1 Preliminaries - CiteSeerX

Stone Duality for Lattices Chrysa s Hartonas and J. Michael Dunn

Abstract

We present a new topological representation and Stone-type duality for general lattices. The dual objects of lattices are triples ( ? ), where are the lter and ideal spaces of the lattice, endowed with a natural topology, and ? is a relation from to . X;

;Y

X; Y

X

Y

1 Preliminaries This paper is largely motivated by the need to develop the second author's Gaggle Theory (Dunn [4]), aiming at providing a systematic procedure for the semantic analysis of various logical calculi via suitable representation results of their associated Lindenbaum algebras. As various calculi drop the distribution axiom (of meets over joins) the project seemed to stumble on a suitable representation result for general lattices. We rectify this situation with the present report (see also Hartonas [6] and Hartonas [7]). We are of course aware that Urquhart [13] has already proposed a topological representation for general lattices. In fact Allwein and the second author (see [1]) exploited Urquhart's representation theorem to provide Kripke semantics for Linear Logic. Allwein and the rst author (see [2]) extended Urquhart's topological representation to a full duality result including a duality for congruences and sublattices. Independently, Hartung [8] extended Urquhart's representation to a full duality working within R. Wille's framework of concept lattices [14]. We follow a dierent course here mainly for the reason that Urquhart's representation necessarily leads to a 3-valued semantics. We establish a

 Both authors acknowledge Gerard Allwein for a number of useful discussions. The present paper is a shortened version of a paper by the same authors which appeared as a pre-print of the Indiana University Logic Group, IULG-93-26, under the title \Duality Theorems for Partial Orders, Semilatices, Galois Connections and Lattices".

1

new representation result, abstracting the essentials from both Urquhart's representation of general lattices and from Goldblatt's topological representation of ortholattices [5]. We prove functoriality of the representation and extend to a full Stone-type duality which allows for representing the lattices of congruences and of sublattices in a straightforward way. Even though semantically/logically motivated we hope that our result will prove useful for purely lattice-theoretic applications as well.

2 Lattice Representation and Duality By a frame we mean a triple F = (X; ?; Y ) where X; Y are sets (perhaps with additional structure) and ? is a binary relation from X to Y . ? induces a galois connection from subsets of X to such of Y explicitly de ned on U  X and V  Y by (Birkho [3])

U = fy 2 Y jU ? yg V = fx 2 X jx ? V g (1) where U ? y means that 8u 2 U u ? y and similarly for x ? V . Each of  and  is a closure operator and thus the relation ? induces two intersection (closure) systems Cx; Cy cosisting of stable subsets of X and of Y respectively: Cx = fA  X j A = Ag Cy = fB  Y j B = Bg Of particular interest for our purposes are frames equipped with distinguished subfamilies Sx  Cx; Sy  Cy of stable sets closed under nite intersections and such that the galois connection  a  restricts to a dual isomorphism  : Sx  = Syop . This implies that each of Sx ; Sy is a lattice with joins de nable in Sx by

_ \ [ A B = (A B) = (A B) and similarly for joins in Sy . If L is an ortholattice then its dual frame (Goldblatt [5]) consists of two copies of the lter space of L and a relation ?, representing orthonegation, de ned on lters by x ? y i 9a 2 x :a 2 y . In the general case any automorphism  of L istrivially a dual isomorphism of L with its opposite = (Lop )op. In particular, we may take  to be the lattice Lop , i.e.  : L ?! identity on L. The automorphism  induces a binary relation ? . Using the identity on L we obtain the relation ? from the set X of lters to the 2

set Y of ideals of L de ned by x ? y i x \ y 6= ; (which is reminiscent of Urquhart's [13] use of disjoint lter-ideal pairs). We think of L as consisting of a meet and a join semilattice (L; ; ^); (L ; _) and represent each in its set of \ lters" by sending an element a to Xa = fx 2 X ja 2 xg and Ya = fy 2 Y ja 2 Y g. The collections Sx = fXaja 2 Lg and Sy = fYaja 2 Lg are closed under intersection and stable under the galois connection  a  induced by ? in the sense explained above. Furthermore,  : Sx  = Syop is a dual isomorphism and hence the original lattice is Tisomorphic to the lattice Sx via the map a 7! Xa and where a ^ b 7! Xa Xb and a _ b 7! (Xa T Xb) = (Ya T Yb). Furnishing each of X and Y with the natural topologies obtained from the subbases

BX = fXaga2L [ f?Xbgb2L BY = fYaga2L [ f?Yb gb2L we can derive an intrinsic characterization of the family Sx as the family of stable, compact-open subsets of X (and similarly for Sy ). To obtain a duality theorem we need to provide an explicit characterization of the suitable types of frames. We do this in the sequel where we also tighten our de nition of a frame and de ne appropriate frame morphisms. Details of some of the arguments summarily presented above are also given.

2.1 Representation of Lattices in Topological Frames

Definition 2.1 An FSpace X is a partially ordered Stone space such that

1. For any U = fx j 2 Ag  X the greatest lower bound of U exists 2. X has a subbasis of clopen sets S = fXigi2I [ f?Xj gj 2I indexed by some set I , such that for each i 2 I; Xi is a principal upper set, generated by a point xi 2 X 3. The subset fxi ji 2 I g  X is join-dense in X , i.e. every point x is the least upper bound of the xi 's it covers. 4. The collection X  = fXi gi2I is closed under nite intersections.

FSpace morphisms are de ned to be the continuous functions h : Y ! X preserving glb's in X and such that h?1 maps X  into Y  . FSpace denotes the respective category of FSpaces. The following is immediate: 3

2.2 Let X be an FSpace. Then X  is a meet semilattice. Fur h?1  h  thermore, if h : X ! Y is an FSpace morphism, then Y ?! X is a

Lemma

=

meet-semilattice homomorphism.

Definition 2.3 A ?-frame is a triple (X; ?; Y ), where X; Y are FSpaces,

? is a binary relation from X to Y and if (; ) is the induced galois connection from P X to P Y , then it restricts to a galois connection from X  to Y  . The ?-frame is called an L-frame (lattice frame) if  is a dual isomorphism of the intersection semilattices X  and Y  . The galois connection induced by ? is de ned as in (1). In the sequel we establish a Stone-type duality between the category Lat of lattices and a suitable category LFrame of lattice frames. Theorem 2.4 (Representation of Lattices) Let L be a lattice and F = (X; ?; Y ) the structure consisting of the set X = F (L) of lters of L, the set Y = I (L) of ideals of L and the relation x ? y i x \ y = 6 ;. Then F can be given the structure of an L-frame. Furthermore, the map a 7! Xa = fx 2 X ja 2 xg is a lattice isomorphism L = X , with a ^ b 7! Xa \ Xb and a _ b 7! (Xa \ Xb). In addition, X  can be characterized as the collection of all clopen, stable subsets of the lter space F (L) = X , where a set U  X is called stable provided that U = U . Proof: We prove the theorem in a series of lemmas. If (P; ; ^) is a meet semilattice let X = F (P ) be the set of its lters and for each p 2 P let Xp = fx 2 X jp 2 xg. Topologize X via the subbassis S = fXpgp2P [ f?Xp0 gp02P Lemma 2.5 If X is as above then with the topology induced by S X is turned to an FSpace. Furthermore, if f : P ?! Q is a meet semilattice homomorphism, then F (f ) = f ? : F (Q) ?! F (P ) is a morphism of FSpaces. Proof: 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. 1

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 4

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 , xB = Tfa 2 P j(9b1; : : :; bn 2 BS)(b1 ^


^ bn  a)g. Then B  xB , hence xB 2 b2B Xb and thus xB 62 b2SB (?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. Let c = b1 ^


^ bn be such that c  a, for some b1; : : :; bn 2 B. If compactness fails, let x 62 Xa [ (?Xb1 ) [


[ (?Xbn ). Then b1 ^


^ bn = c 2 x and a 62 x, while c  a, contradiction. If f : P ! Q is a meet semilattice homomorphism, then the map F (f ) = ? 1 f takes lters of Q to lters of P , hence F (f ) : F (Q) ! F (P ). Since F (f ) is an inverse map it clearly preserves arbitrary intersections of lters. For continuity, it is enough to verify that subbasis sets in X = F (P ) are taken by F (f )?1 to open sets in Y = F (Q), since open sets in X are unions of basic open sets and the basis generated by a subbasis consists of nite intersections (preserved by inverse maps) of subbasis elements. A direct calculation shows that y 2 F (f )?1 (Xa) i y 2 Yf (a) hence F (f )?1 (Xa) = Yf (a) and F (f )?1 (?Xa ) = ?Yf (a). Thus F (f ) is continuous. Now if (L; ; ^; _) is a lattice, the lter spaces X and Y of the meet semilattices (L; ; ^) and (L; ; _)op, respectively, are FSpaces by the above argument. The subbasis of Y consists of the sets of ideals of the form Ya = fy 2 Y ja 2 yg together with their complements. If ? X  Y is the relation x ? y i x \ y 6= ; let (; ) be the induced galois connection from sets of lters to sets of ideals. Lemma 2.6 For any element a of the lattice L, Xa = Ya and Ya = Xa . Proof: If Xa ? y then let b 2 xa \ y . Since y is an ideal and a  b it follows a 2 y and thereby y 2 Ya . For the converse, if y 2 Ya and x 2 Xa then a 2 x \ y 6= ;, hence x ? y for all x 2 X and therefore y 2 Xa. The argument for Ya = Xa is similar. Where X  = fXaja 2 Lg; Y  = fYa ja 2 Lg Lemma 2.6 implies that  is a dual isomorphism of the intersection semilattices X  and Y . This completes the proof of the claim made in Theorem 2.4 that (X; ?; Y ) is an L-frame. For the representation part of Theorem 2.4 observe that if S and K are meet semilattices and  : S ?! K is a dual isomorphism then each of 5

S and K is a full lattice. Setting  = ?1 joins in S can be de ned by s _S s0 = (s ^K s0). Hence Lemma 2.6 implies that the intersection semilattice X  = fXajaS2 Lg is a lattice with joins de ned by Xa W Xb = T (Xa Xb) = (Xa Xb). The composite  is a closure operator on subsets of X and thus itWde nes a closure system with arbitrary joins de ned S T as closurers of unions i2I Ui = ( i2I Ui ) = ( i2I Ui ). That the map a 7! Xa is a lattice isomorphism L  = X  is straightforward. We leave details

to the interested reader. It now remains to verify that X  can be characterized independently of the representation map as the family of stable compact-opens of the space X . By de nition of the topology on X each set Xa is clopen, hence compactopen since X is a Stone space (Lemma 2.5). Stability of the Xa 's has been veri ed in Lemma 2.6. It remains to establish that every stable clopen subset U  X is of the form Xa for some a 2 L. We rst prove the following lemma, generalizing Goldblatt's representation for ortholattices [5]. Lemma 2.7 Assume S; K are meet semilattices and ` : S ?! K is a dual isomorphism with inverse r = `?1 . Let (X; ?; Y ) be the structure consisting of the lter space X of S , the lter space Y of K and the binary relation ? X  Y de ned by x ? y i 9s 2 x `s 2 y. Then X is an FSpace (by Lemma 2.5), X  = fXs js 2 S g is a lattice isomorphic to S and furthermore X  is the collection of stable clopen subsets of X , stability refering to the galois connection (; ) induced by the relation ?. Proof: Existence of a dual isomorphism between S and K implies that each is a lattice as we have previously pointed out. We will only show that every stable compact-open set U is of the form Xs for some s 2 S . If U is clopen and stable and x 62 U = U let y 2 U such that x 6? y . Since y 2 U S , for any z 2 U; z ? y, so there S is az 2 z such that `az 2 y. Thus U  Xaz and by compactness U  ii==1n Xaiz . Let

ax = a1z _


_ anz = r (`a1z ^


^ `anz ) Then for each i, Xaiz  Xax , hence U  Xax . Now notice that ax 62 x. If it were, then `ax = `r (`a1z ^


^ `anz ) = `a1z ^


^ `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 . We apply this lemma to the case where S is a lattice furnished with a dual isomorphism. Trivially, the identity `a = a on L is such a dual isomorphism and the relation ? of the lemma is the relation x ? y i x \ y = 6 ;. In fact 6

the argument is elementary for this case. Indeed, choose ax as in the proof of Lemma 2.7. Since each aiz 2 y and y is an ideal it follows directly that ax 2 y hence ax 62 x sinceSx 6? y, i.e. x \ y = ;. Then ?U  ?Xax and by aiz  ax we have U  1in Xaiz  Xax . This completes the proof of Theorem 2.4. Remark 2.8 1. Lattices can be concretely identi ed as subsystems of closure systems. What Theorem 2.4 shows is that every lattice is isomorphic to a subsystem of a closure system on some set X , where the closure system in question is obtained via the closure operator ? = . 2. The representation theorem also shows that every lattice can be considered as a diagram consisting of two intersection semilattices S; S 0 connected by a dual isomomorphism  : S  = (S 0)op. This reduces the study of lattices to that of galois connected semilattices. We have not explored the consequences of this reduction but think it of potential usefulness.

2.2 Duality

The idea for our representation is to reduce the problem to that of representing semilattices and galois connections. For the lattice case we have considered the trivial galois connection idL : L ?! (Lop)op , viewing L and Lop as meet semilattices. The galois connection induces a binary relation from lters to ideals from which, as in Birkho [3], we extract a new galois connection from sets of lters to sets of ideals. Morphisms of ?-frames will then have to interact with this galois connection in the right way. Definition 2.9 A morphism (f; h) : (X1; Y1; ?1) ?! (X2; Y2; ?2) of ?frames, is de ned to be a pair of FSpace-morphisms f : X1 ! X2 and h : Y1 ! Y2 such that f  = f ?1 : X2 ! X1 and h = h?1 : Y2 ! Y1 make both squares in Figure 1 commute. ^ ?^ ; Y^ ) is a morphism of L-frames Lemma 2.10 If (f1; f2 ) : (X; ?; Y ) ?! (X; ? 1    ^ then f1 := f1 : X ?! X is a lattice homomorphism. (Similalry, f2 : Y^  ?! Y  is a lattice homomorphism).

Proof: Since f1 is an Tinverse function it must preserve intersections, T  ^ ^ i.e. f1 (Xa Xb ) = f1 X^ a f1 X^ b . Preservation of joins follows from the

commutativity of the square in Figure 1. Indeed 7

f

Figure 1: Morphisms of ?-Frames

 X1 1 6 1

-(Y )op 6h

2 2

-(Y )op

X2

1

2

^X^b) f1(X^a W X^b) = f1^(^X^a T T = f2(^X^ a T ^ X^ b ) = (f2^ X^ a T f2 ^ X^ b ) = (f1W X^a f1X^b)  = f1 X^ a f1 X^ b

(by de nition of joins) (by commutativity in Figure 1) (because f2 = f2?1 ) (by commutativity in Figure 1) (by de nition of joins)

Proposition 2.11

1. Every FSpace arises as the space of lters of a meet semilattice. 2. Furthermore, if h : Y ! X is an FSpace map, where X = FA, Y = FB, then there is a unique semilattice morphism h^ : A ! B such that h = h^ ?1 and for any a 2 A, h^ a = b i h?1 (Xa) = Yb .

Proof:

(1) If X is an FSpace with subbasis sets indexed by the set A, partially order A by a  a0 i Xa  Xa0 . Closure of X  under intersection implies that A is a meet semilattice in the de ned ordering. We show that X  F (A). 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: 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, ^?1 (Xa ) = Fa = f 2 F ja " g. Applying ^?1 = ^ on both sides we also get ^?1 (Fa ) = Xa, so that ^ is continous and an open map, hence a homeomorphism. 8

P f



?

P0 

Figure 2: Morphisms of Galois Connections

` - op Q r g `0- ?0 op (Q ) r0

(2) If h : Y ! X is an FSpace morphism, by the previous argument we may assume X = F (A); Y = F (B ), where A and B are the index sets for the respective subbases of X and Y . De ne the map h^ : A ! B by h^ a = b i h?1 (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?1 y . By total separation of X , let ^h?1 y 2 Xa; hy 62 Xa . Thus a 2 ^h?1 y and so h^a 2 y. If h^ a = b, then y 2 Yb and, by de nition of h^, h?1(Xa) = Yb. So y 2 h?1(Xa), hence hy 2 Xa follows. Thus h = F (^h) = ^h?1 as claimed. Next we verify that L-frames are precisely what they are expected to be, namely frames arising from lattices. Given a galois connection1 G = (P; Qop; `; r), 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 ;  ;  ). In the sequel we restrict to considering only galois connections between semilattices. To be able to decide when two galois connections are basically the same (isomorphic) we use the natural notion of morphism for galois connections as a pair of meet semilattice homomorphisms such that both squares in Figure 2 commute. We will then \identify" two galois connections if both f and g are isomorphisms (and the squares commute). Proposition 2.12 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 ; ;  ?1). 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 ;. 1

The notation we use means that ` a r, where ` : P ?! Qop and r : Qop ?! P .

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Proof: If L and K are the index sets of the subbases of X and Y , respectively, we may assume (Proposition 2.11) that X = F (L) and Y = F (K ), where F (L); F (K ) are the lter spaces of L and K . 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 dual isomorphism  : 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 . In view of Proposition 2.12 the sets X; Y in an L-frame (X; ?; Y ) can be thought of as sets of sets (in fact as sets of lters of semilattices). Hence we may restrict to canonical L-frames where ? is the relation x ? y i x \ y 6= ;. Furthermore, if (g; h) : (X; ?; Y ) ?! (X 0; ?0; Y 0 ) is a ?- frame morphism, by Proposition 2.11 X and X 0 are the lter spaces of semilattices A; A0 while Y; Y 0 are their ideal spaces and both g and h arise from semilattice homomorphisms g^ : A ?! A0 and ^h : A ?! A0 . We now let LFrame be the category of canonical L-frames with frame morphisms and observe the following: ^ ?^ ; Y^ ) be a morphism of canonical Lemma 2.13 Let (f; g ) : (X; ?; Y ) ?! (X; L-frames. By Proposition 2.11 we may assume that X; X 0 are the lter spaces of semilattices A; A0 while Y; Y 0 are their ideal spaces and both f and g arise from semilattice homomorphisms f^ : A ?! A0 and g^ : A ?! A0. The assumption that the L-frames are canonical implies that f^ = g^. Proof: If  and ^ are the dual isomorphisms induced by the relations ?; ?^ then f X^a = Xfa^ = Yfa^ while also g^X^a = Yg^a. By the com^ ^. mutativity condition of Figure 1 Yfa ^ = Yg ^a for any a and hence f = g

Before turning to duality we verify that lattice homomorphisms can be represented as morphisms of L-frames. Proposition 2.14 Let f : L ?! K be a lattice homomorphism, ( XL ; ?L ; YL ) and (XK ; ?K ; YK ) the dual frames of L and K . Then f is represented as a frame homomorphism (g; h) : (XK ; ?K ; YK ) ?! (XL ; ?L ; YL ). 10

Proof: Let f^ ; f_ be f regarded as a meet and as a join homomorphism, respectively and set g = F (f^ ) = f^?1 ; h = F (f_ ) = f_?1 . Then g : XK ?! XL and h : YK ?! YL are FSpace morphisms by the argument in the proof of Lemma 2.5. Furthermore g  = g ?1 : XL ?! XK and h = h?1 : YL ?! YK are semilattice homomorphisms since inverse functions preserve intersections. If L ; K are the dual isomorphisms L : XL  = (YL )op   op    and K : XK = (YK ) we need to verify that K g = h L . Since g = F (f^ )?1, by the argument in the proof of Lemma 2.5 we have, given Xa 2 XL, g(Xa) = Xfa 2 XK . By Lemma 2.6, K (Xfa) = Yfa. Similarly L h Y . Hence the pair (g; h) is a morphism of the dual frames Xa ?! Ya ?! fa of L and K . We conclude the paper with a duality theorem. Theorem 2.15 The functors F, sending a lattice L to its dual L-frame F(L) = (XL ; ?L ; YL ), and G, sending an L-frame F = (X; ?; Y ) to its dual lattice G(F ) = X  form a duality of categories Lat = LFrameop. Proof: In Theorem 2.4 we veri ed that a lattice L is isomorphic to its second dual L  = X , where X  = GF(L). If F = (X; ?; Y ) is an L-frame we showed in Proposition 2.12 that F is the dual frame of some lattice L  = X . Since the frame is canonical the automorphism  in the proof of Proposition 2.12 is the identity i. Hence F  = FG(F ). Assume f : L ?! K is a lattice homomorphism and let f^?1 = F (f^ ) and f_?1 = F (f_ ). Then (F (f^ ); F (f_)) is an L-frame morphism. In addition F (f^ )?1 : XL ?! XK is a lattice homomorphism commuting with the isomorphisms L  = XL and K  = XK since (see the proof of Lemma 2.5) F (f^)?1(Xa) = X^fa. ^ ?^ ; Y^ ) is a morphism of LFinally, suppose (f; g ) : (X; ?; Y ) ?! (X; frames and let f  = f ?1 . Then by Lemma 2.10 f  is a lattice homomorphism f  : X^  ?! X . Furthermore, by Proposition 2.11 we may assume that X and X^ are the lter spaces of semilattices A; B and that f is induced by a semilattice homomorphism f^ : A ?! B by f = F (f^) = f^?1 . By the same token g = g^?1 for some g^ : A0 ?! B 0 where Y; Y^ are the lter spaces of A0 and B 0 respectively. Furthermore, by Lemma 2.13 f^ = g^. Since the frames are canonical L-frames the semilattices A; B are in fact lattices and then f^ = g^ = h : A ?! B is a lattice homomorphism. It is then clear that (f; g ) = (F (h^ ); F (h_)).

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References

[1] G. Allwein and J. M. Dunn (1993), \Kripke Models for Linear Logic", The Journal of Symbolic Logic 58-2, 514-545. [2] G. Allwein and C. Hartonas (1993), \Duality for Bounded Lattices", Indiana University Logic Group, Preprint Series, IULG-93-25. [3] G. Birkhoff (1940), Lattice Theory, reprinted 1979, Amer. Math. Soc. Colloquium Publications XXV

[4] 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. [5] R. I. Goldblatt (1975), \The Stone Space of an Ortholattice", Bull. London Math. Soc. 7, 45-48. [6] C. Hartonas (1996), \Order-Duality, Negation and Lattice Representation", in Negation: Notion in Focus, ed. H. Wansing, De Gruyter, Berlin, 27-36. [7] C. Hartonas, \Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics". To appear in Studia Logica, 1997. [8] G. Hartung (1992), \A Topological Representation for Lattices", Algebra Universalis 17, 273-299. [9] P. T. Johnstone (1986), Stone Spaces, Cambridge University Press. [10] H. A. Priestley (1970), \Representation of Distributive Lattices by means of Ordered Stone Spaces", Bull. Lond. Math. Soc. 2, 186-90. [11] M. H. Stone (1938), \The Representation of Boolean Algebras", Bull. Amer. Math. Soc. 44, 807-16. [12] M. H. Stone (1937) \Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25. [13] A. Urquhart (1979), \A Topological Representation Theorem for Lattices" Algebra Universalis 8, 45-58. [14] 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.

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