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Duality Theorems for Partial Orders, Semilattices, Galois Connections and Lattices Chrysa s Hartonas

Departments of Mathematics and Philosophy Indiana University

and J. Michael Dunn

Departments of Philosophy and Computer Science Indiana University

Indiana University Logic Group Preprint No. IULG-93-26 August 1993

Abstract Lattice-Ordered Stone Spaces are shown to be the dual spaces of partial orders or meet semilattices. These results are subsequently extended to obtain a duality between galois connections and ?-frames. Galois connections are viewed as negation-like operators. The representation of galois connections leads us to a representation theorem for lattices, using the identity homomorphism on a lattice L. Rephrased, the representation theorem for lattices asserts that for any lattice L there is a complete, concrete Boolean algebra B , and a closure operator c : B ! B , such that L can be imbedded in the complete lattice of stable elements of B . B can be taken to be the powerset of a lattice-ordered Stone Space, in which case L is identi ed as the lattice of all clopen and stable subsets of the space. Representation is subsequently extended to a duality theorem for lattices and canonical L-frames. .

Duality Theorems for Partial Orders, Semilattices, Galois Connections and Lattices Chrysa s Hartonas

J. Michael Dunn

1 Preliminaries In a standard possible world semantics it is typical to interpret a sentence ' as a set of worlds (a UCLA proposition, as the second author calls it), namely the worlds w such that w makes ' true. Algebraically, as is well known, a world is a lter of the Lindenbaum algebra of the logic. It is desirable for semantical purposes to have a uniform treatment, using the lter space, of the representation problem for a logic presented either as a consequence system L = (L; `), or with connectives added, merely conjunction (L; `; ^), or with a full lattice structure (L; `; ^; _), with or without negation operators of one sort or another, and possibly with additional operators. Stones's duality theorems for Boolean algebras and distributive lattices ([30], [31], [32]) have initiated a well established tradition of representation and duality theorems in terms of compact, totally separated spaces (Stone spaces). Priestley ([28, [29]), in this tradition, proved a duality theorem for distributive lattices using partially ordered Stone spaces. In the case of arbitrary lattices, however, one invariably stumbles over the representation of joins. Urquhart [33] proved an objects-only duality theorem for bounded lattices using the doubly-ordered space X = (X;  ;  ) of disjoint, maximal lterideal pairs. To represent joins, Urquhart de ned a pair of maps r and ` on sets of disjoint, maximal lter-ideal pairs increasing in the 1-order and 2-order, respectively. In the distributive lattice case, r and ` are the set-complement 1

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Chrysa s Hartonas and J. Michael Dunn

operations and a disjoint, maximal lter-ideal pair is a pair consisting of a prime lter and its complement, a prime ideal, so that he obtains, as a special case, Priestley's theorem. r and ` form a galois connection from the lattice of 1-increasing subsets of X to the lattice of 2-increasing subsets. With representation de ned by a 7! ua = A = fx 2 X ja 2 x g, where a is an element of the lattice and x is the lter part of the lter-ideal pair x, Urquhart topologized X via the subbasis f?uaja 2 Lg [ f?rubjb 2 Lg and characterized the image of the representation function u as the collection of all doubly-closed (both A and rA closed in the topology) stable sets (stability meaning that `rA = A). In the distributive case, the doubly closed sets are simply the clopen sets, since in that case rA is the complement of A. What the reader wants to notice about Urquhart's theorem is that what allows for the representation of joins is that r and ` form a duality between Im(u) and Im(ru). To obtain our duality results, we merge the traditions of Stone spaces, on the one hand, and a tradition of applications of polarities, on the other (for our purposes, polarities will be Kripke-style semantic structures). Birkho [9] abstracted a notion of polarity from the construction of the polar space of a closed subspace of projective n-space and pointed out a variety of other instances of the construction. A polarity is a triple (X; Y; ) where X and Y are classes and   X Y is a relation from X to Y . A polarity (X; Y; ) gives rise to a galois connection from subsets of X to subsets of Y in a canonical way. Birkho and von Neumann [8] proposed their by now well known semantics for quantum logic in terms of Hilbert spaces, with the orthogonality relation on the closed subspaces used to interpret negation. Goldblatt [20], also in the second tradition, used a polarity (X; ?), which he calls an orthogonality space, for the representation of ortholattices. Orthogonality frames have been subsequently used for a Kripke-style semantics of quantum logic (Dalla Chiarra, [10]). (Ortho)negation, of course, is an instance of a galois connection. Dunn [18] pointed out that every galois connection on a meet semilattice gives rise to a polarity (X; ?), in a canonical way. 1

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A duality, in the case of partial orders, is a galois connection f : P Qop ; g : Qop P such that gfa = a and fgb = b for all a P; b Q. We also use the term \duality" in its general sense, as an equivalence of a category C with the oppposite category Dop , for some category D (of course the rst sense is a special case of the second). 1

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We prove here a duality result for galois connections and polarities of a certain kind, which we call ?-frames. In preparation for this result we need to develop the representation and duality theory of partial orders and semilattices (meet or join semilattices). In the spirit of our primary objective of a uniform semantics for a logic presented either as a consequence system or with a variety of operators added, we use the lter space of the original structure (partial order or semilattice). The lter space is a certain kind of lattice-ordered Stone space, whose natural description is given in the next section. We also point out here that, just like in the presence of an orthonegation the problem of representing joins is reduced to that of getting a representation of meets and of the orthonegation operator (Goldblatt, [20]), in the more general situation of two meet semilattices related by a duality, i.e. a galois connection S  f - K op g such that fg = idK and gf = idS , joins can be de ned by a _ b = g(fa ^ fb) (in S , and similarly for K ), so that the representation problem reduces to the representation of meets and of the duality (f; g). When K = S , the duality appears as a generalized notion of negation on the self-dual lattice S , which we call a split negation. We give a couple of motivating examples and point out its prospective usefulness in the context of substructural logics . Even though it is not true that every lattice L has a self-duality on it, L and K = Lop are trivially dual via any lattice automorphism  : L ! L. In particular, via the identity map . The representation of the trivial duality (along the lines of the duality theorem for galois connections) is, itself, nontrivial, resulting in a pair of functions f and g relating the space of lters and the space of ideals of the lattice and allowing for the interpretation of joins. We conclude our paper by introducing lattice-frames, i.e. triples (X; Y; ?) where X is the space of lters of L and Y the space of \ lters" of Lop , both viewed as meet semilattices. With an appropriate notion of a canonical lattice-frame morphism, the duality for galois connections is modi ed to 2

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Logics dropping one structural rule or other in their Gentzen system presentation. We somehow managed to get ourselves to confuse this obvious fact with self-duality on a lattice! We take this opportunity to thank Gerard Allwein whose insistence on the obvious made us straighten things out again and maintain the desired simplicity. 2 3

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yield a duality theorem for lattices and canonical lattice-frames.

2 Lattice-Ordered Stone Spaces A lattice-ordered Stone Space is a partially ordered, compact, totally separated space, whose partial order is a full lattice. For our purposes, we wish, however, that the ordering arising from the lattice structure be related to the topological structure. Accordingly, we de ne De nition 2.1 1. An F Space X is a lattice-ordered Stone space such that (a) As a lattice, X is complete (b) X has a subbasis of clopen sets S = fXa ga2A [ f?Xb gb2A indexed by some set A, such that for each a 2 A; Xa is a principal upper set, generated by a point xa (c) The subset fxaja 2 Ag  X is join-dense in X , i.e. every point x W is the join of the xa it covers: x = xa x xa. 2. X is an F Space if the collection X  = fXa ga2A is closed under intersection By an FSpace we shall refer indiscriminately to any of the above types of space. 2 FiSpaces are designed to be, as the reader may have noticed, the lter spaces of partial orders (i = 0) and meet semilattices (i = 1). FSpace morphisms are de ned to be the continuous, complete lattice homomorphisms h : Y ! X such that h? maps X  into Y . F Space, F Space denote the respective categories of FSpaces (the families of FSpaces with their appropriate notion of morphism as de ned above). 0

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2.1 Duality Theorems

Pos and SLat denote, respectively, the categories of partial orders and meet semilattices . By a lter in the case of partial orders we shall mean an 4

What is developed in this section applies, of course, equally well to join semilattices, simply by replacing lters by ideals. 4

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upper set (a cone). If (P; ) is a partial order or meet semilattice, we let FP = X be the set of lters of P . De ne a representation map P ! P X by p 7! Xp = fx 2 X jp 2 xg. By the representation topology on X we mean the topology induced by the subbasis S = fXaga2P [ f?Xb gb2P Proposition 2.2 FP = X is an FSpace. Furthermore, if f : P ! Q is a poset or meet semilattice morphism, then F (f ) = f ? : FQ ! FP is an FSpace morphism. In both cases, total separation is immediate since if x 6= y, say x 6 y, let p 2 x but p 62 y. Then x 2 Xp and y 2 (?Xp). The space of lters has a complete lattice structure and, setting xp = p", the principal lter generated by p, the xp`s generate the lattice. For compactness, using the Alexander subbasis lemma, it is enough to prove that any cover by subbasis sets has a nite subcover. Let A; B  P and C = fXaja 2 Ag [ f?Xb jb 2 B g a cover of X by subbasis elements and suppose no nite subcover exists. Let xB be the lter generated by B . Then T S B  xB , hence xB 2 b2B Xb andSthus xB 62 b2B (?Xb). Remains to show that xB 62 a2A Xa either. This follows from the fact that xB \ A = ;. To see why, suppose a 2 xB , for some a 2 A. If P is a partial order, then b  a for some b 2 B . Assuming compactness fails, there is some x 2 X such that x 62 (?Xb ) [ Xa. Hence b 2 x while a 62 x. But x is an increasing set and we supposed b  a, contradiction. For the meet semilattice case, let c = b ^


^ bn be such that c  a, for some b ; : : :; bn 2 B . If compactness fails, let x 62 Xa [ (?Xb1 ) [


[ (?Xbn ). Then b ^


^ bn = c 2 x and a 62 x, while c  a, contradiction. If f : P ! Q is a poset or meet semilattice map, then the map F (f ) = f ? takes lters of Q to lters of P , hence F (f ) : FQ ! FP . Joins are preserved, since inverse maps preserve intersections and _fxjx 2 W  FP g = \fu 2 FP j [ x  ug 1

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x2W

In the partial order case, xB = B is the upper closure of B . For the meet semilattice case the lter xB is explicitly de ned by xB = a P ( b1; : : :; bn B )(b1 bn a) 5

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^


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Chrysa s Hartonas and J. Michael Dunn

For continuity, it is enough to verify that subbasis sets in X = FP are taken by F (f )? to open sets in Y = FQ, since open sets in X are unions of basic open sets and the basis generated by a subbasis consists of nite intersections of subbasis elements (preserved by inverse maps). A direct calculation shows that y 2 F (f )? (Xa ) i y 2 Yf a hence F (f )? (Xa) = Yf a and F (f )? (?Xa) = ?Yf a . Thus F (f ) is continuous. 2 1

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Proposition 2.3

1. Every FSpace arises as the space of lters of a poset or meet semilattice.

2. Furthermore, if h : Y ! X is an FSpace map, where X = FA, Y = FB , then h = F (^h) for some morphism h^ : A ! B .

(1) If X is an FSpace with subbasis sets indexed by the set A, partially order A by a  a0 i Xa  Xa0 . If X is an F Space, closure of X  under intersection implies that A is a meet semilattice in the de ned ordering. We show that X  FA. If F is the lter space of A and  : A ! X is the map a 7! xa,  induces a complete lattice homomorphism ^ : F ! X , de ned by _ _ ^ fai" ji 2 I g = faiji 2 I g: 1

Similarly, the map  : xa 7! a" extends to a complete lattice homomorphism ^ : X ! F , de ned by _ _ ^ fxaja 2   Ag = fxaja 2 g: Since ^^ = 1 and ^^ = 1, ^ : F ! X is a complete lattice isomorphism. Furthermore, by direct calculation, ^? (Xa) = Fa = f 2 F ja" g. Applying ^? = ^ on both sides we also get ^? (Fa) = Xa, so that ^ is continous and an open map, hence a homeomorphism. (2) If h : Y ! X is an FSpace morphism, by the previous argument we may assume X = FA; Y = FB , where A and B are the index sets for the respective subbases of X and Y . 1

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De ne the map ^h : A ! B by h^ a = b i h? (Xa) = Yb. It is clear that h^ is a homomorphism . Now h = F (^h), for if not, let y 2 Y such that hy 6= ^h? y. By total separation of X , let ^h? y 2 Xa; hy 62 Xa. Thus a 2 h^ ? y and so h^ a 2 y. If h^ a = b, then y 2 Yb and, by de nition of h^ , h? (Xa) = Yb. So y 2 h? (Xa), hence hy 2 Xa follows. Thus h = F (^h) = ^h? as claimed. 2 The proof of the following lemma is immediate from de nitions. Lemma 2.4 Let X be an FiSpace (i = 0; 1) on the subbasis 1

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S = fXaga2A [ f?Xb gb2A Then DX = X  is a partial order or a meet semilattice, for the respective values of i = 0; 1. Furthermore, if h : Y ! X is an FSpace morphism, =h?1 then DX Dh?! DY is a partial order or meet semilattice morphism, as appropriate. 2 If Ci is the category Pos or SLat, for i = 0; 1 respectively, then the previous discussion has provided functors

Ci DF - FiSpaceop

Proposition 2.5 For any P in Ci (i.e. P a poset or meet semilattice) and X in FiSpace, P = DF (P ) and X  FD(X )

i.e. D and F form a duality of categories. That P  = DF (P ) is immediate. For the other half, if X is an FSpace with subbasis indexed by a set A, by the argument of proposition 2.3 we may assume that X = FA. Since A  = DF (A) = X , there is an FSpace  homeomorphism X = FA  FX as argued in proposition 2.3. 2 To characterize DF (P ), de ne a coarse open set on X to be an open set U that can be written as the union U = SfXajXa  U g7. of the appropriate kind, depending on whether h is an F0Space, or an F1Space morphism. 7 In the meet semilattice case, such a set is open in X under the coarser topology induced by the basis X  , which is the reason for our terminology. Incidentally, the lter space of 6

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Lemma 2.6 Let X be an FSpace, with topology induced by the subbasis of clopen sets

S = fXaga2A [ f?Xbgb2A

Then X  is the family of all coarse-open, principal upper sets in the ordering of X induced by its lattice structure. If U = SfXajXa  U g = u ", then for some xa 2 X , xa  u  xa, hence U = Xa . 2 Remark 2.7 We will later have the opportunity to give an alternative characterization of X  as the collection of all clopen and stable sets, if X is the dual space of a meet semilattice with a duality, or of a lattice.

3 The Semantics of Galois Connections and Negation We embark in this section in a study of the semantics of galois connections and of a broadened concept of negation. In connection to the duality theorem of the previous section we may furthermore observe that Lemma 3.1 Let (P; ) , (Q; ) be partial orders and assume there is a pair of maps : : P ?! Qop; : Qop ?! P forming a galois connection, i.e. such that for any a 2 P; b 2 Q a  b ( in P ) i b  :a ( in Q) Letting F (P ) = X; F (Qop) = Y be the lter spaces and X   = Qop = P; Y   the families of coarse-open, principal upper sets, the induced maps F (:)? = : : X  ?! Y ; F ()? =  : Y  ?! X  form a galois connection. If P; Q are meet semilattices and the pair (:; ) is a duality (i.e.  :a = a and :  b = b for all a 2 P; b 2 Q), then the same conclusion is true. 1

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a lattice in the topology induced by the collection of Xa 's is of independent interest in topology (Banaschewski, [6]). Turning to naturally arising kinds of space for our duality results, rather than to some forced kind of space designed exclusively for the needs of representation, is an advantage, we think, of the approach we take here.

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From proposition 2.2, we have that F (:)? (Xa ) = :(Xa) = Y:a, and similarly F ()? (Yb) = (Yb) = Xb . If Xa  Xb , let xa be the principal lter generated by a 2 P . Then  b 2 xa , hence a  b. Consequently, b  :a so that :a 2 Yb. Thus Yb  Y:a. The other direction is similar. In the meet semilattice case, the assumption of a duality makes the maps into homomorphisms and the same argument goes through. 2 The functorial representation gives only partial results, it should be noted, as it cannot treat, for example, the case of a galois connection of meet semilattices (the galois maps are not meet semilattice homomorphisms, unless they form a duality). We will thus shift our representation technique to the use of relational structures, more familiar in the context of interpreting the negation operation. 1

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It is well-known that negation is a special case of galois connection . And there is a tradition of interpreting negation by means of an incompatibility relation whose underlying ideas can be traced back to at least Birkho . Birkho [9] introduced a notion of polarity, as a generalization from the construction of polar spaces in analytic geometry and showed that every such polarity gives rise to a galois connection in a canonical way. Speci cally, a polarity is a triple (X; Y; ?), where X; Y are two classes and ? X  Y is a relation from X to Y . De ning :U = fy 2 Y jU ? yg and  V = fx 2 X jx ? V g where U  X; V  Y and U ? y means that for all x 2 U; x ? y (and similarly for x ? V ), it is not hard to see that (; :) is a galois connection from the powerset of X to the powerset of Y . We may call a set U  X stable (Birkho uses the term \closed") if  :U = U . In many cases of interest, X = Y and ? is a symmetric relation, in which case :U = U . This is the case in Birkho's primary example of subsets of projective nspace. If A = (aij ) is a xed symmetric, nonsingular n  n matrix and y = (y ; : : :; yn) are two vectors, de ne x ? y to mean that Px =i;j x(xiaij; :y:j:;=xn0.); The stable subsets in this case are points, lines, planes etc. For a stable space U; U  is its polar space (using Birkho's notation, rather than our :U ). 8

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with the exception of subminimal negation (Hazen, [24]), which is merely an antitone map. 8

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Chrysa s Hartonas and J. Michael Dunn

A relation similar to that of a space to its polar space holds in the case of the closed subspaces of a Hilbert space, namely the orthogonality relation. Where (x; y) is the scalar product of vectors in a Hilbert space, x ? y is de ned by (x; y) = 0. The orthogonal of a set S of vectors is then given by S  = fxj (x; s) = 0 for all s 2 S g. Goldblatt, in his representation of ortholattices [20], takes ? to be the relation de ned on the space of lters by x ? y i 9a 2 x; a0 2 y, where a0 is the orthocomplement of a. Dunn [18] takes a more general approach and uses ?-frames to interpret a galois connection on a meet semilattice. Speci cally, given a galois connection on a meet semilattice (S; ; ^), he considers the set of lters X and de nes ? by x ? y i 9a 2 x; :a 2 y for two lters x and y of S . ? then induces a galois connection on subsets of X as above, which he denotes by ?U and U ? (:U and  U in the notation above, respectively). The representation function h : a 7! ha = fx 2 X ja 2 xg, he shows, preserves the galois connection, in the sense that h(:a) = ? (ha) and h( a) = (ha)?. Another familiar application of the galois connection induced by a binary relation is the Dedekind-McNeile completion of a partial order (P; ) . Taking ?=, and using our notational convention, we can de ne the operations Au = fp 2 P jA  pg and B ` = fq 2 P jq  B g giving the upper bounds of A and lower bounds of B , respectively. The stable sets are, by de nition, the subsets A  P such that A = Au`. The composite of the two maps of the galois connection is a closure operator (Birkho [9]), hence the stable subsets form a complete lattice. One then proves that the lower cones p#; p 2 P are stable and that the map p 7! p# imbeds the partial order in the complete lattice of its stable subsets. Hardegree [21] made yet another use of polarities, which he calls NK structures (Natural Kind structures). An NK -structure, he proposes, is a triple (I; T;
) where, intuitively, I is a set of individuals, T is a set of what he calls traits, and
is a relation from I to T (intuitively, the instantiation relation). The relation induces a galois connection assigning to a set U of individuals the set tr(U ) of traits they share, and to a set V of traits the set in(V ) of individuals that have every trait in V . He then proposes to formally model a natural kind as a stable set U = in(tr(U )) of individuals. Very similar to Hardegree's application of polarities is the development in Davey and Priestley [11] of so called concept lattices. Davey and Priestley

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credit the initiation of the subject to R. Wille [34]. A similar application has been investigated by Jon Barwise [7], where the relation  involved is between tokens and types, in the context of his currently under development theory of information ow. We return here to Birkho's original notion of a polarity as a triple (X; Y; ?), where ? X  Y is a relation from X to Y . Lemma 3.2 Let P and Q be posets or meet semilattices and

P

: - Qop 

a galois connection. Let X; Y be the lter spaces of P and Q and ? X  Y the relation from X to Y de ned by x ? y i 9a 2 x; :a 2 y. The relation ? induces a galois connection from P X to P Y de ned on subsets U  X and V  Y by

:U = fy 2 Y jU ? yg and  V = fx 2 X jx ? V g In particular, :Xa = Y:a and  Yb = Xb . That (; :) is a galois connection was pointed out by Birkho, as already mentioned. For the second part, we do only :Xa = Y:a, the other being similar. By de nitions, this is equivalent to :a 2 y i Xa ? y. From left to right it is obvious, since for any x 2 Xa ; a 2 x. For the other direction, we have in particular xa ? y, where xa = a". If b 2 xa such that :b 2 y, then from a  b we have :b  :a and so :a 2 y, since y is a lter. 2 Starting with a galois connection from (P; ) to (Q; ) , we obtain a polarity (X; Y; ?), where X and Y are their respective sets of lters which we regard from now on as their dual FSpaces. Conversely, any polarity induces a galois connection. Moreover, if X and Y are lter spaces, the galois connection from P X to P Y restricts by the above lemma to a galois connection (:; ) from X  to Y .

We wish to express the connection between galois connections and their associated polarities as a duality result, showing that the dual galois connection of a polarity is isomorphic, in an approrpriate sense, to the galois

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Chrysa s Hartonas and J. Michael Dunn

connection that gave rise to the polarity. And conversely, that the dual polarity of a galois connection is isomorphic, in an appropriate sense, to the polarity that gave rise to that galois connection. We will basically need such a result in extending the representation theorem for lattices to a full duality theorem. Naturally, some restrictions need to be imposed on the kinds of polarities we consider. We call the restricted notion a ?-frame. De nition 3.3 A ?-frame is a triple (X; Y; ?), where X; Y are FSpaces , and if (:;  ) is the induced galois connection from P X to P Y , then it restricts to a galois connection from X  to Y  . 2 In order to obtain some appropriate notion of isomorphisms of galois connections and ?-frames, we turn each of these classes to a category with morphisms de ned as follows. If G = (P; Qop; :; ) and G 0 = (P 0; (Q0)op; :0;  0) are galois connections a morphism G ! G 0 is a pair of poset or meet semilattice maps (as appropriate) f : P ! P 0 and g : Q ! Q0 such that both squares below commute. 9

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P  :- Qop

f



g

? 0 ? P 0  :-0 (Q0)op 

We will then \identify" two galois connections if both f and g are isomorphisms (and the squares commute). Similarly, a morphism (f; h) : (X ; Y ; ? ) ?! (X ; Y ; ? ) 1

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of ?-frames, is a pair of FSpace-morphisms f : X ! X , h : Y ! Y such that the maps Df = f ? : X  ! X  and Dh = h? : Y  ! Y  give a 1

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We are really de ning two notions of -frame here, accordingly as the spaces are the lter spaces of partial orders or meet semilattices. 10 The notation we use is to be understood as meaning that is left-adjoint to , i.e. : P Qop and : Qop P so that a op b (in Qop ) i a b (in P ). 9

?

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morphism of the induced galois connections, in the sense that the squares below commute

:

X   -(Y )op 6 6Dh Df 1

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Given a galois connection G = (P; Qop; :; ), its dual ?-frame is the triple F = (X; Y; ?) where X; Y are the lter spaces of P and Q, respectively, and ? is the relation ? X  Y de ned by x ? y i 9a 2 x; :a 2 y. Similarly, the dual galois connection of a ?-frame F = (X; Y; ?) is the induced galois connection (X ; (Y )op; :;  ).

Proposition 3.4 The assignments of their dual frames to galois connections

and their dual galois connections to frames are cofunctorial. Furthermore, if

Galois GFr - Frameop

G and Fr are the respective cofunctors, then GFr(G )  = F. = G and FrG(F )  Given (f; h) : G ! G , the map (F (f ); F (h)) : Fr(G ) ! Fr(G ) is a frame morphism, by lemma 3.2 and proposition 2.2. If (f; h) : F ! F is a frame morphism, then by its de nition, the map (Df; Dh) : G(F ) ! G(F ) is a morphism of galois connections. If G = (P; Qop; :; ) is a galois connection, then the dual galois connection of its dual frame F = (X; Y; ?) is GFr(G ) = (X ; (Y )op; :; ). By the duality of proposition 2.5, the maps a 7! Xa and b 7! Yb are isomorphisms P = Y , respectively. In lemma 3.2 we proved that the relevant = X  and Q  squares commute, i.e. that :Xa = Y:a and  Yb = Xb . Hence, G  = GFr(G ). Now let F = (X; Y; ?) be a ?-frame and G(F ) = (X  ; (Y )op; :; ) its dual galois connection. By the argument of proposition 2.3, we may assume that X and Y are the lter spaces of the sets A; B indexing their respective subbases. The galois connection G = (A; B op; :; ), where we de ne :a = b i :Xa = Yb and, similarly,  b0 = a0 i  Yb0 = Xa0 is 1

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Chrysa s Hartonas and J. Michael Dunn

isomorphic to G(F ) by the argument of the previous paragraph. Hence their dual frames are isomorphic, i.e. Fr(G ) = F  2 = FrG(F ). Turning to the logical signi cance of our present discussion, if

P

: - P op 

is a galois connection, we will regard it as a split negation on P . There seem to be two kinds of reasons why one might want to consider a generalization of negation as the one we are proposing here. First, with an underlying intuition of negation being closely related to a relation of icompatibility, it seems to be an arti cial restriction to impose a symmetry condition on incompatibility. It seems to us that there are natural situations where the incompatibility relation is nonsymmetric. We give below a couple of examples. If (L; `) is a consequence system, suppose we informally understand sentences as standing for events (or facts), and assume an interpretation of sentences as sets of possible worlds, with an extensional understanding of a world as a set of sentences (events), intuitively the events that obtain in this world. If X is the set of events, the relation ? on X de ned by e ? e0 if and only if e0 prevents e, is clearly nonsymmetric . The relation ? on events induces now a galois connection on sets of events, i.e. on worlds (interpreting sentences). If w(p) is the world interpreting a sentence p, then 11

12

:w(p)=the world of events preventing every event in w(p) and

 w(p)=the world of events prevented by every event in w(p) It is natural then to assume logical operators such that

:w(p) = w(:p) and  w(p) = w( p): Dunn [17] refers to the pair ( ; ) as \galois connected negations", thinking of each of and as a subminimal negation, as explained in the sequel. 12 My not having at least 50 dollars prevents me from buying a copy of a book worth 50 dollars. My buying a copy of a book worth 50 dollars does not necessarily prevent me from having at least 50 dollars in my pocket. 11



 :

:

Duality Theorems

15

There is no implied commitment, however, to an ontology of events in order to make sense of a nonsymmetric incompatibility relation. For another example, any attempt to a logic of action, it seems, will sooner or later want to consider a variety of such relations, from a strong relation of preventing to weaker relations, such as merely obstructing . A second kind of motivation for considering a negation split in two comes from recent interest in substructural logics . With either of contraction or weakening dropped, conjunction splits in two operations, the usual conjunction and a new product operation (intensional conjunction, fusion, tensor product). With the exchange law in place, the new product operation is commutative and otherwise it leads to considering an implication relation split in two . Assuming a dualizing object 0, negation should split in two as well, its two components de ned by :a = a & 0, and  a = 0 . a . 13

14

15

16

17

The traditionally considered negation operations are instances of galois connections, where the two maps of the galois connection coincide. One notable exception is Hazen's subminimal negation in [24] which, algebraically, is merely an antitone map. A split negation comes in varieties, too. A minimal split negation is a pair of maps (; :) forming a galois connection, which is the variety considered in Dunn [17], called, as already mentioned, a pair of galois connected (subminimal) negations. We call it here a minimal split negation as it extends the familiar minimal negation resulting when the two maps in (; :) are the same. With intuitions more based on the connection between negation and incompatibility, we extend here further to some varieties of split negation that An obviously nonsymmetric relation: Jon's practicing his saxophone obstructs me from reading a technical paper while, unfortunately, my doing so does not seem to obstruct him at all from carrying on with his practice. 14Relevance [4],[5] and Linear Logic [19] are probably the best known examples of substructural logics, dropping weakening and both weakening and contraction, respectively. Other examples include BCC and BCK logics [27]. A survey of the subject can be found in [14]. 15There is currently interest in Noncommutative Linear Logic, where the exchange law is also dropped. See, for example, [26]. 16Implication is residuated with the product operation. With a noncommutative product, one is naturally led to considering a left and a right residual. 17To the best of our knowledge, logics with a split negation have not been investigated prior to the initiation of the subject in Dunn [17]. 13

16

Chrysa s Hartonas and J. Michael Dunn

seem to be of some signi cance, namely the De Morgan split negation and split orthonegation. Algebraically, the rst is a self-duality operation, i.e. a galois connection with the stability requirements that, for all a, :a = a = :  a. We rst list the following more general fact. Lemma 3.5 If (L; ; ^) and (K; ; ^) are meet semilattices and there exists a duality

L then setting

: - K op 

a _ b = (:a ^ :b) in L and c _ d = :( c^  d) in K both L and K are full lattices. 2 The proof is straightforward, consisting in verifying that the expressions on the right are truely the least upper bounds, and will be ommitted. It should be noted that by a similar argument the corresponding claim for join semilattices is true. We will have further use of this lemma for representation purposes. For the moment we note the following consequence. Corollary 3.6 A self-dual semilattice (taking L = K in the lemma above) is a full lattice. If

L

: - Lop 

is the self-duality, then by

a _ b = (:a ^ :b)

a ^ b = :( a_  b)

meets and joins are interde nable. 2 Note that, by the above lemma, when L = K , joins and meets can be alternatively de ned by

a _ b = :( a^  b) and a ^ b = (:a _ :b) Consequently,

Duality Theorems

17

Corollary 3.7 If (; :) is a self duality on L, then :(a _ b) = :a ^ :b and :(a ^ b) = :a _ :b and similarly

 (a _ b) = a^  b and  (a ^ b) = a_  b i.e. each of  and : satis es the De Morgan laws. 2 This corollary is our justi cation for calling a self-duality (; :) a De Morgan

split negation. By a split orthonegation we mean a galois connection (; :) which is a self duality and satis es, in addition,  a ^ :a = 0 and a^  a = 0, or equivalently, :a _ a = 1 and  a _ a = 1 Needless to say, there are more variants of split negation than we have mentioned. We will reserve a more thorough investigation for another paper and move now to one of our primary tasks, namely the duality for arbitrary lattices.

4 Duality for Lattices In section 2.1 we dealt with duality theorems for partial orders or meet semilattices (therefore, also, join semilattices). In section 3 we treated the semantics of galois connections, proved a duality theorem for ?-frames and introduced generalized notions of negation. Before turning to duality for lattices, we rst note that with a duality

S

: - K op 

of the semilattices S and K , and where X = FS; Y = FK are their dual Stone spaces, the collection of sets X  = DX (and similarly for DY ) can be alternatively characterized as the collection of stable sets in a way that extends Goldblatt's representation of ortholattices . 18

Our argument in the proof of the following proposition is a small generalization of Goldblatt's argument in [20]. 18

18

Chrysa s Hartonas and J. Michael Dunn

Given such a duality, the representation (functorial or relational) produced a galois connection (:; ) from P X to P Y . Call a subset U  X a stable set if :U = U . Note also that, in view of our previous duality results (semilattices) and the discussion on split negation, the following proposition may be taken as concluding the representation of lattices with a De Morgan split negation (or a split orthonegation). Proposition 4.1 With S; X and DX = X  as above, DX can be alternatively characterized as the collection of all clopen, stable subsets of X . If U is clopen and stable and x 62 U = :U let y 2 :U such that x 6? y. Since y 2 :SU , for any z 2 U; z ? y, so there is az 2 z such that :az 2 y. S i n Thus U  Xaz and by compactness U  i Xaiz . Let ax = az _


_ anz = (:az ^


^ :anz ) Then for each i, Xaiz  Xax , hence U  Xax . Now notice that ax 62 x. If it were, then :ax = :  (:az ^


^ :anz) = :az ^


^ :anz 2 y, since each :aiz 2 y. This contradicts the assumption that x 6? y. Hence x 2 ?Xax and thereby ?U  ?Xax . Given also U  Xax , it follows that U = Xax . 2 Corollary 4.2 (Goldblatt, 1975) Let X be the Stone Space of an ortholattice L. Then L is isomorphic to the lattice of all clopen, stable subsets of X. 2 = =1

1

1

1

1

The main subject of the present section is to conclude by extending the previous results to the case of full lattices. The idea is to use galois connections for the represnetation of joins, with an incompatibility relation interpreting the former. Lemma 3.5 and corollary 3.6 show that the representation problem for lattices can be reduced to that of meet semilattices, assuming a duality on the semilattice. In the case of a full lattice the trivial galois connection L  i - (Lop )op i is a duality between L and Lop, where i is the identity map . We think of 19

19

In fact, any lattice automorphism  : L = L is such a duality of L and Lop . 

Duality Theorems

19

L and Lop as meet semilattices and represent them with their lter spaces X and Y as in section 2. The lter space of Lop is of course the space of ideals of L. In the induced ?-frame (X; Y; ?), ? is the relation ? X  Y from ters to ideals of L de ned by x ? y () 9a 2 x(ia 2 y) () x \ y 6= ; for a lter x 2 X and an ideal y 2 Y . Using ? we de ne the galois connection from subsets U  X to subsets V  Y by :U = fy 2 Y jU ? yg = fy 2 Y j(8x 2 U )(x \ y 6= ;)g and similarly V = fx 2 X jx ? V g = fx 2 X j(8y 2 V )(x \ y 6= ;)g From lemma 3.2 we obtain :Xa = Ya and  Ya = Xa . In this particular case, the identity :Xa = Ya, for example, amounts to verifying that for all ideals y, :a 2 y () Xa ? y. From left to right it is trivial. For the other direction, we have in particular xa ? y, where xa = a" is the principal lter generated by a. Thus for some b  a, b 2 y. Since y is an ideal, it follows that a 2 y as well. Then the pair (:; ) forms an adjoint coequivalence (a duality) X 

: - (Y )op 

so that joins in X  can be de ned by

_

Xa Xb = (:Xa \ :Xb) Notice also that proposition 4.1 applies in the case of the identity as duality between L and Lop (naturally, the argument is elementary in this particular case), allowing for a characterization of the representation in terms of clopen, stable sets. Hence we have the following Proposition 4.3 (Representation of Lattices) Let L be a lattice. If X is the space of lters of L and Y its space of ideals, there is a galois connection (: ; ) from sets of lters to sets of ideals, de ned as above via the relation ? X  Y , where x ? y i x \ y 6= ; for a lter x and an ideal

20

Chrysa s Hartonas and J. Michael Dunn

y. Furthermore, the map a 7! Xa = fx 2 X ja 2 xg is a lattice isomorphism L  = DF (L), with a ^ b 7! Xa \ Xb and a _ b 7! (:Xa \ :Xb ), where DF (L) is the collection of all clopen, stable subsets of the lter space F (L) = X . 2 Remark 4.4 The composite of the galois connection c = : : P X ! P X is a closure operator and joins in the complete lattice of stable subsets of X are also de ned as the closure of unions. We may therefore consider the lter space alone as the dual space of the lattice and rephrase the representation theorem as follows: Theorem 4.5 (Representation of Lattices) For every lattice L there is a concrete, complete Boolean algebra B , and a closure operator c : B ! B , such that L can be imbedded in the complete lattice of stable elements of B . More speci cally, B may be taken to be B = P X for a lattice-ordered Stone Space X , in which case L can be identi ed with the lattice of clopen, stable subsets of X . 2 Let us notice now that we might have used in the representation theorem above not the identity i : L ! L but rather any automorphism  2 Aut(L). In our representing the lattice L what we have in eect being doing is provide a representation of the galois connection G = (L; (Lop)op; i; i? ), where we think of each of L and Lop as a meet semilattice. If ; 2 Aut(L) are two automorphisms, then the galois connections G = (L; (Lop)op; ; ? ) and G = (L; (Lop)op; ; ? ) are isomorphic via the map (; ) : G  = G . This follows by merely inspecting the diagram L  (Lop)op ?   ? ?op op L  ? (L ) Consequently, for any automorphism , the isomorphism G  = G entails 20

1

1

1

1

1

20

We use the term \stable" where one normally uses \closed" to avoid confusion of sets

U closed in the topological sense with sets \closed" in the sense that cU = U .

Duality Theorems

21

that the dual frames are isomorphic, i.e. (X; Y; ?)  = (X; Y; ?), where ? is de ned by x ? y i 9a 2 x; a 2 y. To obtain a duality theorem for lattices we introduce lattice-frames and provide them with an appropriate notion of morphism. De nition 4.6 An L-frame (lattice-frame) is a ?-frame F = (X; Y; ?) where X and Y are F Spaces (i.e. lter spaces of meet semilattices) such that the induced galois connection (: ; ) is a duality of X  and Y . 2 L-frames are indeed what we need them to be, namely frames that arise canonically from lattices, in the sense of the following Proposition 4.7 Let F = (X; Y; ?) be a lattice-frame. There is a lattice L and an automorphism  : L ! L such that F is (up to frame isomorphism) the dual frame of the galois connection (L; (Lop)op ; ;  ? ). In fact, we may assume that  is the identity on L and that ? is the canonical relation from lters to ideals of L de ned by x ? y i x \ y 6= ;. If L and K are the index sets of the subbases of X and Y , respectively, we may assume (proposition 2.3) that X = FL and Y = FK . The hypothesis implies that K  = Lop, so that we may assume that X and Y are the spaces of lters and ideals, respectively, of L. The duality 1

1

X 

: - (Y )op 

induces a lattice automorphism  : L ! L, de ned by a = b i :Xa = Yb . Let F = (X; Y; ? ), where x ? y i 9a 2 x; a 2 y. Then F  = F via the identity (IX ; IY ) : F ! F. In fact, since (i; ) : Gi  = G is an isomorphism of galois connections (where i : L ! L is the identity), it is clear that Fi = (X; Y; ?i)  = F , where ?i is the canonical relation x ?i y i x \ y 6= ;. Hence we may always assume that in a lattice frame, the relation ? X  Y is the canonical relation ?i.2 There is a choice to be made as to how to de ne L-frame morphisms. One choice, as will be explained, would lead us to identify lattice homomorphisms

22

Chrysa s Hartonas and J. Michael Dunn

f ; f : L ! K if there are automorphisms  : L ! L and  : K ! K such that f = f . Even though this is not such be a signi cant restriction, we make an alternative choice in the de nition below. ^ Y^ ; ?^ ); (X; Y; ?) be L-frames in the sense of de niDe nition 4.8 Let (X; tion 4.4. If f^ : X^ ! X and f^ : Y^ ! Y are FSpace homomorphisms, then f1 ;f2 ^ Y^ ; ?^ ) ?! (X; (X; Y; ?) is a canonical L-frame morphism provided that the 1

2

1

2

1 (^ ^)

2

cubes in the diagram below commute (i.e. all faces of the cubes commute).

: 

X^ 

6 ? ? f? ?

- (Y^ )op ? 6 ? f? 0 ?

2

2

1

? 6

X 

  f??? ? ? L

K

2

:  1

1

1

- (Y?)op 6 Id (K op)op Id  0 f??? ? ? - (Lop)op 2

Id Id

In the diagram,  : L  = X  and : K  = X^  are the isomorphisms a 7! Xa and b 7! X^b , respectively, and similarly for 0 and 0. By Proposition 2.3, the maps f^ and f^ are the images under the functor F of semilattice morphisms f and f as indicated in the diagram. f  and f  are the morphisms fi = F (fi)? = D(f^i ). The maps :i and i are the induced galois connections. Commutativity of the top square is the condition for a frame morphism. Commutativity of the left and right sides of the cube expresses the fact that fi = DF (fi). We may think of commutativity of the front and back faces as expressing the fact that ? is the canonical relation. Finaly, commutativity of the bottom square implies that f = f . Note that f and f are originally viewed as meet and join, respectively, semilattice morphisms from L to K . The latter are of course full lattices, by the conditions of the de nition of a lattice-frame. Hence, what the de nition of a canonical L-frame morphism forces is that every such morphism is obtained from a lattice homomorphism f : L ! K. 1

1

1

2

2

1

1

2

2

1

2

Duality Theorems

23

The alternative choice we could have made is to replace the identities in the bottom square with any automorphisms  : L ! L and  : K ! K . From commutativity of the bottom square we now only obtain f = f , where f is a meet semilattice and f a join semilattice morphism. The condition however implies each is a full lattice homomorphism since, for example f (a _ b) =  ? f (a _ b) = f a _ f b. This would, however, have us identify two lattice morphisms if such a condition as above holds, which we nd an unnecessary even though possibly not important restriction. 1

1

2

2

1

1

2

1

1

With this notion of a canonical frame homomorphism we can now state our nal result. Theorem 4.9 The categories Lat of lattices and CLFrame of canonical L-frames are dual. A lattice L is mapped to its canonical frame (X; Y; ?), where X is the lter space of L and Y its space of ideals. Conversely, a canonical L-frame is mapped to the lattice X  of clopen sets that are stable under the duality induced by the relation ? (the latter, recall, is de ned by x ? y i x \ y 6= ;). 2 The duality, it should be noted, of Lat and CLFrame is a full duality. A canonical L-frame map was forced, by de nition to be obtained from a lattice homomorphism. Conversely, given a homomorphism f : L ! K , regard it as a meet semilattice homomorphism and a join semilattice homomorphism f^ : (L; ; ^) ! (K; ; ^) and f_ : (L;  _) ! (K; ; _) It is then immediate that the appropriate cubes commute, hence that f gives rise to a canonical lattice-frame morphism. We add here the following observation, for reasons of completeness. Proposition 4.10 Let L be a lattice and F = (X; Y; ?) its associated canonical L-frame. Let X p  X be the subset of prime lters in the subspace topology. Then X p is compact if and only if L is distributive. If X p is compact, then it is a CTOD space, in Priestley's sense. Hence its dual lattice is distributive. Conversely, if L is distributive, then since the subbasis inducing its topology is the same as in Priestley's theorem, the same argument shows X p to be compact. 2

24

Chrysa s Hartonas and J. Michael Dunn

Remark 4.11 Urquhart's [33] is the rst topological representation theorem

for general lattices. Allwein [1] showed that the restriction to maximal lterideal pairs is unnecessary, indeed that it gets in the way of a full, functorial duality theorem. Allwein and Hartonas [3] report the full duality theorem for lattices, extending the duality to one for congruences and sublattices, and obtaining a sheaf representation of lattices. Allwein and Dunn [2] use Urquhart's representation theorem as the basis for Kripke models for Linear Logic and related non-distributive substructural logics. The present paper may be seen as an application of an extension of Dunn's representation of galois connections in [18] and is part of the rst author's dissertation research [22] in progress. In [23], the rst author presents some further results on lattice representation and extends the representation and duality for lattices to lattice-ordered structures of logical signi cance. The results in [23] allow for a uni ed approach to the semantics of substructural logics, such as linear logic, while furnishing a framework signi cantly simpler than that presented in [2].

References [1] G. ALLWEIN (1992), The Duality of Algebraic and Kripke Models for Linear Logic, Doctoral Dissertation, Indiana University. [2] G. ALLWEIN and J. M. DUNN (1993), \Kripke Models for Linear Logic", The Journal of Symbolic Logic 58-2, 514-545. [3] G. ALLWEIN and C. HARTONAS (1993), \Duality for Bounded Lattices", Indiana University Logic Group, Preprint Series, IULG-93-25. [4] A. R. ANDERSON and N. D. BELNAP, Jr., (1975), Entailment: The Logic of Relevance and Necessity vol I, Princeton University Press, Princeton [5] A. R. ANDERSON, N. D. BELNAP, Jr., J. M. DUNN et al (1992), Entailment: The Logic of Relevance and Necessity vol II, Princeton University Press, Princeton

Duality Theorems

25

[6] B. BANASCHEWSKI (1973), \The Filter Space of a Lattice: Its Role in General Topology", Proc. Univ. Houston, Lattice Theory Conference, Houston 1973 147-155. [7] J. BARWISE (1993), private communication. [8] G. BIRKHOFF and J. VON NEUMANN (1936), \The logic of Quantum Mechanics", Ann. Math. 37, 823-843. [9] G. BIRKHOFF (1940), Lattice Theory, reprinted 1979, Amer. Math. Soc. Colloquium Publications XXV [10] M. L. DALLA CHIARA (1986), \Quantum Logic", in D. Gabbay and F. Guenthner (eds) Handbook of Philosophical Logic vol III, D. Reidel Publishing Company, 427-469. [11] B. A. DAVEY and H. A. PRIESTLEY (1990), Introduction to Lattices and Order, Cambridge University Press [12] K. DOSEN (1988), \Sequent-Systems and Groupoid Models I", Studia Logica XLVII, 4, 353-385. [13] K. DOSEN (1989), \Sequent-Systems and Groupoid Models II", Studia Logica XLVIII, 1, 41-65.  [14] K. DOSEN and P. SCHRODER-HEISTER (1993), eds, Substructural Logics, Oxford Press. [15] J. M. DUNN (1986), \Relevance Logic and Entailment", in D. Gabbay and F. Guenthner (eds) Handbook of Philosophical Logic vol III, D. Reidel Publishing Company, 117-224. [16] J. M. DUNN (1991), \Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operators", Logics in AI, Proceedings European Workshop JELIA 1990, LNCS 478, Springer Verlag. [17] J. M. DUNN (1993), \Star and Perp: Two treatments of Negation", Indiana University Logic Group, Preprint Series, IULG-93-21. Forthcoming in Philosophical Perspectives (Philosophy of Language and Logic),8, ed. J. Thomberlin.

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[18] J. M. DUNN (1993), \Partial-Gaggles applied to Logics with Restricted Structural Rules", Indiana University Logic Group, Preprint Series, IULG-93-22. Forthcoming in [13]. [19] J. Y. GIRARD (1987), \Linear Logic", Theoretical Computer Science 50, 1-102. [20] R. I. GOLDBLATT (1975), \The Stone Space of an Ortholattice", Bull. London Math. Soc. 7, 45-48. [21] G. M. HARDEGREE (1982), \An approach to the Logic of Natural Kinds", Paci c Phil. Quarterly 63, 122-132. [22] C. HARTONAS, Semantic Aspects of Substructural Logics, dissertation research in progress, Indiana University, Departments of Mathematics and Philosophy. [23] C. HARTONAS (1993), \Representation and Duality Theorems for Lattices with Additional Operators", forthcoming in the IULG Preprint Series. [24] A. HAZEN (1992), \Subminimal Negation", Philosophy Department Preprint 1/92, University of Melbourne, Melbourne. [25] P. T. JOHNSTONE (1986), Stone Spaces, Cambridge University Press. [26] P. LINCOLN, J. MITCHELL, A. SCEDROV and N. SHANKAR (1990), \Decision Problems for Propositional Linear Logic", SRI-CSL 90-08. [27] H. ONO and Y. KOMORI (1985), \Logics Without the Contraction Rule", The Journal of Symbolic Logic 50, 1, 169-201. [28] H. A. PRIESTLEY (1970), \Representation of Distributive Lattices by means of Ordered Stone Spaces", Bull. Lond. Math. Soc. 2, 186-90. [29] H. A. PRIESTLEY (1972), \Ordered Topological Spaces and the Representation of Distributive Lattices", Proc. Lond. Math. Soc. (3) 24, 507-30. [30] M. H. STONE (1936), \The Theory of Representations for Bollean Algebras", Trans. Amer. Math. Soc. 40, 37-111.

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[31] M. H. STONE (1938), \The Representation of Boolean Algebras", Bull. Amer. Math. Soc. 44, 807-16. [32] M. H. STONE (1937) \Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25. [33] A. URQUHART (1979), \A Topological Representation Theorem for Lattices" Algebra Universalis 8, 45-58. [34] R. WILLE (1987), \Bedeutungen von Begrisverbanden", Beitrage zur Begrisanalyse, ed. by B. Ganter, R. Wille and K. E. Wol, B. I. Wissenschaftsverlag, Mannheim, Wein, Zurich, 161-211.