF~ a hyperfilter frame iff for every x~ C: (6) (VBr ~- B=~Br 226 K. DoWn The converse of (6) holds trivially. (Note that a hyperfilter frame is not an exact analogue of what is called an augmented frame in [1], p. 220, because 0 N(x) need not belong to D r, and hence it need not belong to N(x); in other words, N(x) is not necessarily a complete filter.) It is easy to check the following lemma: LEMMA 2. Every hyperfilter frame is a filter frame. The converse of this lemma does not hold necessarily if N(x) is infinite (a counterexample from [1], pp. 220-221, takes F with C the set of real numbers, D r = t~C, and N(x) = (B ~_ C: (3y > x) the open interval (x, y) ___ B}; this F is a filter frame which is not a hyperfilter frame). The following lemmata show that reducible frames are intertranslatable with hyperfilter frames: LEMMA 3.1. If in a hyperfilter frame F = (C, N, DF) we define R by: (7) x1Rx2c:,x2E('~N(Xl), then Fr = ( C, R, Dr) is a reducible frame in which we have: (8) N(x) = {BeDF: S(x) =__ B). LEMMA 3.2. If in a reducible frame Fr = ( C, R, DF) we define N by (8), then F = (C, N, Dr) is a hyperfilter frame in which we have (7). It is clear that with the interdefinability of R and S in relational frames, clause (7) amounts to: (7') S(x) = ~ N(x). It is also clear that descriptive hyperfilter frames are intertranslatable with reducible frames where p is one-one and onto, i.e. with descriptive reducible frames. These last frames are the frames which Goldblatt calls descriptive frames (see [4], pp. 63ff; Axioms I and III of [4] state that p is one-one and onto, whereas Axiom II is built into the definition of reducible frames). Although not every filter frame is a hyperfilter frame, for descriptive frames we have the following lemma: LEMMA 4. Every descriptive filter frame is a hyperfilter frame. PROOF. Suppose F = (C, N, D F) is a descriptive filter frame, and for some x e C and some BeD r we have Br The set N(x)w (-B} has the finite intersection property, i.e. every finite subset of this set has a nonempty intersection (otherwise, since N(x) is a filter, for some B~ e N(x) we would have B 1 c~ -B = 0, and this yields BeN(x), a contradiction). So, N(x)w {-B} can be extended to an ultrafilter X 1 of ~r Since F is descriptive, for some x t e C we have X 1 =p(xl). It remains to check that xle~N(x ) and xl~B. So, AN(x) ~=B. q.e.d. (Note that this proof shows that every filter frame where p is onto is a hyperfilter frame.) Duality between modal algebras ... 227 Hence, descriptive filter frames are intertranslatable with descriptive reducible frames. I0. Frame homomorphisms and frame morphisms. Next we show that Goldblatt's frame homomorphisms between descriptive reducible frames amount to our frame morphisms between descriptive filter frames. In [4] (p. 53) aflame homomorphism between Fr 1 = (C1, R1, D F'> and Fr 2 = (C2, RE, D F2> is a mapping f:Cl-,C 2 such that for every BzeD F', every xl, x2~C 1, and every xeC2: (i) (~ff) (B2) e D F', (ii.i) x I R 1 x 2 =~ f (xl) R2f (x2) , (ii.ii) f(xl)R 2 x =*.3x2(x -- f(x2)&x 1 R 1 x2); where zlf and z~tFr are defined analogously to what we had in Sections 2 and 3. Clause (i) above is exactly as in our definition of frame morphisms in Section 3, whereas clauses (ii.i) and (ii.ii) can be replaced by: (ii) f(xl)g2xc:,3x2(x = f(x2)&x 1 R 1 x2). Given the interdefinability of R and S, we can replace (ii) by: (ii') S2(f(xl) ) = {f(x2):x2~Sl(xl)}, which can be compared with the following reformulation of clause (ii) in the definition of frame morphisms: N2(_f(x,) ) = {B2ED/~2: (~]f)(B2)eN,(x,) }. For frame homomorphisms we can prove the following analogue of Theorem 3A: LEMMA 5. IfFr 1 = (C 1, R 1, Dr'> and Fr 2 = LEMMA 6.2. Let Fr I = (C t, R1, D r') and Fr 2 = (C2, R E, D r2) be descrip- tive reducible farmes, let N 1 in Fr t and N 2 in Fr 2 be defined by (8) of Lemma 3.1, and let f: C l ~ C 2 be a frame homomorphism from Fr I to Fr 2. Then f is a .frame morphism from the descriptive filterframe F t = (Cl, N 1, D ~'~) to the descriptive filter frame F 2 = (C 2, S 2, Dr2). (As a matter of fact, Lemma 6.2 holds if we replace "descriptive reducible" by "reducible", and "descriptive filter" by "hyperfilter', as can easily be checked by a direct proof of (,z/f)(B2) r N l(xl)~B 2 r N2(f(xl)). ) So, the category DFF of descriptive filter frames with frame morphisms is isomorphic with the category DRF of descriptive reducible frames with frame homomorphisms. The functors by which this categorial isomorphism is established are the functors whose definition we can infer from Lemmata 3.1 and 3.2; according to Lemmata 6,1 and 6.2, these functors assign to frame morphisms and frame homomorphisms the very same mapping f:C 1 --,C 2. Now, for the category DRF Goldblatt has shown in [4] (pp. 69-72) that it is dual to the category NMA of normal modal algebras of Section 8. The functors by which this duality is established are the functor d and a functor ~- which is like ~-, save that we have: (3,) X IRa,X2c~{bI:DbleXI} ~_X2, (4,) (VbED)LA,(q(b)) = {X~C~: SA,(X) ~_ q(b)}. The functor ~. conceived as a functor which assigns to a normal modal algebra A a descriptive filter frame, rather than a descriptive reducible frame, has as the corresponding clauses: (3.) NA.(X) = {q(b):S~(X) c_ q(b)} = {q(b):(VX~ eC'4)('Cbt(Rb, eX~b~ ~ X,)~beX~)}, (4,) (VbeD)L~(q(b)) = {X eCa: O NA,(X) ~ q(b)}" Let N'* and LA be defined as for o~(see Section 2) by: (3) na(x) = {q(b): lNbeX}, (4) (VbeD)LA(q(b)) = {XeCa:q(b)eNa(X)}. The following lemma shows that Na(X)= N~ (X): Duality between modal algebras ... 229 LEMMA 7. In every normal modal algebra A, for every beD, and every X e CA: Db ~ X c~(V X 1 e CA)(Vbl ([]bl e X =:'b I ~ X l)=~b ~ X l). PROOF. From left to right the proof is obvious. For the other direction the proof is analogous to the proof of Lemma 4. Suppose [] b r X. Then we show that {b~: l-lbl e X} u {,-~b} has the finite intersection property. The ultrafilter X t to which this set can be extended is such that Vb~([]b~ eX~b~ eXx) and b r X 1. q.e.d. Since NA(X) = Na,(X), and since in .~"~, A, which is a hyperfilter frame, we have (]Na(X)~ q(b)~',,q(b)eSa(X), it follows that La(q(b))= L~(q(b)). So, the functors ~ and .~-, do not differ, and our duality result for NMA and DFF in Section 8 is essentially the same as Goldblatt's duality result for NMA and DRF. Full neighbourhood frames 11. The category CAA. A modal algebra A = (D, ^, v, ---, 1, 0, F-q) is complete iff for every X _ D the infimum of X, denoted by A X, and the supremum of X, denoted by v X, belong to D. An element b e D is an atom iff b -r 0 and (Vb I <~ b)(b I = b or b 1 = 0). For the atoms of a modal algebra we shall use a, al, a2, ..., and for sets of atoms we shall use Z, Z 1, Z 2 ..... A modal algebra is atomic iff (Vb # O)(3a)a <<. b. A complete homomorphism from A 1 to A 2 is a homomorphism h:D~--.D 2 which satisfies: h(^ {bFi~l})= ^ {h(b,):iEl}, h(v {b,:i~I})= v {h(bi):i~l}, The category CA A of complete atomic modal algebras is defined by the following: objects: complete atomic modal algebras, morphisms: complete homomorphisms. To abbreviate, we shall call the objects of CAA full algebras. 12. Full neighbourhood flames. A general neighbourhood frame F = (C, N, D r) is full iff D r = ~C. To abbreviate, we shall call such frames full frames. Full frames coincide with the usual notion of neighbourhood frames. If dF is defined as in Section 2, it is clear that the following holds: THEOREM 7A. If F is a full frame, ~r is a full algebra. Over a full algebra A we define a frame ffA = (C A, N A, D~A), where: (1) CA= {a:aED and a is an atom}, (2) O ~A = ~C A, (3) NA:CA-,~(~C A) is defined by NA(a) = {Z ~_ CA:a ~< [] v Z}, (4) (VZ ~_ CA)LAZ = {a~CA:Zr 230 K. DoWn We can immediately infer the following: THEOREM 7F. If A is a full algebra, ~A is a full frame. The mapping s:D-~D ~A from A to M(ffA), analogous to q, is defined by s(b) = {a: a <<. b}. We easily infer the following theorem: THEOREM 8A. If A is full, s is a complete isomorphism from A to d(~A). The mapping r:C~ C ~'F from a full F to ff(,~'F), analogous to p, is defined by r(x) = {x}. We immediately have that r is one-one and onto. (So, every full frame is "descriptive" with respect to r.) For a frame morphismf:C~-,C 2 from a full frame F 1 to a full frame F2, clause (i) of Section 3 is trivially satisfied, and we easily obtain the following theorem: THEOREM 9A. The mappingf:C~ ~ C 2 is a frame morphismfrom a full F 1 to a full F 2 iff df is a complete homomorphism from s~F 2 to dF 1. Every one-one frame morphismffrom a full F~ to a full F 2 is an embedding, sinceJ~Bl] =f[C1] c~ f[Bl]. So, by Lemma l, a frame morphismffrom a full F 1 to a full F 2 is a frame isomorphism ifffis one-one and onto. Then we prove the following theorem: THEOREM 8F. If F is full, r is a frame isomorphism from F to (#(~F). PROOF. It is clear that r:C~C ~r is one-one and onto. It remains only to show that for every W_~ C ~r we have (dr)(W)~ S(x),:~ We N~e(r(x)). Since r(x) = {x} and (.~r)(W) = {x:{x}eW} = U W, we need to show U WeN(x)c*,WeNdr({X}), and for that we have: L(U W) L( U W) ~ WeN(x). q.e.d. 13. The category FNF. The category FNF of full neighbourhood frames is defined by the following: objects: full neighbourhood frames, morphisms: frame morphisms. It is clear that ~1, defined as in Sections 2 and 3, determines a contravariant functor from FNF to CAA. 14. The functor f~. In Section 12 we have shown how to associate with each full algebra A a full frame (~A. It remains to show how we associate with each complete homomorphism h from a full A 1 to a full A 2 a frame morphism @h. For a 1 ~ C A' and a 2 e C A" let (C~h) (a2) = a I r 2 ~ h(al). First we show that this equivalence defines a genuine function (~h from C A2 to CA': LEMMA 8. lf h is a complete homomorphism from a full A I to a full A 2, then for every atom a2~D 2 there is a unique atom aleD 1 such that a 2 <<. h(al). Duality between modal aloebras ... 231 PROOF. Suppose h is as in the lemma, and there is no a IeD x such that a 2 <~ h(al). We infer that (Va I eD1)a 2 <~ ~h(at), since for every atom a 2 either a 2 ~< b or a 2 ~< -,-b. So, a 2 ~< A {~h(at):a leDt}, and then by using the fact that h is a complete homomorphism and that A 2 is full, we obtain a 2 = 0, which is a contradiction. Suppose now for some a'~,a'~eD 1 such that a'~ # a'~ we have a 2 ~< h(a'O and a 2 ~< h(a~). We again easily infer the contradiction a 2 = 0. q.e.d. Then we have the following theorem, for which, using Theorem 9A, we can give a proof analogous to the proof of Theorem 3F: THEOREM 9F. The mappin9 h:D x -*D 2 is a complete homomorphism from a full A x to a full A 2 iff ~h is a frame morphism from ffA 2 to ffA 1. We can also easily check the following theorem: THEOREM 10F. (a) ~Id a = Idea; (b) (~(h2ohl) = (~hl)~ So, ~ defines a contravariant functor from CAA to FNF. Next we prove the following analogue of Theorem 5F for complete homomorphisms h from a full A I to a full A2: THEOREM llF. (a) If h is onto, then @h is one-one. (b) If h is one-one, then ~h is onto. In order to prove (a) of this theorem, we first establish the following lemma: LEMMA 9. If h is onto, then (Va2~D2)(]a I EOl)a 2 = h(at). PROOF. Since h is onto, (Vaz6D2)(3bt~D1)a 2 = h(bl). It easily follows that a 2 = v {h(aa): a t ~< b~}. Let X 2 = {h(at): a t <<. bt&h(at) # 0}. It is easy to see that X 2 # O and that a z = .V X 2. So, for an h(at)EX 2 we have h(at) ~< a 2. Since h(al)# 0, and a 2 is an atom, h(ax)= 02. q.e.d. PROOF OF THEOREM 11F. (a) Suppose h is onto, and suppose (ffh)(a~) = = (~h)(a'~). It follows that (Vat~D1)(a'2 ~ h(at)~,a~ <<. h(at)). By Lemma 8, this means that (Va26D2)(a'2 ~ a2*x,a~ ~ a2), and this implies a~ = a'~. (b) Suppose h is one-one. We want to show that (Vat,Dr) (3a2~D2) a z ~ h(a O. First we show that (Va t ~Dt)h(al) ~ 0 (otherwise, h(at) = h(0), and since h is one-one, a t =0, a contradiction). Since A 2 is atomic, (']a2~D2) a z <<. h(al), q.e.d. The analogue for M of Theorem 10F, and the analogue for f of Theorem l IF, hold in virtue of Theorems 4A and 5A. 15. Duality between CAA and FNF. Now we can prove the duality between CAA and FNF: 232 K. Do~en THEOREM 12. The categories CAA and FNF are dual by the functors and f#. PROOF. We have to show that the following diagrams commute: h f A 1 A 2 F 1 " F 2 / s 1 s 2 r rl r2 ) ) ,.~,(~A2,) ~(4F1, } ~(4f] By Theorem 8A we have that s I and s z are complete isomorphisms, and by Theorem 8F we have that r~ and r 2 are frame isomorphisms, q.e.d. Analogously to what we had in Section 8, we can establish duality between the following two categories. On the algebraic side we have the category defined by the following: objects: complete atomic normal modal algebras, morphisms: complete homomorphisms, and on the frame side we have the category defined by the following: objects: full filter neighbourhood frames, morphisms: frame morphisms. To prove this duality, which holds by the functors M and ~, we use the following analogue of Theorem 7F: THEOREM 13F. For a full A we have that (~A is a filter frame iff A is normal. 16. Full relational flames and full hyperfilter frames. A general relational frame (see Section 9)-Fr = (C, R, Dr> is full iff D r = .~C. These frames are the usual Kripke frames for modal logic. Every full relational frame is reducible, since we have: (V B E ~C)(x I ~ LB=~ x 2 e B) =~(x I ~ L(S(Xl))=~ x 2 e S(xt) ) ='(S(xl) =- S(xl)= x2 e S(xl)) ~ x1Rx 2. In a full hyperfilter frame the clause: (6) (VBe;~c)(ON(x) c_ B=~ BeN(x)) is replaceable by the requirement that our frame be a filter frame in which for every x we have ~ N(x)eN(x) (so, full hyperfilter frames are augmented in the sense of [1], p. 220; i.e. for every x we have that N(x) is a complete filter). However, a full filter frame need not be a hyperfilter frame (the counterexample mentioned after Lemma 2 in Section 9 is based on a full filter frame). According to Lemmata 3.1 and 3.2, full relational frames are intertrans- latable with full hyperfilter frames. It is now easy to prove directly Lemmata 6.1 Duality between modal algebras ... 233 and 6.2 where "descriptive filter frame" is replaced by "full hyperfilter frame", and ,,descriptive reducible frame" by "full relational frame". So, the category of full hyperfilter frames with frame morphisms is isomorphic to the category of full relational frames with frame homomorphisms. For this last category Thomason asserts in [!1] that it is dual to the category whose objects are completely normal modal algebras, i.e. complete atomic normal modal algebras which satisfy: []^ {b,:iel}= ^ { and whose morphisms are complete homomorphisms. The functors by which this duality holds are ar and a functor fq,, which is like f#, save that we have: (3,) a,R'~a2c~a , ^ -..[].,. a2 :/:O r <~ ~ [] ... az, (4,) LA, Z={a:Sa,(a) cZ}. The functor N, conceived as a functor which assigns to a completely normal modal algebra A a full hyperfilter frame, rather than a full relational frame, has as the corresponding clauses: (3,) N'~(a) = {Z:Val(a <~ .,~ [] ~ a I =*.a I eZ)}, (4,) L,AZ= {a:~Na,(a) cZ}. Now, N a and LA for ~ were defined in Section 12 by: (3) N*(a)= {z: a.< [] vZ}, (4) LAZ = {a:ZeNA(a)}. The following lemmata show that NA(a)= N,a(a): LEMMA 10.1. In every complete atomic normal modal algebra A, for every aeD, and every Z c D: a <~ [] v Z=>gal(a <.N ~ [] "~al=~al~Z). PROOF. Suppose a~<[]vZ, a~<,-~[].--a 1 and alq~Z. Since a I is an atom, from a I~Z it follows that a I ~< ~ v Z, and this entails [] v Z ~< []~ al. Hence, a ~< []--- a~, which is impossible since a~< ~[]--.a I. q.e.d. LEMMA 10.2. In every completely normal modal algebra A, for every a ~ D, and every Z c D: Val(a <~ .,. [] .,~ at=a.al ~Z)=a,a <~ [] v Z. PROOF. Suppose not a<<. [] v Z. Next suppose not (3al <~ ~ v Z) a ~< .-- [] ~ a t. Then (Va I ~< --- v Z)a <~ [] .,. al, and hence, a <<. ^ {[] ",- a~:a~ ~< --. v Z}. Since our algebra is completely normal, we have a ~< [] ^ {~a,: a~ ~< --. v Z}, and this easily yields a ~< [] v Z, a contradic- tion. So, for some al ~< ,-- v Z we have a ~< -,- [] -,- a I . This al does not belong to Z (otherwise, a 1 ~< v Z). q.e.d. 234 K. Do~en (The proof of these lemmata is essentially the proof we must go through in order to show that s is a homomorphism from a completely normal A to M(~,A).) Since NA(a)= N2(a), and since in ~,A, which is a hyperfilter frame, we have NNA(a)~_ Z<::.Z~NA(a), it follows that LAZ = L,Az. So, the functors fq and @, are essentially the same functor, and Thomason's duality result mentioned above amounts to the assertion that the category of completely normal modal algebras with complete homomorphisms is dual by the functors and ~ to the category of full hyperfilter neighbourhood frames with frame morphisms. References [1] B. F. CHELLAS, Modal Logic:. An Introduction, Cambridge University Press, Cambridge, 1980. [2] L. L. ESAKIA,On topological Kripke models (in Russian), Doklady Akademii Nauk SSSR 214 (1974), pp. 298301. [3] L. L. ESAKtA, Heyting Algebras !: Duality Theory (in Russian), Metsniereba, Tbilisi, 1985. [4] R. I. GOLDaI.ATT, Metamathematics of modal logic, Reports on Mathematical Logic 6 (1976), pp. 41-77; 7 (1976), pp. 21-52 (all page references in the text are to the first part, in 6 (1976)). [5] G. HANSOUL, A duality for Boolean algebras with operators, Algebra Universalis 17 (1983), pp. 34 49. [6] B. J6NSSON and A. TARSKI, Boolean algebras with operators: Part I, American Journal of Mathematics 73 (1951), pp. 891 939. [7] B. PAREIGIS, Categories and Functors, Academic Press, New York, 1970. [8] R. S. PIEgCE, Compact Zero-Dimensional Metric Spaces of Finite Type, Memoirs of the American Mathematical Society 130 (1972). [9] H. RASIOWA and R. SIKORSKI, The Mathematics of Metamathematics, Pafistwowe Wydawnictwo Naukowe, Warsaw, 1963. FI0] K. SEGERBERG, An Essay in Classical Modal Logic, University of Uppsala, Uppsala, 1971. El 1] S. K. TSOMASON, Categories of frames for modal logic, The Journal of Symbolic Logic 40 (1975), pp. 439-442. [I 2] J. WILLIAMS,Structure diagrams for primitive Boolean algebras, Proceedings of the American Mathematical Society 47 (1975), pp. 1-9. MATEMATI~KI INSTITUT KNEZ MIHAII.OVA 35 11001 BEOGRAO, P.F. 367 YUGOSLAVIA Received October 7, 1987 Studia Logica XLVIII, 2