Duality and Canonical Extensions of Bounded Distributive Lattices ...

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Oct 19, 1998 - D(kA)(F) = Ank?1. A (F) = fa 2 A jkA(a) 62Fg: If f 2 Jh has arity n (i.e. fA : An !A is a join-hemimorphism), D(fA) D(A)n+1 is de ned by. D(fA)(F1;::: ...
Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I Viorica Sofronie-Stokkermans Max-Planck-Institut fur Informatik, Saarbrucken, Germany e-mail: [email protected] October 19, 1998 Abstract

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that nitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes.

Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of de ning interpretations or models. One possibility is to use algebras { usually lattices with operators { as models. Propositional variables are interpreted over elements of these algebraic models, and the logical connectives correspond to the algebra operators. The other way is to use Kripke-style semantics. Here the models are known as frames, and usually carry a relational rather than an algebraic structure. Formulae are interpreted as subsets of the frame in a way that depends on the particular structure on these frames. Usually, the algebraic models have the advantage that soundness and completeness results are easier to prove, which is not always the case with the Kripke-style models. Kripke-style models are, on the other hand, more comfortable to use because of their often much simpler structure. This is one of the reasons why the most substantial progress in the study of modal logics took place since the introduction of relational structures in the study of the semantics of modal logics by Kripke [Kri63]. The ultimate goal of this paper is to give a better understanding of the link between the algebraic models and the Kripke-style models for certain classes of propositional logics based on distributive lattices with operators (thus, we go one step beyond the modal case). Our choice is motivated, on the one hand, by the fact that many non-classical logics that occur in a natural way in practical applications can be proved to be sound and complete with respect to certain classes of distributive lattices with operators, and, on the other hand, by the fact that distributive lattices (with well-behaved operators) usually have good and economical representation theorems. In this rst part of the paper we present the algebraic results by which we can reach this goal: We start by extending the Priestley duality for distributive lattices to distributive lattices with operators satisfying certain preservation properties, in particular various types of \antimorphisms", thus extending the duality given by Goldblatt in [Gol89]. We then show that nitely-generated varieties of distributive lattices with operators (lattice homomorphisms and antimorphisms; join- and meet-hemimorphisms, and join- and meet-hemiantimorphisms) have the closure property under canonical embedding algebras (which will be formally de ned in Section 4). In the second part of the paper [SS98] we use these results in analyzing the link between the algebraic and Kripke-style models of certain non-classical logics. The idea of using representation theorems for establishing a link between the algebraic and relational semantics of non-classical logics goes back to Jonsson and Tarski [JT51, JT52], who for this purpose used 1

an extension of Stone's representation theorem for Boolean algebras with operators. Every relational structure X = (X; fRgR2R) comprising a collection R of nitary relations on a set X gives rise to a Boolean algebra with operators X + = (P (X ); ffR gR2R) in a canonical way. Jonsson and Tarski showed that for every Boolean algebra with operators A there exists a relational structure C (A) of appropriate type (the canonical structure of A), and an embedding of A into C (A)+ . The algebra C (A)+ is, by de nition, the canonical extension or canonical embedding algebra of A. Similar results are used in the study of modal systems by Lemmon [Lem66a, Lem66b] and Goldblatt [Gol89]. More general duality theorems (for partial orders, semilattices, Galois connections and lattices) are used by Hartonas and Dunn in [HD93]. Representation theorems are also used by Allwein for giving a Kripke-style semantics for linear logics [All93]. Also, similar ideas occur in correspondence theory [vB84], where the link between axioms and theorems of a given logic and properties of the corresponding Kripke models of the logic is investigated. Our work is in uenced by the results of Goldblatt [Gol89], who showed that the \modal case" can be seen as a simple illustration of more general results from universal algebra. Goldblatt establishes various relations between properties of a given class K of relational structures and properties of the class SK + of all algebras which are isomorphic to subalgebras of X + for some relational structure X in K , and studies the general structure of so-called complex varieties, i.e. varieties of Boolean algebras with operators that are representable in the form SK + for some class K of relational spaces. One of the properties of varieties of Boolean algebras with operators investigated in [Gol89] is the closure under canonical embedding algebras. One of its possible applications is in giving an algebraic formulation and an alternative proof of the fact that if a modal logic is characterized by an elementary class of Kripke structures (i.e. de ned by a set of rst-order sentences) then it is validated by its Henkin structures [Gol89]. This result was rst proved by van Benthem in [vB80], by model-theoretical arguments. Although in [Gol89] only the case of Boolean algebras and the corresponding relational structures are considered in detail, the results presented there suggest that similar methods can also be applied to more general classes of non-classical propositional logics, whose algebraic models are bounded distributive lattices endowed with (well-behaved) operators. The closure under canonical embedding algebras for varieties of distributive lattices with additive operators was studied by Gehrke and Jonsson in [GJ94]. In this paper we extend both the results in [Gol89], by considering distributive lattices with operators instead of Boolean algebras with operators, and those in [GJ94], in that we also consider certain classes of antimorphisms. In the second part of this paper [SS98] we will show that, under certain circumstances, closure under canonical embedding algebras can be used to establish, by purely algebraic means, soundness and completeness results for certain logics with respect to classes of Kripke-style models obtained in a canonical way from their algebraic models. The paper is structured as follows: In Section 1 we brie y introduce the notations and the main notions that will be used later. In Section 2 we present an extension of the Priestley duality to distributive lattices with operators in the following classes: lattice homomorphisms and antimorphisms; join- and meet-hemimorphisms; and join- and meet-hemiantimorphisms. In Section 3 ordered relational structures are de ned and some of their properties are studied. In Section 4 we present some cases of varieties that are closed under embedding algebras. By a result of [GJ94], all varieties of distributive algebras with additive operators are closed under canonical embedding algebras; an essential assumption in the proof is the fact that all operators are order-preserving. We prove that this also holds for certain classes of distributive algebras with operators which are not necessarily order-preserving, namely we show that such a closure condition holds for nitely generated varieties of distributive lattices with operators in the classes mentioned in Section 2. In Section 5 we present some conclusions and plans for future work. In the second part of this paper [SS98] we will show how the results presented here can be applied in studying the link between algebraic and Kripke-style models, for certain classes of propositional logics.

1 Preliminaries We will now brie y review the main concepts and notations that will be used in our work. Order theory. A partially-ordered set is pair (P; ), where P is a set and   P  P is a re exive, transitive and antisymmetric relation (i.e. a partial order on P ). Given a partially-ordered set (P; ), a subset J of P is an order-ideal if for every x; y 2 P , if x  y and y 2 J then x 2 J (i.e. it is downwardsclosed with respect to ). A subset F  P is an order- lter if for every x; y 2 P , if x  y and x 2 F 2

then y 2 F (i.e. it is upwards-closed with respect to ). In what follows we will denote by O(P ) the set of order- lters of a partially-ordered set P ; for every subset A of P we will use the following notations: " A = fx 2 P j 9a 2 A; x  ag, and # A = fx 2 P j 9a 2 A; a  xg. In particular, if a 2 P then " a = fx 2 P j x  ag and # a = fx 2 P j a  xg. For further informations on partially-ordered sets we refer to [DP90]. Lattices. A lattice is a partially-ordered set (L; ) with the property that every two elements x; y 2 L have a supremum (denoted x _ y) and an in mum (denoted by x ^ y) in L. Alternatively, a non-empty set L together with two binary operations _ and ^ on L is called lattice if _; ^ are commutative, associative, idempotent, and satisfy the absorption law. A distributive lattice is a lattice that satis es either of the distributive laws (which are equivalent in a lattice). We say that a lattice L has a rst (smallest) element if there is an element 0 2 L such that 0  x for every x 2 L. A lattice L has a last (greatest) element if there is an element 1 2 L such that x  1 for every x 2 L. A lattice having both a rst and a last element is called bounded. If not speci ed otherwise, in what follows we will denote the rst element in a lattice (if any) by 0 and the last element (if any) by 1. Bounded lattices will be represented as algebras (L; _; ^; 0; 1). (For instance, for every partially-ordered set (P; ), (O(P ); [; \; ;; P ) is a bounded distributive lattice with 0 = ; and 1 = P .) An element x 2 L is called join irreducible if x 6= 0 (in case L has a 0) and x cannot be expressed as the join of two other elements in L, i.e. if it has the property that x = y _ z implies y = x or z = x. An element x 2 L is called meet irreducible if x 6= 1 (in case L has a 1) and it has the property that x = y ^ z implies y = x or z = x. Let L be a lattice. A non-empty subset J of L is called an ideal if for every x; y 2 J , x _ y 2 J , and for every x; y 2 L, if y 2 J and x  y then x 2 J . Dually, a non-empty subset F of L is called a lter if for every x; y 2 F , x ^ y 2 F , and for every x; y 2 L, if x 2 F and x  y then y 2 F . Thus, an ideal is a non-empty downwards-closed set closed under join, and a lter is a non-empty upwards-closed set closed under meet. An ideal or lter is called proper if it does not coincide with L. Let L be a lattice and J a proper ideal in L. The ideal J is said to be prime if for every x; y 2 L, if x ^ y 2 J then x 2 J or y 2 J . The set of prime ideals of L is denoted by Ip (L). A prime lter is de ned dually, i.e. it is a proper lter F with the property that if x; y 2 L and x _ y 2 F then x 2 F or y 2 F . The set of prime lters is denoted by Fp (L). Ordered topological spaces. An ordered topological space is a triple (X; ;  ) where  is a partial order on X and  is a topology on X . The set of clopen (i.e. closed and open) order- lters of such a space will be denoted ClopenOF(X; ;  ), or simply by ClopenOF(X ) if the order and topology are supposed to be known. ClopenOF(X; ;  ) is a sublattice of the lattice of order- lters of the partially-ordered space (X; ), where the join is the union and the meet is the intersection. An ordered topological space is totally order-disconnected if, whenever x 6 y, there exists a clopen order- lter U 2 O(X ) with x 2 U and y 62 U . A Priestley space is a totally order-disconnected space that is also compact. We note that in a Priestley space the clopen order- lters and their complements form a subbasis for the topology. It is easy to see that this topology is the join of two topologies, namely: (i) the upper topology, generated by the set of clopen order- lters as a basis, and (ii) the lower topology, generated by the set of complements of clopen order- lters as a basis. Priestley Duality. In what follows we will often refer to the Priestley duality theorem for distributive lattices; for details see e.g. [Pri70, Pri72, DP90, Gol89]. For the convenience of the reader we mention the main results. The Priestley representation theorem states that every distributive lattice L is isomorphic to the lattice of clopen order lters of the ordered topological space D(L) = (Fp (L); ;  ) having as points the prime lters of L, ordered by inclusion, and the topology  generated by the sets of the form Xa = fF j F prime lter a 2 F g and their complements as a subbasis. Moreover, it was shown that a dual equivalence exists between the category D01 of bounded distributive lattices (with homomorphisms of bounded distributive lattices as morphisms) and the category P of Priestley spaces (with continuous, order-preserving maps as morphisms). This dual equivalence is de ned by two contravariant functors D : D01 ! P and E : P ! D01 . These functors are de ned on objects as follows1: for every bounded distributive lattice L, D(L) = (Fp (L); ;  ) (which will be called in what follows the Priestley dual of L) is the ordered topological space de ned as explained above, and for every Priestley space X , E (X ) is its set of clopen order- lters, on which a bounded distributive lattice structure is de ned by taking 1 Actually, in [Pri70, Pri72] the functors are de ned, equivalently, as follows: for every bounded distributive lattice L, ( ) = fh : L ! f0; 1g j h 0,1-lattice homomorphismg and for every Priestley space X , E (X ) = ff : X ! f0; 1g j f continuous, order-preservingg, where f0; 1g is regarded as a bounded lattice in the rst case, and as an ordered set (with 0 < 1) endowed with the discrete topology in the second case.

D L

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0 = ;; 1 = X and de ning joins and meets as unions resp. intersections of clopen order lters. (The set of clopen order- lters of D(L) is the set fXa j a 2 Lg.) The functors D and E are de ned on morphisms as follows: for every morphism of bounded distributive lattices f : L1 ! L2 , D(f ) : D(L2 ) ! D(L1 ) is de ned by D(f )(F ) = f ?1 (F ) for every prime lter F of L2 , and for every continuous, order-preserving map h : X1 ! X2 , E (h) : E (X2 ) ! E (X1 ) is de ned by E (h)(U ) = h?1 (U ) for every clopen order- lter U of X2 . The functors D and E establish a dual equivalence between the categories D01 and P: for every bounded distributive lattice L there is a 0,1-lattice isomorphism L : L ' E (D(L)), and for every Priestley space X , there exists an order-preserving homeomorphism X : X ' D(E (X )). In particular, for every nite lattice L, D(L) = f" i j i join-irreducible in Lg, the order is de ned by " i  " j if and only if i  j in L, and the Priestley topology on D(L) is discrete. Thus, if L is nite, then L ' O(D(L)). Universal algebra. For the basic notions of universal algebra needed in what follows we refer for instance to [BS81]. For every signature  and every arity function a :  ! N , a -algebra is a structure (A; fA g2 ), where for every  2 , A : Aa() ! A. If the signature  is assumed to be known we will sometimes use the notation A for the -algebra (A; fA g2 ). We say that a -algebra has a bounded distributive lattice reduct if there exist operation symbols _; ^; 0; 1 in  such that (A; _A ; ^A ; 0A ; 1A) is a bounded distributive lattice. A variety is a class of algebras which is closed under homomorphic images, subalgebras and direct products. The variety generated by a class K of algebras is the class of all algebras that are homomorphic images of a subalgebra of a product of elements in K; it will be denoted HSP (K).

2 Priestley duality for distributive lattices with operators In [Gol89] the Priestley duality for distributive lattices is extended to a dual equivalence between the category DLO of distributive lattices having as operators join- and meet-hemimorphisms and a suitable category RPS of relational Priestley spaces. In what follows we extend the results of [Gol89] in that besides considering distributive lattices endowed with join- and meet-hemimorphisms we also consider separately operators that are homomorphisms, antimorphisms, join- and meet-hemiantimorphisms. These results are then used in [SS98] in order to de ne a general notion of relational (not necessarily topological) models for certain logics, and a notion of satis ability in such models. This type of results was used in [SS97] in the study of ( rst-order) nitely-valued logics. The Priestley dual of the algebra of truth values was used for obtaining ecient translations to clause forms and an automated theorem proving procedure. De nition 1 Let A be an algebra with a bounded lattice reduct. A lattice antimorphism on A is a function k : A ! A with k(0) = 1; k(1) = 0; k(a1 _ a2 ) = k(a1 ) ^ k(a2 ) and k(a1 ^ a2 ) = k(a1 ) _ k(a2 ). A join-hemimorphism2 on A is a function f : An ! A such that for every i; 1  i  n, (Jh1) f (a1 ; : : : ; ai?1 ; 0; ai+1 ; : : : ; an ) = 0; (Jh2) f (a1 ; : : : ; ai?1 ; b1 _ b2 ; ai+1 ; : : : ; an ) = = f (a1 ; : : : ; ai?1 ; b1 ; ai+1 ; : : : ; an ) _ f (a1 ; : : : ; ai?1 ; b2 ; ai+1 ; : : : ; an ): A meet-hemimorphism on A is a function g : An ! A such that for every i; 1  i  n, (Mh1) g(a1 ; : : : ; ai?1 ; 1; ai+1 ; : : : ; an ) = 1; (Mh2) g(a1 ; : : : ; ai?1 ; b1 ^ b2 ; ai+1 ; : : : ; an ) = = g(a1 ; : : : ; ai?1 ; b1; ai+1 ; : : : ; an ) ^ g(a1 ; : : : ; ai?1 ; b2 ; ai+1 ; : : : ; an ): A join-hemiantimorphism on A is a function f 0 : An ! A such that for every i; 1  i  n, (Ja1) f 0 (a1 ; : : : ; ai?1 ; 1; ai+1 ; : : : ; an ) = 0; (Ja2) f 0 (a1 ; : : : ; ai?1 ; b1 ^ b2 ; ai+1 ; : : : ; an ) = = f 0 (a1 ; : : : ; ai?1 ; b1 ; ai+1 ; : : : ; an ) _ f 0 (a1 ; : : : ; ai?1 ; b2; ai+1 ; : : : ; an ): A meet-hemiantimorphism on A is a function g0 : An ! A such that for every i; 1  i  n, (Ma1) g0 (a1 ; : : : ; ai?1 ; 0; ai+1 ; : : : ; an ) = 1; (Ma2) g0 (a1 ; : : : ; ai?1 ; b1 _ b2 ; ai+1 ; : : : ; an ) = = g0 (a1 ; : : : ; ai?1 ; b1; ai+1 ; : : : ; an ) ^ g0 (a1 ; : : : ; ai?1 ; b2 ; ai+1 ; : : : ; an ):

2 The term \hemimorphism" was introduced by Halmos in [Hal55]; the concept was used for the representation of the necessity operator of modal logic.

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De nition 2 Let (X; ) be a partially ordered set and let R  X n+1 be a (n + 1)-ary relation on X . (1) R is an increasing relation if it has the property that for all x 2 X n and every y; z 2 X , if R(x; y) and y  z then R(x; z ). (2) R is a decreasing relation if it has the property that for all x 2 X n and every y; z 2 X , if R(x; y) and z  y then R(x; z ). De nition 3 Let R  X n+1 be a (n +1)-ary relation, and let y 2 X . The inverse image of y with respect to R is the set R?1 (y) = f(x1 ; : : : ; xn ) j R(x1 ; : : : ; xn ; y)g.

We will start by de ning the category of distributive lattices with operators which will be studied in what follows, as well as a corresponding category of Priestley spaces with additional functions and relations. Then, based on previous papers in which the link between the algebraic and the relational semantics for modal logics is analyzed (as [JT51, JT52, Lem66a, Lem66b], as well as the general study in [Gol89]) we will indicate a canonical way in which one can associate functions or relations in the dual space and vice-versa, and will show that a dual equivalence between these categories can be thus de ned.

2.1 The categories DLO and RPS

Let  be a signature containing function symbols in the classes mentioned above. In order to distinguish these classes, we will write  = Lh [ La [ Jh [ Mh [ Ja [ Ma, where Lh; La; Jh; Mh; Ja; and Ma can possibly be empty. As a convention, in what follows (if not explicitly speci ed otherwise) the name h will be used for an operation symbol in Lh, k for an operation symbol in La, f for one in Jh, g for one in Mh, f 0 for one in Ja, and g0 for one in Ma. Similarly, on the corresponding Priestley spaces, we will denote by H (resp. K ) the map corresponding to operation symbols in Lh (resp. La), by R (resp. R0 ) the relation corresponding to a symbol in Jh (resp. Ja), and by Q (resp. Q0 ) the relation corresponding to a symbol in Mh (resp. Ma). Let DLO be the category of bounded distributive lattices with operators in  having:

Objects:

(A; _; ^; 0; 1; fhA gh2Lh; fkA gk2La ; ffAgf 2Jh; fgAgg2Mh ; ffA0 gf 02Ja ; fgA0 gg0 2Ma ) where (A; _; ^; 0; 1) is a bounded distributive lattice, and for every h 2 Lh, k 2 La, f 2 Jh, g 2 Mh, f 0 2 Ja, g0 2 Ma: hA is a lattice homomorphism, kA is a lattice antimorphism, fA is a join-hemimorphism, gA is a meet-hemimorphism, fA0 is a join-hemiantimorphism, and gA0 is a meet-hemiantimorphism.

Morphisms: Lattice morphisms that preserve 0; 1 and the operators in Lh; La; Jh; Mh; Ja; and Ma. Let RPS be the category of relational Priestley spaces, having:

Objects:

Relational Priestley spaces, i.e. ordered topological spaces endowed with relations, (X; ; ; fHX gH 2Lh; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma ); where: (1) (X;   ) is a Priestley space, (2) for every H 2 Lh, HX : X ! X is a continuous order-preserving map, (3) for every K 2 La, KX : X ! X is a continuous order-reversing map, (4) for every R 2 Jh with arity n, RX  X n+1 is an increasing relation such that: (4a) 8y 2 D(A), RX?1 (y) is closed in the product on D(A)n of the upper topology, (4b) 8Y1; : : : ; Yn  X clopen, fy j 9x1 2 Y1 ; : : : ; xn 2 Yn : RX (x1 ; : : : ; xn ; y)g is clopen. (5) for every Q 2 Mh with arity n, QX  X n+1 is a decreasing relation such that: (5a) 8y 2 D(A), Q?X1 (y) is closed in the product on D(A)n of the lower topology, (5b) 8Y1; : : : ; Yn  X clopen, fy j 8x1 ; : : : ; xn ; QX (x1 ; : : : ; xn ) ) 9i : xi 2 Yi g is clopen. (6) for every R0 2 Ja with arity n, RX0  X n+1 is an increasing relation such that: (6a) 8y 2 D(A), RX0?1 (y) is closed in the product on D(A)n of the lower topology, (6b) 8Y1; : : : ; Yn  X clopen, fy j 9x1 62 Y1 ; : : : ; xn 62 Yn : RX0 (x1 ; : : : ; xn ; y)g is clopen. (7) for every Q0 2 Ma with arity n, Q0X  X n+1 is a decreasing relation such that: 1 n (7a) 8y 2 D(A), Q0? X (y ) is closed in the product0 on D(A) of the upper topology, (7b) 8Y1; : : : ; Yn  X clopen, fy j 8x1 ; : : : ; xn ; QX (x1 ; : : : ; xn ) ) 9i : xi 62 Yi g is clopen.

Morphisms continuous bounded morphisms, see the following de nition. 5

De nition 4 (Bounded morphism) Let X1 and X2 be two relational Priestley spaces, X1 = (X1; ; 1; fHX1 gH2Lh; fKX1 gK2La; fRX1 gR2Jh; fQX1 gQ2Mh; fRX0 1 gR02Ja; fQ0X1 gQ0 2Ma); X2 = (X2; ; 2; fHX2 gH2Lh; fKX2 gK2La; fRX2 gR2Jh; fQX2 gQ2Mh; fRX0 2 gR02Ja; fQ0X2 gQ0 2Ma): (1) A map  : X1 ! X2 is a morphism if it preserves the order , the operations in Lh; La and the relations in Jh; Mh; Ja; and Ma in passing from X1 to X2 , i.e. it satis es the following conditions: (M1) x  y implies (x)  (y), (M2) for every H 2 Lh [ La, x = HX1 (y) implies (x) = HX2 ((y)) (M3) for every R 2 Jh [ Mh [ Ja [ Ma, RX1 (x1 ; : : : ; xn ; x) implies RX2 ((x1 ); : : : ; (xn ); (x)). (2) A morphism  : X1 ! X2 is bounded if for all z 2 X1 it satis es: (B1) RX2 (y1 ; : : : ; yn; (z )) implies 9x1 ; : : : ; xn 2 X1 (RX1 (x1 ; : : : ; xn ; z ) and yi  (xi ), for every 1  i  n), for every y1 ; : : : ; yn 2 X2 , and every R 2 Jh [ Ma, (B2) RX2 (y1 ; : : : ; yn; (z )) implies 9x1 ; : : : ; xn 2 X1 (RX1 (x1 ; : : : ; xn ; z ) and (xi )  yi , for every 1  i  n), for every y1 ; : : : ; yn 2 X2 , and every Q 2 Mh [ Ja.

It is easy to see that the composition of bounded morphisms is again a bounded morphism. Note that the operations in Lh and La can be seen as relations in the usual way, i.e. as follows:

H = f(x1 ; x2 ) j x1 = HX (x2 )g; K = f(x1 ; x2 ) j x1 = KX (x2 )g: Conditions similar to (B 1) and (B 2) can be stated also for these relations, namely: (BLh) HX2 (y; (x2 )) implies 9x1 2 X1 : HX1 (x1 ; x2 ) and y = (x1 ), (BLa) KX2 (y; (x2 )) implies 9x1 2 X1 : KX1 (x1 ; x2 ) and y = (x1 ). However, it is not necessary to explicitly specify conditions similar to (B 1) and (B 2) for the operations in Lh and La because they are already satis ed, as shown by Lemma 1. Lemma 1 Let  : X1 ! X2 be a morphism of relational Priestley spaces. Let H 2 Lh [ La. For every y 2 X2 , if HX2 ((x2 )) = y, then there exists a x1 2 X1 such that HX1 (x2 ) = x1 and y = (x1 ). Proof : It suces to take x1 = HX1 (x2 ). Then (x1 ) = y follows from the fact that  is a morphism.2 In [Gol89] it is proved that there is a dual equivalence between the category DLO of distributive lattices with operators (join- and meet-hemimorphisms) and a corresponding category RPS of relational Priestley spaces. We now show that this correspondence can be extended to a dual equivalence between the category DLO of distributive lattices with operators in  (i.e. including lattice morphisms and antimorphisms, and join- and meet-hemiantimorphisms) and the corresponding category RPS of relational Priestley spaces. For this, we de ne two functors, D : DLO ! RPS and E : RPS ! DLO .

2.2 The functor D

In this section we de ne a contravariant functor D : DLO ! RPS . We present the de nition of D on objects and on morphisms, as well as the main properties of D which are needed later.

2.2.1 De nition of D on objects

For every distributive lattice with operators A in DLO, let D(A) be the space (Fp (A); ; ; fD(hA )gh2Lh; fD(kA )gk2La ; fD(fA )gf 2Jh; fD(gA )gg2Mh ; fD(fA0 )gf 0 2Ja ; fD(gA0 )gg0 2Ma ); where (Fp (A); ;  ) is the Priestley space associated with A (consisting of the set of all prime lters of A, ordered by inclusion, endowed with the Priestley topology), and the additional mappings and relations are de ned as follows. If h 2 Lh (i.e. hA : A ! A is a lattice homomorphism), D(hA ) : D(A) ! D(A) is de ned by D(hA )(F ) = hA?1 (F ) = fa 2 A j hA (a) 2 F g: 6

If k 2 La (i.e. kA : A ! A is a lattice antimorphism), D(kA ) : D(A) ! D(A) is de ned by

D(kA )(F ) = AnkA?1 (F ) = fa 2 A j kA (a) 62 F g: If f 2 Jh has arity n (i.e. fA : An ! A is a join-hemimorphism), D(fA )  D(A)n+1 is de ned by D(fA )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1; : : : ; an (ai 2 Fi ; 8i 2 f1; : : : ; ng implies fA (a1 ; : : : ; an ) 2 Fn+1 ): If g 2 Mh has arity n (i.e. gA : An ! A is a meet-hemimorphism), D(gA )  D(A)n+1 is de ned by D(gA )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1; : : : ; an (gA (a1 ; : : : ; an ) 2 Fn+1 implies 9i 2 f1; : : :; ng with ai 2 Fi ): If f 0 2 Ja has arity n (i.e. fA0 : An ! A is a join-hemiantimorphism), D(fA0 )  D(A)n+1 is de ned by D(fA0 )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1; : : : ; an (ai 62 Fi ; 8i 2 f1; : : : ; ng implies fA0 (a1 ; : : : ; an ) 2 Fn+1 ): If g0 2 Ma has arity n (i.e. gA0 : An ! A is a meet-hemiantimorphism), D(gA0 )  D(A)n+1 is de ned by D(gA0 )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1; : : : ; an (gA0 (a1 ; : : : ; an ) 2 Fn+1 implies 9i 2 f1; : : :; ng with ai 62 Fi ): Lemma 2 Let A be a distributive lattice with operators in the classes Lh, La, Jh, Mh, Ja, Ma. Let (D(A); ;  ) be the Priestley dual of A. The following holds: (1) If h 2 Lh then D(hA ) : D(A) ! D(A) is order-preserving, and continuous with respect to  . (2) If k 2 La then D(kA ) : D(A) ! D(A) is order-reversing, and continuous with respect to  . (3) If f 2 Jh then D(fA )  D(A)n+1 is an increasing relation such that for every F 2 D(A), D(fA )?1 (F ) is closed in the product topology on D(A)n of the upper topology, and moreover, for every U1; : : : ; Un 2 ClopenOF(D(A)), the set fF j 9F1 2 U1 ; : : : ; Fn 2 Un : D(fA )(F1 ; : : : ; Fn ; F )g

is clopen. (4) If g 2 Mh then D(gA )  D(A)n+1 is a decreasing relation such that for every F 2 D(A), D(gA )?1 (F ) is closed in the product topology on D(A)n of the lower topology, and moreover, for every U1 ; : : : ; Un 2 ClopenOF(D(A)), the set fF j 8F1 ; : : : ; Fn ; D(gA )(F1 ; : : : ; Fn ; F ) ) 9i : Fi 2 Ui g is clopen. (5) If f 0 2 Ja then D(fA0 )  D(A)n+1 is an increasing relation such that for every F 2 D(A), D(fA0 )?1 (F ) is closed in the product topology on D(A)n of the lower topology, and moreover, for every U1 ; : : : ; Un 2 ClopenOF(D(A)), the set fF j 9F1 62 U1 : : : 9Fn 62 Un : D(fA0 )(F1 ; : : : ; Fn ; F )g is clopen. (6) If g0 2 Ma then D(gA0 )  D(A)n+1 is a decreasing relation such that for every F 2 D(A), D(gA0 )?1 (F ) is closed in the product topology on D(A)n of the upper topology, and moreover, for every U1 ; : : : ; Un 2 ClopenOF(D(A)), the set fF j 8F1 ; : : : ; Fn ; D(gA0 )(F1 ; : : : ; Fn ; F ) ) 9i : Fi 62 Ui g is clopen.

Proof : (1) has been proved by Priestley [Pri70], see also [Pri72], p.515 and [DP90], p.207 . (2) Let kA : A ! A be a lattice antimorphism. Let F1 ; F2 2 D(A) be such that F1  F2 . Then for every a 2 D(kA )(F2 ) = AnkA?1 (F2 ), we have kA (a) 62 F2 , hence, since F1  F2 , kA (a) 62 F1 . Hence, a 2 D(kA )(F1 ) = AnkA?1 (F1 ). Thus, D(kA )(F2 )  D(kA )(F1 ). In order to show that D(kA ) is continuous it is sucient to show that D(kA )?1 (Xa ) and D(kA )?1 (D(A)nXa ) are open. This holds, because D(kA )?1 (Xa ) = fF j D(kA )(F ) 2 Xa g = fF j a 62 kA?1 (F )g = fF j kA (a) 62 F g = D(A)nXkA (a) and D(kA )?1 (D(A)nXa ) = fF j D(kA )(F ) 62 Xa g = fF j a 2 kA?1 (F )g = fF j kA (a) 62 F g = XkA (a) .

The properties (3) and (4) are analyzed in [Gol89]. For details concerning these proofs we refer to Theorem 2.2.1 and Theorem 2.2.2 in [Gol89], pp.187-190. (5) Let f 2 Ja. By the de nition of D(fA ) we know that for every F1 ; : : : ; Fn ; Fn+1 2 D(A)n+1 , D(fA )(F1 ; : : : ; Fn ; Fn+1 ) if and only if 8a1 ; : : : ; an ; ((a1 62 F1 ^ : : : ^ an 62 Fn ) ) fA(a1 ; : : : ; an ) 2 Fn+1 ). 7

It is easy to see that D(fA ) is increasing. Let F  G and assume that D(fA )(F1 ; : : : ; Fn ; F ). Let a1 ; : : : ; an be such that a1 62 F1 ; : : : ; an 62 Fn . From the fact that D(fA )(F1 ; : : : ; Fn ; F ) it follows that fA(a1 ; : : : ; an ) 2 F  G. Thus we proved that D(fA )(F1 ; : : : ; Fn ; G). We now show that for every F 2 D(A), D(fA )?1 (F ) is closed in the product topology on D(A)n of the lower topology. Let (G1 ; : : : ; Gn ) 2 D(A)n be such that (G1 ; : : : ; Gn ) 62 D(fA )?1 (F ). Then there exist a1 ; : : : ; an such that a1 62 G1 ; : : : ; an 62 Gn and f (a1 ; : : : ; an ) 62 F . Hence, there exist a1 ; : : : ; an such that G1 62 Xai ; : : : ; Gn 62 Xan and f (a1 ; : : : ; an ) 62 F . Let N = D(A)nXa1  : : :  D(A)nXan . For every (F1 ; : : : ; Fn ) 2 N we have ai 62 Fi for every 1  i  n. We know that f (a1 ; : : : ; an ) 62 F . It follows therefore that (F1 ; : : : ; Fn ) 62 D(fA )?1 (F ). This shows that there exists an open neighborhood of (G1 ; : : : ; Gn ) (in the product topology on D(A)n of the lower topology) which is contained in the complementary of D(fA )?1 (F ). Thus, D(fA )?1 (F ) is closed in the product topology on D(A)n of the lower topology. In order to show that for every U1 ; : : : ; Un 2 ClopenOF(D(A)) the set fF j 9F1 62 U1 ; : : : ; Fn 62 Un : D(fA )(F1 ; : : : ; Fn ; F )g is clopen, taking into account the form of the clopen order- lters of D(A), it suces to show that the necessary and sucient condition for F 2 XfA (a1 ;:::;an) is (9G1 ; : : : ; Gn 2 D(A) such that a1 62 G1 ; : : : ; an 62 Gn and D(fA )(G1 ; : : : ; Gn ; F )). The suciency of the condition follows easily: assume that there exist G1 ; : : : ; Gn 2 D(A) such that (a1 62 G1 ; : : : ; an 62 Gn and D(fA )(G1 ; : : : ; Gn ; F )): Since D(fA )(G1 ; : : : ; Gn ; F ), it follows that for all b1; : : : ; bn , if b1 62 G1 ; : : : ; bn 62 Gn then fA (b1 ; : : : ; bn) 2 F . In particular taking b1 = a1 ; : : : ; bn = an it follows that fA (a1 ; : : : ; an ) 2 F , i.e. F 2 XfA (a1 ;:::;an ) . In order to prove the necessity, we use an argument similar to the one used by Goldblatt in Theorem 2.2.1 in [Gol89]. We construct inductively a sequence of prime lters G1 ; : : : ; Gn such that for every i  n the following hold: (i) ai 62 Gi , (ii) If yk 62 Gk for all k  i, then fA (y1 ; : : : ; yi ; ai+1 ; : : : ; an ) 2 F . Let j  n be xed and assume that for every i < j , Gi has been de ned such that the conditions (i) and (ii) are satis ed. Then Gj will be de ned such that (i) and (ii) are satis ed for j . Let Hj = fz j 9y1 62 G1 ; : : : ; 9yj?1 62 Gj?1 : fA(y1 ; : : : ; yj?1 ; z; aj+1 ; : : : ; an ) 62 F g. Claim 1. Hj is closed under all nite meets. Proof : Assume that z1 ; z2 2 Hj . Then for all i < j there exist yi1 ; yi2 2 Gi such that fA(y11 ; : : : ; yj1?1 ; z1 ; aj+1 ; : : : ; an ) 62 F and fA(y12 ; : : : ; yj2?1 ; z2; aj+1 ; : : : ; an ) 62 F . Let yi = yi1 _ yi2 for i = 1; : : : ; j ? 1. Since fA is order-reversing in every argument, it follows that fA (y0 ; : : : ; yj?1 ; zk ; aj+1 ; : : : ; an ) 62 F for k = 1; 2. Therefore, A ( 0 j ?1 1 ^ 2 j +1 n) = A( 0 j ?1 1 j +1 n) _ A( 0 j ?1 2 j +1 n ) 62 F and thus, z1 ^ z2 2 Hj . V Note also that 1 2 Hj . It follows that I 2 Hj for every nite (possible empty) subset I  Hj . V Claim 2. Hj is separated from aj , i.e. for every nite subset I  Hj , I 6Vaj . Proof : Assume that thereVexists a nite subset I  Hj , such that I  aj . Since Hj is closed under all nite meets, z = I 2 Hj , and, therefore, there exist y1 62 G1 ; : : : ; yj?1 62 Gj?1 such that fA (y1 ; : : : ; yj?1 ; z; aj+1 ; : : : ; an ) 62 F . V On the other hand, I  aj , it follows that V since fA is order-reversing in every argument and fA (y1 ; : : : ; yj?1 ; I; aj+1 ; : : : ; an )  fA (y1 ; : : : ; yj?1 ; aj ; aj+1 ; : : : ; an ), and since V by the induction hypothesis fA (y1 ; : : : ; yj?1 ; aj ; aj+1 ; : : : ; an ) 2 F it follows that fA (y1 ; : : : ; yj?1 ; I; aj+1 ; : : : ; an ) 2 F , which is a contradiction. f

f

y ;:::;y

;z ;a

y ;:::;y

;z

z ;a

;:::;a

f

y ;:::;y

;z ;a

;:::;a

;:::;a

Since Hj is separated from aj , there exists a prime lter Gj such that Hj  Gj and aj 62 Gj . Thus, (i) is ful lled by Gj . In order to prove that (ii) also holds, let y1 ; : : : ; yj such that yk 62 Gk for every k  j . Thus, yj 62 Gj and therefore yj 62 Hj . It follows that for all y1 62 G1 ; : : : ; yj?1 62 Gj?1 , fA (y1 ; : : : ; yj?1 ; yj ; aj+1 ; : : : ; an ) 2 F , hence, in particular, taking y1 = y1 ; : : : ; yj?1 = yj?1 , fA(y1 ; : : : ; yj ; aj+1 ; : : : ; an ) 2 F . (6) The proof is analogous to that of (5), taking into account the de nition of D(gA ) for g 2 Ma. 2 Lemma 2 shows that D maps objects of DLO to objects of RPS . 8

2.2.2 De nition of D on morphisms For every morphism h : A1 ! A2 in DLO , let D(h) : D(A2 ) ! D(A1 ) be de ned for every F 2 D(A2 ) by D(h)(F ) = h?1 (F ):

Lemma 3 Let h : A1 ! A2 be a morphism of DLO. Then (1) D(h) : D(A2 ) ! D(A1 ) de ned by D(h)(F ) = h?1 (F ) for every F 2 D(A2 ) is continuous and preserves the order , the operations Lh; La and the relations in Jh; Mh; Ja and Ma in passing from D(A2 ) to D(A1 ). Moreover, it satis es the following conditions: (B1) for every f 2 Jh [ Ma and every y1 ; : : : ; yn 2 D(A1 ), if D(fA1 )(y1 ; : : : ; yn ; D(h)(z )) then 9x1 ; : : : ; xn 2 D(A2 ) such that D(fA2 )(x1 ; : : : ; xn ; z ) and yi  D(h)(xi ), for every 1  i  n, (B2) for every f 2 Mh [ Ja and every y1 ; : : : ; yn 2 D(A1 ), if D(fA1 )(y1 ; : : : ; yn ; D(h)(z )) then 9x1 ; : : : ; xn 2 D(A2 ) such that D(fA2 )(x1 ; : : : ; xn ; z ) and D(h)(xi )  yi , for every 1  i  n. (2) If h is injective, then D(h) is surjective. (3) If h is surjective, then D(h) is an order-embedding.

Proof : In order to prove (1) we only have to show that the conditions (B1) and (B2) are satis ed for f 2 Ja resp. f 2 Ma (for the rest we refer to [Gol89], Theorem 2.3.2). Let f 2 Ja be a join-hemiantimorphism. Since the proof is similar to the one for join-hemimorphisms given in [Gol89] up to replacing some of the prime lters (resp. order lters) with their complements, we just present a sketch of the proof. Note rst that if f has arity 0 then we are exactly in the situation of 0-ary meet-hemimorphisms, considered in [Gol89]. Assume that D(fA1 )(G1 ; : : : ; Gn ; D(h)(F )), where G1 ; : : : ; Gn 2 D(A1 ) and F 2 D(A2 ). We will show that there exist F1 ; : : : ; Fn 2 D(A2 ) such that D(fA2 )(F1 ; : : : ; Fn ; F ) and for every i = 1; : : : ; n, D(h)(Fi )  Gi . The construction is carried on inductively such that for every i  n,

(i) Fi \ h(A1 nGi ) = ;, (ii) if yk 62 Fk for k  i, and xp 62 Gp for i < p  n, then fA2 (y1 ; : : : ; yi ; h(xi+1 ); : : : ; h(xn )) 2 F: Let j  n and suppose that F1 ; : : : ; Fj?1 have been inductively de ned to satisfy (i) and (ii). De ne Hj  A2 as follows:

Hj = fz j



9y1 62 F1 ; : : : ; yj?1 62 Fj?1 ; 9xj+1 62 Gj+1 ; : : : ; xn 62 Gn ; s.t. fA2 (y1 ; : : : ; yj?1 ; z; h(xj+1 ); : : : ; h(xn )) 62 F g:

Claim 1. Hj is closed under nite meets. Proof : Let z1; z2 2 Hj . For zi (i 2 f1; 2g); there exist y1i 62 F1 ; : : : ; yji ?1 62 Fj?1 , and there exist xij+1 62 Gj+1 ; : : : ; xin 62 Gn , such that fA2 (y1i ; : : : ; yji ?1 ; zi ; h(xij+1 ); : : : ; h(xin )) 62 F . Let yk = yk1 _ yk2 for every k 2 f1; : : :; j ? 1g and xp = x1p _ x2p for every p 2 fj + 1; : : : ; ng. Then it is easy to see that fA2 (y1 ; : : : ; yj?1 ; zi ; h(xj+1 ); : : : ; h(xn )) 62 F for i = 1; 2, hence fA2 (y1 ; : : : ; yj?1 ; z1 ^ z2 ; h(xj+1 ); : : : ; h(xn )) 62 F . This shows that for z1 ; z2 there exist y1 62 F1 ; : : : ; yj?1 62 Fj?1 ; xj+1 62 Gj+1 ; : : : ; xn 62 Gn such that fA2 (y1 ; : : : ; yj?1 ; z1 ^ z2 ; h(xj+1 ); : : : ; h(xn )) 62 F , i.e. that z1 ^ z2 2 Hj . It is also easy to see that 1 2 Hj . It follows therefore that Hj is closed under all nite (possibly empty) meets. Claim 2. Hj is separated from V h(A W1 nGj ). V Proof : Assume that I  J for some nite I  Hj and J  h(A1 nGj ). Then I 2 Hj , hence there exist y1 62 F1 ; : : : ; yj?1 62 Fj?1 ; xj+1 62 Gj+1 ; : : : ; xn 62 Gn such that ^

fA2 (y1 ; : : : ; yj?1 ; I; h(xj+1 ); : : : ; h(xn )) 62 F: On the other hand, since f 2 Ja, it is order reversing, hence ^

_

fA2 (y1 ; : : : ; yj?1 ; I; h(xj+1 ); : : : ; h(xn ))  fA2 (y1 ; : : : ; yj?1 ; J; h(xj+1 ); : : : ; h(xn )); 9

W

and thus fA2 (y1 ; : : : ; yj?1 ; J; h(xj+1 ); : : : ; h(xn )) 62 F . W Note now that if J  h(A1 nGj ) is a nite subset, then J = h(b) for some b 2 A1 nGj . Thus it follows that fA2 (y1 ; : : : ; yj?1 ; h(b); h(xj+1 ); : : : ; h(xn )) 62 F . If j > 1 this contradicts the induction hypothesis (ii) for j ? 1. If j = 1 then we have h(fA1 (b; x2 ; : : : ; xn )) = fA2 (h(b); h(x2 ) : : : ; h(xn )) 62 F , hence fA1 (b; x2 ; : : : ; xn ) 62 h?1 (F ) = D(h)(F ). Since b 62 G1 and xi 62 Gi for every i  2, this contradicts D(fA1 )(G1 ; : : : ; Gn ; D(h)(F )). Hence, by the prime lter theorem, there exists a prime lter Fj 2 D(A2 ) such that Hj  Fj and Fj \ h(A1 nGj ) = ;. Thus, (i) is ful lled by Fj ; from the fact that Hj  Fj it follows that if z 62 Fj then z 62 Hj , hence for every yi 62 Fi , i < j , every yj 62 Fj , and every xp 62 Gp , j < p  n, fA2 (y1 ; : : : ; yj?1 ; yj ; h(xj+1 ); : : : ; h(xn )) 2 F , i.e. (ii) holds too. The case f 2 Ma can be solved similarly combining arguments from the cases Mh and Ja. For (2) we refer to [Gol89]. (3) Assume that h : A1 ! A2 is onto. Then D(h) is obviously order-preserving. We show that it is an order-embedding. Let F1 ; F2 2 D(A2 ). Assume that D(h)(F1 )  D(h)(F2 ). We want to show that F1  F2 . Let b 2 F1 be arbitrary. From the surjectivity of h, b = h(a) for some a 2 A1 . Thus, b = h(a) for some a 2 h?1 (F1 ) = D(h)(F1 )  D(h)(F2 ). Hence, b = h(a) 2 F2 . This proves that F1  F2 . 2 Lemma 3 shows that D maps morphisms of DLO to morphisms of RPS.

2.3 The functor E

In this section we de ne a contravariant functor E : RPS ! DLO . We present separately the de nition of E on objects and on morphisms, as well as its main properties which are needed later.

2.3.1 De nition of E on objects For every relational Priestley space

(X; ; ; fHX gH 2Lh; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma ) let E (X ) be the bounded lattice of clopen order- lters of X (where 0 = ;; 1 = X , the join is de ned by taking the union, and the meet by taking the intersection), endowed with additional operators de ned as shown in what follows. For every H 2 Lh (i.e. such that HX : X ! X is continuous and order-preserving), E (HX ) : E (X ) ! E (X ) is de ned by E (HX )(U ) = HX?1 (U ) = fx 2 X j HX (x) 2 U g: For every K 2 La (i.e. such that KX : X ! X is continuous and order-reversing), E (KX ) : E (X ) ! E (X ) is de ned by E (KX )(U ) = X nKX?1(U ) = fx 2 X j KX (x) 62 U g: If R 2 Jh has arity n (i.e. RX  X n+1 is an (n + 1)-ary increasing relation satisfying conditions (4a) and (4b) in the de nition of relational Priestley spaces), then E (RX ) : E (X )n ! E (X ) is de ned by

E (RX )(U1 ; : : : ; Un) = fx 2 X j 9x1 2 U1 ; : : : ; xn 2 Un : RX (x1 ; : : : ; xn ; x)g: If Q 2 Mh has arity n (i.e. QX  X n+1 is a (n + 1)-ary decreasing relation satisfying conditions (5a) and (5b)), then E (QX ) : E (X )n ! E (X ) is de ned by

E (QX )(U1 ; : : : ; Un) = fx 2 X j 8x1 ; : : : ; xn (QX (x1 ; : : : ; xn ; x) ) 9i; xi 2 Ui )g: If R0 2 Ja has arity n (i.e. RX0  X n+1 is an (n + 1)-ary increasing relation satisfying conditions (6a) and (6b)), then E (RX0 ) : E (X )n ! E (X ) is de ned by

E (RX0 )(U1 ; : : : ; Un) = fx 2 X j 9x1 62 U1 ; : : : ; xn 62 Un : RX0 (x1 ; : : : ; xn ; x)g: 10

If Q0 2 Ma has arity n (i.e. Q0X  X n+1 is a (n + 1)-ary decreasing relation satisfying conditions (7a) and (7b)), then E (Q0X ) : E (X )n ! E (X ) is de ned by E (Q0X )(U1 ; : : : ; Un) = fx 2 X j 8x1 ; : : : ; xn (Q0X (x1 ; : : : ; xn ; x) ) 9i; xi 62 Ui )g: From the de nition of the objects of RPS it follows immediately that the maps above are well-de ned.

2.3.2 De nition of E on morphisms

For every continuous bounded morphism  : X1 ! X2 let E () : E (X2 ) ! E (X1 ) be de ned for every U 2 E (X2 ) by E ()(U ) = ?1 (U ): Lemma 4 Let  : X1 ! X2 be a bounded morphism. The following hold. (1) For every clopen order- lter U of X2 , ?1 (U ) is a clopen order- lter of X1 . (2) E () : E (X2 ) ! E (X1 ) is a homomorphism in DLO . Proof : (1) follows from the fact that  is continuous and preserves . (2) follows easily from the fact that  is a bounded morphism. 2

2.4 Duality

We now show that the functors D and E de ne a dual equivalence between the categories DLO and RPS. Lemma 5 For every A 2 jDLOj, there exists an isomorphism A : A ! E (D(A)). Proof : Let A : A ! E (D(A)) be de ned by A (a) = Xa . By the Priestley duality for distributive lattices we know that A is an isomorphism of bounded distributive lattices, and using the results from Lemma 2 it is not dicult to see that it preserves the operations, namely for every h 2 Lh; k 2 La; f 2 Jh; g 2 Mh; f 0 2 Ja; g0 2 Ma:

A (hA (a)) = XhA(a) = D(hA )?1 (Xa ) = E (D(hA ))(A (a)); A (kA (a)) = XkA (a) = X nD(kA)?1 (Xa ) = E (D(kA ))(A (a)); A (fA (a1 ; : : : ; an )) = XfA (a1 ;:::;an) = fF j 9F1 2 Xa1 ; : : : ; Fn 2 Xan : D(fA )(F1 ; : : : ; Fn ; F )g = = E (D(fA ))(Xa1 ; : : : ; Xan ) = E (D(fA ))(A (a1 ); : : : ; A (an )); A (gA (a1 ; : : : ; an )) = XgA (a1 ;:::;an) = fF j 8F1; : : : ; Fn ; D(gA )(F1 ; : : : ; Fn ) ) 9i : Fi 2 Xai g = = E (D(gA ))(Xa1 ; : : : ; Xan ) = E (D(gA ))(A (a1 ); : : : ; A (an )); A (fA0 (a1 ; : : : ; an )) = XfA0 (a1 ;:::;an) = fF j 9F1 62 Xa1 ; : : : ; Fn 62 Xan : D(fA0 )(F1 ; : : : ; Fn ; F )g = = E (D(fA0 ))(Xa1 ; : : : ; Xan ) = E (D(fA0 ))(A (a1 ); : : : ; A (an )); 0 A (gA (a1 ; : : : ; an )) = XgA0 (a1 ;:::;an) = fF j 8F1; : : : ; Fn ; D(gA0 )(F1 ; : : : ; Fn ) ) 9i : Fi 62 Xai g = 2 = E (D(gA0 ))(Xa1 ; : : : ; Xan ) = E (D(gA0 ))(A (a1 ); : : : ; A (an )): Lemma 6 Let X 2 jRPSj. Then there exists a bounded homeomorphism X : X ! D(E (X )). Proof : Let X : X ! D(E (X )) be de ned by X (x) = Fx = fY 2 E (X ) j x 2 Y g. From the Priestley duality, X is a isomorphism of ordered spaces and a homeomorphism (see e.g. [DP90, Gol89]). We only have to prove that X preserves the operators. Claim 1. Let H 2 Lh and x 2 X . Then FHX (x) = D(E (HX ))(Fx ). Proof : It follows from the fact that Y 2 FHX (x) if and only if HX (x) 2 Y if and only if x 2 HX?1 (Y ) if and only if x 2 E (HX )(Y ) if and only if E (HX )(Y ) 2 Fx if and only if Y 2 E (HX )?1 (Fx ) = D(E (HX ))(Fx ). Claim 2. Let K 2 La and x 2 X . Then FKX (x) = D(E (KX ))(Fx ). Proof : It follows from the fact that Y 2 FKX (x) if and only if KX (x) 2 Y if and only if x 62 KX?1 (Y ) if and only if x 62 E (KX )(Y ) if and only if E (KX )(Y ) 62 Fx if and only if Y 2 E (KX )?1 (Fx ) = D(E (KX ))(Fx ). 11

Claim 3. Let R 2 Jh with arity n and x1 ; : : : ; xn+1 2 X . Then RX (x1 ; : : : ; xn+1 ) if and only if D(E (RX ))(Fx1 ; : : : ; Fxn+1 ). Proof : see e.g. [Gol89]. Claim 4. Let Q 2 Mh with arity n and x1 ; : : : ; xn+1 2 X . Then QX (x1 ; : : : ; xn+1 ) if and only if D(E (QX ))(Fx1 ; : : : ; Fxn+1 ). Proof : see e.g. [Gol89]. Claim 5. Let R0 2 Ja with arity n and x1 ; : : : ; xn+1 2 X . Then RX0 (x1 ; : : : ; xn+1 ) if and only if D(E (RX0 ))(Fx1 ; : : : ; Fxn+1 ). Proof : First assume that RX0 (x1 ; : : : ; xn+1 ). Let Y1 ; : : : ; Yn 2 E (X ) be such that Y1 62 Fx1 ; : : : ; Yn 62 Fxn (i.e. xi 62 Yi for i = 1; : : : ; n). By using the assumption that RX0 (x1 ; : : : ; xn+1 ), it follows that xn+1 2 E (RX0 )(Y1 ; : : : ; Yn ), i.e. E (RX0 )(Y1 ; : : : ; Yn ) 2 Fxn+1 . Thus, D(E (RX0 ))(Fx1 ; : : : ; Fxn+1 ). For the converse, suppose that not RX0 (x1 ; : : : ; xn ; xn+1 ). Then (x1 ; : : : ; xn ) 62 RX0?1 (xn+1 ). Since RX0?1 (xn+1 ) is closed in the product of the lower topology, there exists a basic open neighborhood N of (x1 ; : : : ; xn ) in this topology, which is disjoint from RX0?1 (xn+1 ). By the de nition of the lower topology, N = X nY1  : : :  X nYn for some clopen sets Y1 ; : : : ; Yn . Since X nY1  : : :  X nYn \ RX0?1 (xn+1 ) = ; it follows that 8z1 62 Y1 ; : : : ; zn 62 Yn ; not RX0 (z1 ; : : : ; zn; xn+1 ), hence xn+1 62 E (RX0 )(Y1 ; : : : ; Yn ), i.e. E (RX0 )(Y1 ; : : : ; Yn ) 62 Fxn+1 . Since (x1 ; : : : ; xn ) 2 N = X nY1  : : :  X nYn, we have on the one hand Yi 62 Fxi for every i = 1; : : : ; n, and on the other hand, by the remarks above, E (RX0 )(Y1 ; : : : ; Yn ) 62 Fxn+1 . Thus there exist Y1 ; : : : ; Yn with Yi 62 Fxi and E (RX0 )(Y1 ; : : : ; Yn ) 62 Fxn+1 . This shows that not D(E (RX0 ))(Fx1 ; : : : ; Fxn ; Fxn+1 ). Claim 6. Let Q0 2 Ma with arity n and x1 ; : : : ; xn+1 2 X . Then Q0X (x1 ; : : : ; xn+1 ) if and only if D(E (QX ))(Fx1 ; : : : ; Fxn+1 ). Proof : The proof combines ideas from the proofs of Claim 4 and Claim 5.

2

Theorem 7 The composites D  E and E  D are naturally isomorphic to the identity functors on DLO and RPS respectively, and so the two categories are dually equivalent.

Proof : It is sucient to show that the isomorphisms A : A ! E (D(A)) and X : X ! D(E (X )) are the components of two natural transformations. For details we refer e.g. to [Gol89]. 2

3 A category of ordered relational structures In what follows, inspired by the de nition of relational Priestley spaces, we introduce the notion of ordered relational structures, and a corresponding notion of morphism of ordered relational structures.

3.1 Ordered relational structures

De nition 5 An ordered -relational structure is a partially ordered set endowed with additional functions and relations X = (X; ; fHX gH2Lh; fKX gK2La; fRX gR2Jh; fQX gQ2Mh; fRX0 gR02Ja; fQ0X gQ0 2Ma) where for every H 2 Lh, HX : X ! X is an order-preserving map, for every K 2 La, KX : X ! X is an order-reversing map, for every R 2 Jh [ Ja with arity n, RX  X n+1 is an increasing relation, and for every Q 2 Mh [ Ma with arity n, QX  X n+1 is a decreasing relation. Remark: Let X = (X; ; fHX gH2Lh; fKX gK2La; fRX gR2Jh; fQX gQ2Mh; fRX0 gR02Ja; fQ0X gQ02Ma ) be an ordered relational structure. Consider the discrete topology  on X . The ordered topological space (X; ;  ) is totally order-disconnected, and it is compact if and only if X is nite.

We now show that for every ordered -relational structure X , its set of order- lters can be endowed with a bounded distributive lattice structure, with additional operators which can be de ned in a canonical way starting from the maps and relations on X . Note rst that, for every partially ordered set (X; ), we can de ne two binary operators on O(X ): 12

_ : O(X ) ! O(X ) de ned for every U; V 2 O(X ) by U _ V = U [ V , ^ : O(X ) ! O(X ) de ned for every U; V 2 O(X ) by U ^ V = U \ V . Thus, (O(X ); _; ^; ;; X ) is a bounded distributive lattice with 0 (= ;) and 1 (= X ). The next lemma shows that O(X ) can be endowed with a DLO -structure. Lemma 8 Let X = (X; ; fHX gH2Lh; fKX gK2La; fRX gR2Jh; fQX gQ2Mh; fRX0 gR0 2Ja; fQ0X gQ02Ma ) be

an ordered -relational structure with operations and relations in the classes Lh; La; Jh; Mh; Ja; Ma. The following hold: (1) Every order preserving operation on X , H 2 Lh induces a lattice morphism hH : O(X ) ! O(X ), de ned for every order- lter U of X by hH (U ) = HX?1 (U ). (2) Every order reversing operation on X , K 2 La induces a lattice antimorphism kK : O(X ) ! O(X ), de ned for every order- lter U of X by kK (U ) = X nKX?1(U ). (3) Every increasing relation RX  X n+1 , R 2 Jh, induces a join-hemimorphism fR : O(X )n ! O(X ), de ned for every U1 ; : : : ; Un 2 O(X ) by

fR (U1 ; : : : ; Un) = fx 2 X j 9x1 2 U1; : : : ; xn 2 Un : RX (x1 ; : : : ; xn ; x)g: (4) Every decreasing relation QX  X n+1, Q 2 Mh, induces a meet hemimorphism gQ : O(X )n ! O(X ), de ned for every U1 ; : : : ; Un 2 O(X ) by

gQ (U1 ; : : : ; Un) = fx 2 X j 8x1 ; : : : ; xn (QX (x1 ; : : : ; xn ; x) ) 9i; xi 2 Ui )g: (5) Every increasing relation RX0  X n+1 , R0 2 Ja, induces a join-hemiantimorphism fR0 0 : O(X )n ! O(X ), de ned for every U1 ; : : : ; Un 2 O(X ) by

fR0 0 (U1 ; : : : ; Un ) = fx 2 X j 9x1 62 U1 ; : : : ; xn 62 Un : RX0 (x1 ; : : : ; xn ; x)g: (6) Every decreasing relation Q0  X n+1 , Q0 2 Ma, induces a meet-hemiantimorphism gQ0 0 : O(X )n ! O(X ), de ned for every U1 ; : : : ; Un 2 O(X ) by

gQ0 0 (U1 ; : : : ; Un ) = fx 2 X j 8x1; : : : ; xn (Q0X (x1 ; : : : ; xn ; x) ) 9i; xi 62 Ui )g: Proof : (1) Let HX : X ! X be order-preserving, and let hH be de ned for every U 2 O(X ) by hH (U ) = HX?1 (U ). We show that for every U 2 O(X ), hH (U ) 2 O(X ). Let x  y. Assume that x 2 hH (U ) = HX?1 (U ). Then HX (x) 2 U . Since HX is order-preserving and U an order- lter it follows that HX (y) 2 U , i.e. that y 2 hH (U ). This proves that hH (U ) 2 O(X ). The fact that hH is a lattice homomorphism follows immediately. (2) Let KX : X ! X be order-reversing, and let kK be de ned for every U 2 O(X ) by kK (U ) = X nKX?1(U ). We show that for every U 2 O(X ), kK (U ) 2 O(X ). Let x  y. Assume that x 2 kK (U ). Then x 62 KX?1(U ), hence KX (x) 62 U . Since KX is order-reversing and U an order- lter it follows that KX (y) 62 U , i.e. that y 2 kK (U ). This proves that kK (U ) 2 O(X ). The fact that kK is a lattice antimorphism follows immediately. (3) Let RX  X n be an increasing relation. We show that for every U1 ; : : : ; Un 2 O(X ), fR (U1 ; : : : ; Un ) 2 O(X ). Let x  y. Assume that x 2 fR (U1 ; : : : ; Un ). Then for some x1 2 U1 ; : : : ; xn 2 Un , RX (x1 ; : : : ; xn ; x). Since RX is an increasing relation it follows that RX (x1 ; : : : ; xn ; y). Hence we proved that there exist x1 2 U1 ; : : : ; xn 2 Un such that RX (x1 ; : : : ; xn ; y), i.e. that y 2 fR (U1 ; : : : ; Un ). For every i 2 f1; : : : ; ng, fR (U1 ; : : : ; Ui?1 ; ;; Ui+1; : : : ; Un ) = ;. Also, fR (U1 ; : : : ; Ui?1 ; Ui [ Vi ; Ui+1 ; : : : ; Un ) = = fx 2 X j 9x1 2 U1 ; : : : ; xi 2 Ui [ Vi ; : : : ; xn 2 Un : RX (x1 ; : : : ; xn ; x)g = = fx 2 X j 9x1 2 U1 ; : : : ; xi 2 Ui ; : : : ; xn 2 Un : RX (x1 ; : : : ; xn ; x)g[ [fx 2 X j 9x1 2 U1 ; : : : ; xi 2 Vi ; : : : ; xn 2 Un : RX (x1 ; : : : ; xn ; x)g = = fR (U1 ; : : : ; Ui?1 ; Ui ; Ui+1 ; : : : ; Un ) [ fR (U1 ; : : : ; Ui?1 ; Vi ; Ui+1 ; : : : ; Un).

13

(4) Let QX  X n be a decreasing relation. We show that for every U1 ; : : : ; Un 2 O(X ), gQ (U1 ; : : : ; Un ) 2 O(X ). Let x  y. Assume that x 2 gQ (U1 ; : : : ; Un), i.e. for every x1 ; : : : ; xn , if QX (x1 ; : : : ; xn ; x), then there exists an i 2 f1; : : :; ng with xi 2 Ui . Let x1 ; : : : ; xn be such that QX (x1 ; : : : ; xn ; y). Then, by the fact that QX is decreasing, QX (x1 ; : : : ; xn ; x), hence xi 2 Ui for some i 2 f1; : : : ; ng. This shows that y 2 gQ (U1 ; : : : ; Un ). It is easy to see that for every 1  i  n, gQ (U1 ; : : : ; Ui?1 ; X; Ui+1 ; : : : ; Un ) = X . Moreover, gQ (U1 ; : : : ; Ui?1 ; Ui \ Vi ; Ui+1 ; : : : ; Un ) = = fx 2 X j 8x1 ; : : : ; xn if QX (x1 ; : : : ; xn ; x) then either xj 2 Uj for some j 6= i; or xi 2 Ui \ Vi g = = fx 2 X j 8x1 ; : : : ; xn if QX (x1 ; : : : ; xn ; x) then xj 2 Uj for some j g \ \ fx 2 X j 8x1; : : : ; xn if QX (x1 ; : : : ; xn ; x) then either xj 2 Uj for some j 6= i; or xi 2 Vi g = = gQ (U1 ; : : : ; Ui?1 ; Ui ; Ui+1 ; : : : ; Un ) \ gQ (U1 ; : : : ; Ui?1 ; Vi ; Ui+1 ; : : : ; Un ). The proofs of (5) and (6) are similar. 2 This shows that for every ordered -relational structure (X; ; fHX gH 2Lh ; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma );

(O(X ); _; ^; ;; X; fhH gH 2Lh ; fkK gK 2La; ffRgR2Jh ; fgQ gQ2Mh ; ffR0 0 gR0 2Ja ; fgQ0 0 gQ0 2Ma ) is a bounded distributive lattice with operators in  which will be denoted O(X ).

Remark 9 Note that every relational Priestley space (X; ; ; fHX gH 2Lh; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma ) can be in particular regarded as an ordered -relational space, if we ignore its topology. It is easy to see that there the inclusion ClopenOF(X )  O(X ) is an injective homomorphism of bounded distributive lattices with operators in , i.e. ClopenOF(X ) is a subalgebra of O(X ).

3.2 Morphisms of ordered -relational structures

Bounded morphisms of ordered -relational structures are bounded morphisms as de ned in De nition 4.

Lemma 10 Let  : X1 ! X2 be a bounded morphism of ordered -relational spaces. Then (1) O() : O(X2 ) ! O(X1 ) de ned by O()(U ) = ?1 (U ) for every U 2 O(X2 ) is a homomorphism in DLO . (2) If  is order-embedding, then O() is surjective. (3) If  is surjective, then O() is injective.

Proof : (1) follows immediately from the de nition of a bounded morphism. (2) We assume that  is an order-embedding and show that O() is surjective. Let U 2 O(X1 ). Let " (U ) 2 O(X2 ) be the order- lter generated by (U ). We show that O()(" (U )) = U . By de nition, O()(" (U )) = ?1 (" (U )) = fx j (x) 2 " (U )g = fx j (x)  (x1 ) for some x1 2 U g. It is easy to see that for every x 2 U; (x)  (x1 ) with x1 = x 2 U . Thus, U  O()("(U )). Conversely, let x 2 O()("(U )). Then (x)  (x1 ) for some x1 2 U . From the fact that  is an order-embedding it follows that x  x1 , hence, since U is an order- lter, x 2 U . Thus, O()("(U )) = U . (3) Assume that  is surjective. We show that O() is injective. Let U1 ; U2 2 O(X2 ) be such that O()(U1 ) = O()(U2 ). Then ?1 (U1 ) = ?1 (U2 ). Let x 2 U1 . Since  is onto, x = (y) for some y 2 X1 . Thus, x = (y) for some y 2 ?1 (U1 ) = ?1 (U2 ). It follows therefore that x = (y) 2 U2 . The converse inclusion follows similarly. Thus U1 = U2 . 2

14

4 Canonical embedding algebras In the introduction we mentioned that in the study of modal logics a link can be established between algebraic and Kripke-style models of these logics in the case when the class of their algebraic models are varieties closed under canonical embedding algebras.

De nition 6 Let V be a variety of bounded distributive lattices with operators in the classes considered above. We say that the variety V is closed under embedding algebras if it has the property (CE ) for every algebra A 2 V , O(D(A)) 2 V . In this section we present some of the cases when a variety is closed under canonical embedding algebras. We rst point out that, due to a result of Gehrke and Jonsson, every variety of distributive lattices with additive operators is closed under canonical embedding algebras. Then, we show that nitely-generated varieties of distributive lattices with more general classes of operators (which need not be order-preserving) are closed under canonical embedding algebras.

4.1 Distributive lattices with additive operators

We brie y present here the results contained in [GJ94] for distributive lattices with additive operators. For every bounded distributive lattice A with additive operators let D(A) be its Priestley dual. The canonical extension of A is de ned in [GJ94] as being the extension A of A (unique up to equivalence) such that the natural isomorphism from A to E (D(A)) can be extended to an isomorphism from A onto the lattice of all isotone maps from D(A) into the two-element ordered set f0; 1g, with 0  1. It follows therefore that A is isomorphic to O(D(A)). The following theorem is proved in [GJ94].

Theorem 11 ([GJ94]) For every distributive algebra A with additive operators, A and its canonical extension A satisfy the same identities.

The case of maps which are anti-isotone, or isotone in some of their arguments and anti-isotone in others is not considered in [GJ94]. In what follows we also take into account antimorphisms, joinhemiantimorphisms and meet-hemiantimorphisms as operators.

4.2 Distributive lattices with non-additive operators

In this section we will provide a partial answer to the question whether, given a variety V of bounded distributive lattices with operators (not necessarily additive), V is closed under canonical embedding algebras (i.e. for every A 2 V , O(D(A)) 2 V ). From Lemma 8 it follows that for every algebra A in DLO, O(D(A)) is again in DLO. In what follows we will show that if A 2 HSP (A1 ; : : : ; An ), where Ai is nite for every i = 1; : : : ; n, then O(D(A)) 2 HSP (A1 ; : : : ; An ). The proof will proceed as follows: 1. We show that if for every C 2 ISP (A1 ; : : : ; An ), O(D(C )) 2 ISP (A1 ; : : : ; An ), then HSP (A1 ; : : : ; An ) is closed under canonical embedding algebras. 2. We show that, in order to prove thatPfor every C 2 ISP (A1 ; : : : ; An ), O(D(C )) 2 ISP (A1 ; : : : ; An ), it is sucient to show that if X = j2J D(Bj ) for a family fBj j j 2 J g such that for every j 2 J , Bj 2 fA1 ; : : : ; An g then: (a) there is an ultrapower X I =U of X having a bounded morphism which is onto,  X I =U ! D(O(X ));

(b) if Y is an ultrapower of X then O(Y ) 2 ISP (A1 ; : : : ; An ). (The sums and ultrapowers are de ned in De nition 7 and resp. De nition 8.) 3. We show that (a) and (b) hold. 15

The next sections will be dedicated to proving these facts. Our goal is to prove that if A1 ; : : : ; An are nite distributive lattices with operators and A 2 HSP (A1 ; : : : ; An ) then O(D(A)) 2 HSP (A1 ; : : : ; An ). In what follows, unless explicitly speci ed otherwise, A1 ; : : : ; An denote nite distributive lattices with operators. Let A 2 HSP (A1 ; : :Q : ; An ). Then there exists C 2 ISP (A1 ; : : : ; An ) (i.e. there exists an injective homomorphism i : C ! j2J Bj where for every j 2 J; Bj 2 fA1 ; : : : ; An g), and a surjective morphism f : C ! A. Q C i / j2J Bj f 

A If we apply the functor D to the diagram above, by Lemma 3 the following holds: Q D(OC ) o D(i) D( j2J Bj )

D(f )

D(A) where D(i) is a surjective bounded morphism and D(f ) a bounded morphism which is an order-embedding. By Lemma 10 we have: Q O(D(C )) O(D(i)) / O(D( j2J Bj )) O(D(f )) 

O(D(A))

where O(D(i)) is an injective homomorphism and O(D(f )) is a surjective homomorphism.

Consequence 12 Assume that for every C 2 ISP (fA1; : : : ; Ang), O(D(C )) 2 ISP (fA1; : : : ; Ang). Then for every A 2 HSP (A1 ; : : : ; An ), O(D(A)) 2 HSP (A1 ; : : : ; An ). This proves 1. on page 15. Our goal is now to show that for every C 2 ISP (A1 ; : : : ; An ), O(D(C )) 2 ISP (fA1 ; : : : ; An g). Q

Let C 2 ISP (A1 ; : : : ; An ). Then there exists an injective homomorphism i : C ! j2J Bj , where for every j 2 J , Bj 2 fA1 ; : : : ; An g. Since A1 ; : : : ; An are all nite, we Q know that for every j 2 J , Bj ' O(D(Bj )). Thus, there exists an injective homomorphism i : C ! j2J O(D(Bj )).

De nition 7 Let fXi j i 2 I g be a family of ordered -relational structures, Xi = (Xi; i; fHXi gH2Lh; fKXi gK 2La; fRXi gR2Jh; fQXi gQ2Mh ; fRX0 i gR0 2Ja ; fQ0Xi gQ0 2Ma ). The sum Pi2I Xi of this family is the relational structure (X; ; fHX gH 2Lh ; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma ), where:

   

`

X = i2I Xi (the disjoint union), for every x; y 2 X x  y if and only if x; y 2 Xi and x i y for some i 2 I , for every x 2 X and every H 2 Lh [ La, HX (x) = HXi (x) where i is such that x 2 Xi , for every x1 ; : : : ; xn 2 X n , and every R 2 Jh [ Mh [ Ja [ Ma, RX (x1 ; : : : ; xn ) if and only if for some i 2 I , x1 ; : : : ; xn 2 Xi and RXi (x1 ; : : : ; xn ).

16

It is easy to see that if for every i 2 I , HXi is order-preserving, then also HX is order-preserving; if for every i 2 I , KXi is order-reversing, then also KX is order-reversing; if for every i 2 I , RXi is increasing, then also RX is decreasing. then also RX is increasing; if for every i 2 I , RXi is decreasing, ` Also, for every i 2 I , the canonical embedding ji : Xi ! i2I Xi de ned by ji (x) = (x; i) is a bounded morphism (which is an order-embedding).

Lemma 13 For every family fXi j i 2 I g of relational spaces, there exists an isomorphism between O(Pi2I Xi ) and Qi2I O(Xi ). P P Proof : Let ji : Xi ! i2I Xi be the canonical embedding and let i = O(ji ) : O( i2I Xi ) ! O(Xi ) be de ned by Pi (U ) = ji?1 (UQ). By Lemma 10, O(ji ) is onto. k Let k : O( i2I Xi ) ! i2I O(Xi ) be de ned by k(U ) = (i (U ))i2I = (ji?1 (U ))i2I . Obviously,  ?S is a homomorphism. It is onto because for every (Ui )i2I , where Ui P2 O(Xi ), (Ui )i2I = k i2I Ui . Moreover, k is injective: assume that k(U ) = k(V ) for some U; V 2 O( i2I Xi ). It follows that for every i 2 I and every x 2 Xi , (x; i) 2 U if and only if x 2 ji?1 (U ) = i (U ) = i (V ) = ji?1 (V ) if and only if (x; i) 2 V . 2 As a consequence, it immediately follows that, for the family fBj j j 2 J g considered before, there exists an isomorphism

k : O(

X

j 2J

D(Bj )) '

Y

j 2J

O(D(Bj )):

Lemma 14 Let C 2 ISP (A1 ; : : : ; An ) (i.e. such that there exists an injective homomorphism i : C ! Q j 2J Bj , where P fBj j j 2 J g  fA1 ; : : : ; An g). Then the algebra O(D(C )) is isomorphic to a subalgebra of O(D(O( j2J D(Bj )))). Q Proof : From Lemma 13 and the fact that there exists an injective homomorphism i : C ! j2J O(D(Bj )) Q k?1 O(P D(B )) is an injective homomorphism. Thereit follows that k?1  i : C !i j2J O(D(Bj )) ?! j j 2J P ? 1 fore, by Lemma 3 and Lemma 10, O(D(k  i)) : O(D(C )) ?! O(D(O( j2J D(Bj )))) is an injective 2

homomorphism.

P

The following considerations will help us to prove that O(D(O( j2J D(Bj )))) 2 ISP (A1 ; : : : ; An ). Let P X = j2J D(Bj ). (a) We rst show that there is an ultrapower X I =U of X having a bounded morphism  which is onto,  X I =U ! D(O(X )). Therefore there exists an injective homomorphism

O(D(O(X ))) O() / O(X I =U ): P

(b) We then show that if Y is an ultrapower of X = j2J D(Bj ), where Bj 2 fA1 ; : : : ; An g for every j 2 J , then O(Y ) 2 ISP (A1 ; : : : ; An ).

Lemma 15 Assume that the conditions (a) and (b) above hold. Then for every C 2 ISP (A1; : : : ; An), O(D(C )) 2 ISP (A1 ; : : : ; An ). Proof : Follows from the the remarks above, from the fact that, by Lemma 14 and the assumption (a), there exists an injective homomorphism

O()  k?1  i : O(D(C )) k

?1 i /

O(D(O(Pj2J D(Bj ))))

P

O() / O((Pj2J D(Bj ))I =U );

2 and from the fact that, by (b), O(( j2J D(Bj ))I =U ) 2 ISP (A1 ; : : : ; An ). This proves 2. on page 15. The fact that (a) and (b) hold will be proved in Section 4.2.3. In order to prove these assertions we start by de ning ultraproducts of ordered relational structures. 17

4.2.1 Ultraproducts of relational spaces Let J be an index set and let U  P (J ) be an ultra lter (i.e. a maximal lter). Let fXj j j 2 J g be a family of relational structures of theQsame type R (where for every j , Xj = (Xj ; fRj gR2R )). Let U be the relation de ned for every x; y 2 j2J Xj by: x U y if and only if fj 2 J j xj = yj g 2 U: Q Q It is easy to see that the relation U is an equivalence relation on j2J Xj . For every x 2 j2J Xj let [x]Q U denote the equivalence class of x with respectQto the equivalence relation U . For every relation R on j2J Xj let RU be a relation on the quotient ( j2J Xj )= U de ned by RU ([x1 ]U ; : : : ; [xn ]U ) if and only if fj 2 J j Rj (x1j ; : : : ; xnj )g 2 U: De nition 8 (Ultraproducts) For every family of relational spaces fXj j j 2 J g (where for every j , Xj = (Xj ; fRj gR2R)), and for every ultra lter U  P (J ), the quotient structure Y (( Xj )= U ; fRU gR2R ) j 2J

is called ultraproduct of the family fXj j j 2 J g with respect to U . Q

Q

In what follows we will denote j2J Xj = U by j2J Xj =U . If Xj = X for every j 2 J , X J =U is called an ultrapower of X . Ordered -relational structures are in particular relational structures: the functions corresponding to the operation symbols in Lh and La can in particular be seen as relations. Thus, ultraproducts of ordered -relational structures can be de ned in the usual way. The next lemma shows that an ultraproduct of ordered -relational structures is again an ordered -relational structure.

Lemma 16 Let fXj j j 2 J g be a family of ordered -relational structures of the form Xj = (Xj ; j ; fHXj gH 2Lh; fKXj gK 2La; fRXj gR2Jh ; fQXj gQ2Mh ; fRX0 j gR0 2JaQ; fQ0Xj gQ0 2Ma ). Let U  P (J ) be an ultra lter. Let the following relations be de ned on the quotient j2J Xj = U :  [x]U U [y]U if and only if fj j xj j yj g 2 U ,  for every H 2 Lh [ La, HU : Qj2J Xj ! Qj2J Xj is de ned by HU ([x]U ) = [(HXj (xj ))j2J ]U ,  for every R 2 Jh [ Mh [ Ja [ Ma, RU ([x1 ]U ; : : : ; [xn ]U ) if and only if fj j RXj (x1j ; : : : ; xnj )g 2 U . Q Then j2J Xj = U is again an ordered -relational structure. Proof : We rst show that U is a partial order. For every x, [x]U U [x]U , since fj j xj j xj g = J 2 U ; assume now that [x]U U [y]U and [y]U U [z ]U . It follows that fj j xj j yj g 2 U and fj j yj j zj g 2 U . We know that fj j xj j yj g \ fj j yj j zj g  fj j xj j zj g. It follows therefore that fj j xj j zj g 2 U . Antisymmetry is proved analogously. In order to show that for every H 2 Lh, HU is order-preserving, let H 2 Lh and let x; y be such that [x]U U [y]U . It follows that fj j xj j yj g 2 U , and since fj j xj j yj g  fj j HXj (xj ) j HXj (yj )g it follows that HU ([x]U ) U HU ([y]U ). Similarly it can be shown that for every K 2 La, KU is orderreversing. We now show that for every R 2 Jh [ Ja, RU is increasing. Let R 2 Jh. Assume that [x]U U [y]U and RU ([x1 ]U ; : : : ; [xn ]U ; [x]U ). It follows that RU ([x1 ]U ; : : : ; [xn ]U ; [y]U ), since fj j xj j yj g \ fj j RXj (x1j ; : : : ; xnj ; xj )g  fj j RXj (x1j ; : : : ; xnj ; yj )g, hence fj j RXj (x1j ; : : : ; xnj ; yj )g 2 U . The fact that for every Q 2 Mh [ Ma, QU is decreasing can be proved in a similar way. 2

Lemma 17 Let fXj j j 2 J g be a nite set of nite ordered -relational structures (such that, as a set, fXj j Q j 2 J g = fY1 ; : : : ; Ym g, where for every i = 1; : : : ; m, Yi is nite). Let U be an ultra lter over J . Then j2J Xj =U is isomorphic to one of the Yi , namely to that Yi such that fj 2 J j Xj = Yi g 2 U . Proof : It follows from the fact that, since J is nite, the ultra lter U is Q generated by some k 2 J , 2 and therefore, by classical properties of ultraproducts (cf. e.g. [BS71], p.124), j2J Xj =U ' Xk . 18

4.2.2 Elements of model theory

The proof of Theorem 18, inspired by the idea used by Goldblatt in [Gol89], requires some elements of model theory, namely the use of !-saturated elementary extensions. In this section we brie y summarize the main de nitions and results in model theory necessary for this. For further details we refer to [CK73]. A rst-order language L consists of a set of a set R of relation symbols, a set F of function symbols and a set C of constants. (In what follows, following [CK73], we do not allow relation or function symbols with arity 0.) The power or cardinal of a language L, denoted by kLk, is de ned by kLk = ! _ card(L). A model for a rst-order language L is a pair A = (A; I ), where A is a non-empty set and I is an interpretation function, mapping the relation symbols (resp. function symbols, constant symbols) of A to relations (resp. functions, constants) with the same arity on A. The cardinal or power of a model A = (A; I ) is the cardinal of its domain A. If we start with a model A for the language L we can expand it to a model for a language L0 = L[ S , by giving appropriate interpretations for the symbols in S . Given a model A and a subset Y  A, the expanded model obtained by adding constants for all elements a 2 Y will be denoted by AY , and its language by LY . If L is a rst-order language and X is a set (members of X are called variables) the terms of type L over X are the terms in X over the signature F , the atomic formulae are expressions of the form t1 ' t2 , where t1 ; t2 are terms of type L over X , and r(t1 ; : : : ; tn ), where r 2 R of arity n, and t1 ; : : : ; tn are terms of type L over X . The set of ( rst-order) formulae of type L over X is inductively de ned starting from the atomic formulae by using the logical operators _; ^; :; ) and the quanti ers 8 and 9. A rst-order theory T of a language L is a collection of sentences over L. The theory Th(A) of a model A is the set of all sentences which hold in A. A formula (x1 ; : : : ; xn ) is said to be consistent with respect to a theory T if there exists a model A of T which realizes . Given a cardinal , a model A is said to be -saturated if and only if for every subset Y  A with fewer than elements, the expanded model AY realizes every set of formulae ?(x) of LY which is consistent with the theory Th(AY ) of AY . The following results are known (cf. e.g. [CK73]): (1) For any set I of power ! there exists a countably incomplete ultra lter (i.e. an ultra lter which is not closed under countable intersections) U over I (by Proposition 4.3.4 and Proposition 4.3.5 in [CK73], p.249). (2) Let I be a set of power !. Let L be a language such that card(L)  !. Let A be a model for L, and let U be a countably incomplete ultra lter of I . Then the ultraproduct AI =U is !+-saturated (and, hence, !-saturated) (by Theorem 6.1.1, [CK73], p.384). More precisely the following holds: For every set (x) of formulae in L, if each nite subset of (x) is satis able in AI =U , then (x) is satis able in AI =U . (3) For every model A, non-empty index set I , and ultra lter U over I , the natural embedding  : A ! AI =U that associates with every element a 2 A the equivalence class of (a)i2I with respect to U is an elementary embedding (i.e. for every formula (x1 ; : : : ; xn ) of L and any a1 ; : : : ; an 2 A, a1 ; : : : ; an satisfy  in A if and only if (a1 ); : : : ; (an ) satis es  in AI =U ). It follows therefore that if L is a language such that card(L)  !, A a model for L, and I a set of power !, then there exists a countably incomplete ultra lter U over I such that (i) AI =U is an elementary extension of A; and (ii) for every set (x) of formulae in L, if each nite subset of (x) is satis able in AI =U , then (x) is satis able in AI =U . In conclusion, if L is a language such that card(L)  !, then every model A of L has a !-saturated elementary extension A1 , which is an ultrapower of A. This result is used in the proof of Theorem 18.

4.2.3 Closure under canonical embedding algebras

The following theorem extends the results of Goldblatt [Gol89]: we show that the arguments used in the proof of Theorem 3.6.1 in [Gol89] (for the case of relational structures corresponding to modal logic with one n-ary join-hemimorphic modality) can be extended to the more general relational structures considered here.

Theorem 18 Let X be an ordered -relational structure. Then there exists an ultrapower X J =U of X having a bounded morphism onto the canonical structure D(O(X )) of X ,  : X J =U ! D(O(X )): 19

Proof : The main idea of the proof is to associate with every ordered -relational structure X an extension XL by unary operators associated to elements of O(X ), and, further, an !-saturated elementary extension X1 of XL , which is actually an ultrapower of XL . We show that a map  : X1 ! D(O(X )) can be de ned in a canonical way. The !-saturation of X1 and the fact that the extension is elementary are then used in order to prove that  is a bounded morphism of relational spaces which is onto. Let X = (X; ; fHX gH 2Lh ; fKX gK 2La; fRX gR2Jh; fQX gQ2Mh ; fRX0 gR0 2Ja ; fQ0X gQ0 2Ma ) be an ordered -relational structure. Let L be the rst-order language with a binary relation symbol , a binary relation symbol =, a unary function symbol H for every H 2 Lh [ La, an (n + 1)-ary relation symbol R for every R 2 Jh [ Mh [ Ja [ Ma with arity n, and a unary relation symbol Y for every order- lter Y 2 O(X ). Let X1 = (X1 ; 1 ; =1 ; fHX1 gH 2Lh ; fKX1 gK 2La; fRX1 gR2Jh ; fQX1 gQ2Mh ; fRX0 1 gR0 2Ja ; fQ0X1 gQ0 2Ma ; fY  j Y 2 O(X )g) be an !-saturated elementary extension of XL = (X; ; =; fHX gH 2Lh ; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 1 gR0 2Ja ; fQ0X1 gQ0 2Ma ; fY j Y 2 O(X )g). Such an extension exists (by arguments of model theory [CK73]; see also [Gol89] or the remarks in Section 4.2.2). X1 may be realized as an ultrapower of XL , such that (X1 ; 1 ; fHX1 gH 2Lh ; fKX1 gK 2La; fRX1 gR2Jh ; fQX1 gQ2Mh ; fRX0 1 gR0 2Ja ; fQ0X1 gQ0 2Ma ) is an ultrapower of X = (X; ; fHX gH 2Lh; fKX gK 2La; fRX gR2Jh ; fQX gQ2Mh ; fRX0 1 gR0 2Ja ; fQ0X1 gQ0 2Ma ). Note that the construction presented here di ers from that in [Gol89] only in the fact that the additional unary symbols added to the language are indexed here by the elements of O(X ) rather than P (X ) as in the boolean case considered in [Gol89]. In what follows, the case of R 2 Jh is essentially given in [Gol89]; here we show that similar arguments can be used in order to extend the results to the cases when the signature also contains function symbols H 2 Lh, K 2 La and relation symbols Q 2 Mh, R0 2 Ja and Q0 2 Ma. For every x 2 X1 let Fx = fY 2 O(X ) j Y  (x)g. The sentences 8vX (v), :9v;(v), 8v; Y \ Z (v) , Y (v) ^ Z (v), 8v; Y [ Z (v) ) Y (v) _ Z (v) are all true in XL for every Y; Z 2 O(X ), thus they are true in the elementary extension X1 of XL . Thus, X 2 Fx , ; 62 Fx , Y \ Z 2 Fx if and only if Y 2 Fx and Z 2 Fx , Y [ Z 2 Fx implies Y 2 Fx or Z 2 Fx . It follows that Fx is a prime lter of O(X ). Therefore we obtain a map  : X1 ! D(O(X )) de ned for every x 2 X1 by (x) = Fx . We show now that  is a bounded morphism of ordered -relational structures from X1 onto D(O(X )). (In what follows we will use the same notational conventions for the operations in O(X ) as used in Section 3.1.) Claim 1.  is onto. Proof : Let G 2 D(O(X )). Let ?G = fY (v) j Y 2 Gg [ f:Y (v) j Y 62 Gg. We rst show that ?G is nitely satis able in XL . Indeed,Tlet I S and J be two nite subsets of O(X ) with T I  G Sand J  O(X )nG. Since G is a prime lter, I  6 J . S(In order to show this, assume T that I  J . Then, since I  G, it follows that I 2 G, hence J 2 G. It follows that j 2 G for some j 2 J , which contradicts the fact that J  O(X )nG.) Thus, there exists y 2 X such that T S y 2 I and y 62 J , i.e. ?G is nitely satis able in XL . This shows that ?G is nitely satis able in X1 . By !-saturation it follows that there exists x 2 X1 that satis es ?G in X1 , i.e. such that Fx = G. Claim 2.  is a morphism. Proof : We show that  preserves all the functions and relations. (1) (HX1 (x)) = D(hH )((x)) for H 2 Lh. Proof : We show that Y 2 FHX1 (x) if and only if Z = hH (Y ) = HX?11 (Y ) 2 Fx . If Y 2 FHX1 (x) and Z = hH (Y ), then the following hold in XL , and are therefore true also in X1 :

8v18v2 (Y (v1 ) ^ H (v2 ) = v1 ) Z (v2 )); 20

8v18v2 (Z (v2 ) ^ H (v2 ) = v1 ) Y (v1 )):

Taking in particular v1 = HX1 (x) and v2 = x, it follows that Y 2 FHX1 (x) if and only if hH (Y ) 2 Fx , i.e. if and only if Y 2 D(hH )(Fx ). (2) (KX1 (x)) = D(kK )((x)) for K 2 La. Proof : We show that Y 2 FKX1 (x) if and only if Z = kK (Y ) 62 Fx . (Then, Y 2 FKX1 (x) if and only if Y 2 D(kK )(Fx ).) If Y 2 FKX1 (x) and Z = kK (Y ), then the following hold in XL , and are therefore true also in X1 : 8v18v2 (Y (v1 ) ^ K (v2 ) = v1 ) :Z (v2 ));

8v18v2 (:Z (v2 ) ^ K (v2 ) = v1 ) Y (v1 )): (3) Let R be a relation in Jh. We show that if RX1 (x1 ; : : : ; xn ; xn+1 ) then D(fR )(Fx1 ; : : : ; Fxn+1 ). Proof : Assume that RX1 (x1 ; : : : ; xn ; xn+1 ) for x1 ; : : : ; xn+1 2 X1 . We want to show that for every Yi 2 Fxi , i = 1; : : : ; n, fR (Y1 ; : : : ; Yn ) 2 Fxn+1 . Let Yi 2 Fxi for every i 2 f1; : : : ; ng. If Y = fR (Y1 ; : : : ; Yn ) then

8v1; : : : ; 8vn+1 (Y 1 (v1 ) ^ : : : ^ Y n (vn ) ^ R(v1 ; : : : ; vn+1 ) ) Y (vn+1 )) is true in XL , hence it is also true in X1 . Therefore, since RX1 (x1 ; : : : ; xn+1 ), and Yi (xi ) for all i = 1; : : : ; n, it follows that Y  (xn+1 ), where Y = fR (Y1 ; : : : ; Yn ). Thus, fR (Y1 ; : : : ; Yn ) 2 Fxn+1 . This proves that D(fR )(Fx1 ; : : : ; Fxn+1 ). (4) Let Q be a relation in Mh. We show that if QX1 (x1 ; : : : ; xn ; xn+1 ) then D(gQ )(Fx1 ; : : : ; Fxn+1 ). Proof : Assume that QX1 (x1 ; : : : ; xn ; xn+1 ) for x1 ; : : : ; xn+1 2 X1 . We want to show that for every Y1 ; : : : ; Yn , if gQ (Y1 ; : : : ; Yn ) 2 Fxn+1 then there exists i 2 f1; : : : ; ng with Yi 2 Fxi . Let Y1 ; : : : ; Yn be arbitrary, and assume that Y = gQ (Y1 ; : : : ; Yn ) 2 Fxn+1 . We know that

8v1; : : : ; 8vn+1(Q(v1 ; : : : ; vn+1 ) ^ Y (vn+1 ) ) (Y 1 (v1 ) _ : : : _ Y n (vn ))) is true in XL , hence it is also true in X1 . Therefore, since QX1 (x1 ; : : : ; xn ; xn+1 ) and Y  (xn+1 ) (where Y = gQ (Y1 ; : : : ; Yn )) it follows that there exists i 2 f1; : : : ; ng such that Yi (xi ), i.e. Yi 2 Fxi . This proves that D(gQ )(Fx1 ; : : : ; Fxn+1 ). (5) Let R0 be a relation in Ja. We show that if RX0 1 (x1 ; : : : ; xn ; xn+1 ) then D(fR0 0 )(Fx1 ; : : : ; Fxn+1 ). Proof : The proof proceeds as in (3) taking into account the fact that if Y = fR0 0 (Y1 ; : : : ; Yn ), then the formula

8v1 ; : : : ; 8vn+1 (:Y 1 (v1 ) ^ : : : ^ :Y n (vn ) ^ R(v1 ; : : : ; vn+1 ) ) Y (vn+1 )): holds in XL , hence also in X1 . (6) Let Q0 be a relation in Ja. We show that if Q0X1 (x1 ; : : : ; xn ; xn+1 ) then D(gQ0 0 )(Fx1 ; : : : ; Fxn+1 ). Proof : The proof proceeds as in (4), taking into account the fact that if Y = gQ0 0 (Y1 ; : : : ; Yn ), then the formula

8v1; : : : ; 8vn+1(Q0 (v1 ; : : : ; vn+1 ) ^ Y (vn+1 ) ) (:Y 1 (v1 ) _ : : : _ :Y n (vn )))

holds in XL , hence also in X1 . Claim 3.  is a bounded homomorphism. Proof : We show that  satis es the boundedness condition: (1) Let R 2 Jh. Assume that D(fR )(G1 ; : : : ; Gn ; (z )). We show that there exist x1 ; : : : ; xn 2 X1 such that RX1 (x1 ; : : : ; xn ; z ) and (xi )  Gi for every i 2 f1; : : :; ng. Proof : Let ? = fR(v1 ; : : : ; vn ; z )g [ fY (v1 ) j Y 2 G1 g [ : : : [ fY (vn ) j Y 2 Gn g. If (x1 ; : : : ; xn ) satis es ? in X1 (when vi is interpreted as xi ) then RX1 (x1 ; : : : ; xn ; z ) and for every Y 2 Gi , Y  (xi ) for every i = 1; : : : ; n, i.e. RX1 (x1 ; : : : ; xn ; z ) and Gi  Fxi = (xi ) for every i = 1; : : : ; n. 21

We show that ? is satis able in X1 . Using the !-saturation of X1 and the closure of each Gi under nite intersection it is enough to show that if Yi 2 Gi then ?0 = fR(v1 ; : : : ; vn ; z ); Y 1 (v1 ); : : : ; Y n (vn )g is satis able in X1 . Since D(fR )(G1 ; : : : ; Gn ; Fz ) it follows that for every Yi 2 Gi ; i = 1; : : : ; n, Y = fR (Y1 ; : : : ; Yn ) 2 Fz . But for Y = fR (Y1 ; : : : ; Yn ) the sentence 8v(Y (v) ) 9v1 ; : : : ; 9vn (Y 1 (v1 ) ^ : : : ^ Y n (vn ) ^ R(v1 ; : : : ; vn ; v))): is true in XL , hence it is true in X1 . Hence, there exist x1 ; : : : ; xn 2 X1 such that Yi (xi ) for every i = 1; : : : ; n and RX1 (x1 ; : : : ; xn ; z ). But then x1 ; : : : ; xn satis es ?0 in X1 . From !-saturation of X1 it follows that there exist x1 ; : : : ; xn that satisfy ?. (2) Let Q 2 Mh. Assume that D(gQ )(G1 ; : : : ; Gn ; (z )). We show that there exist x1 ; : : : ; xn 2 X1 such that QX1 (x1 ; : : : ; xn ; z ) and (xi )  Gi for every i 2 f1; : : : ; ng. Proof : Let ? = fQ(v1 ; : : : ; vn ; z )g [ f:Y (v1 ) j Y 62 G1 g [ : : : [ f:Y (vn ) j Y 62 Gn g. If (x1 ; : : : ; xn ) satis es ? in X1 (when vi is interpreted as xi ) then QX1 (x1 ; : : : ; xn ; z ) and for every Y 62 Gi , :Y  (xi ) holds for every i = 1; : : : ; n, i.e. QX1 (x1 ; : : : ; xn ; z ) and (xi ) = Fxi  Gi for every i = 1; : : : ; n. We show that ? is satis able in X1 . Using the !-saturation of X1 and the closure of O(X )nGi under nite union for each Gi , it is enough to show that if Yi 62 Gi , i = 1; : : : ; n, then ?0 = fQ(v1 ; : : : ; vn ; z ); :Y 1 (v1 ); : : : ; :Y n (vn )g is satis able in X1 . Since D(gQ )(G1 ; : : : ; Gn ; Fz ) it follows that for every Y1 ; : : : ; Yn , if Yi 62 Gi ; i = 1; : : : ; n, Y = gQ (Y1 ; : : : ; Yn ) 62 Fz . But for Y = gQ (Y1 ; : : : ; Yn ) the sentence 8v(:Y (v) ) 9v1 ; : : : ; 9vn (:Y 1 (v1 ) ^ : : : ^ :Y n (vn ) ^ Q(v1 ; : : : ; vn ; v))) is true in XL , hence it is true in X1 . Hence, there exist x1 ; : : : ; xn 2 X1 such that :Yi (xi ) for every i = 1; : : : ; n and QX1 (x1 ; : : : ; xn ; z ). But then x1 ; : : : ; xn satis es ?0 in X1 , hence ? is satis able. 2 (3) Let R0 2 Ja. Assume that D(fR0 0 )(G1 ; : : : ; Gn ; (z )). We prove that there exist x1 ; : : : ; xn such that RX0 1 (x1 ; : : : ; xn ; z ) and (xi )  Gi for every i 2 f1; : : :; ng. Proof : Let ? = fR0 (v1 ; : : : ; vn ; z )g [ f:Y (v1 ) j Y 62 G1 g [ : : : [ f:Y (vn ) j Y 62 Gn g. We show that ? is satis able in X1 . Using the !-saturation of X1 and the closure of O(X )nGi under nite union for each Gi , it is enough to show that if Yi 62 Gi , i = 1; : : : ; n, then ?0 = fR0 (v1 ; : : : ; vn ; z ); :Y 1 (v1 ); : : : ; :Y n (vn )g is satis able in X1 . Since D(fR0 0 )(G1 ; : : : ; Gn ; Fz ) it follows that for every Y1 ; : : : ; Yn , if Yi 62 Gi ; i = 1; : : : ; n, then Y = fR0 0 (Y1 ; : : : ; Yn ) 2 Fz (i.e. Y  (z ) holds in X1 ). Let Y1 ; : : : ; Yn be such that for every i = 1; : : : ; n, Yi 62 Gi . Then Y = fR0 0 (Y1 ; : : : ; Yn ) 2 Fz . But for Y = fR0 0 (Y1 ; : : : ; Yn ) the sentence 8v(Y (v) ) 9v1 ; : : : ; 9vn (:Y 1 (x1 ) ^ : : : ^ :Y n (xn ) ^ R0 (v1 ; : : : ; vn ; v))) is true in XL, hence also in X1 . Hence, there exist x1 ; : : : ; xn 2 X1 such that :Yi (xi ) for every i = 1; : : : ; n and RX0 1 (x1 ; : : : ; xn ; z ). Thus, there exist x1 ; : : : ; xn that satisfy ?0 , hence ? is satis able. Moreover, for every Yi 62 Gi , we have Yi 62 Fxi . This shows that (xi ) = Fxi  Gi . (4) Let Q0 2 Ma. Assume that D(gQ0 0 )(G1 ; : : : ; Gn ; (z )). We show that there exist x1 ; : : : ; xn 2 X1 such that Q0X1 (x1 ; : : : ; xn ; z ) and (xi )  Gi for every i 2 f1; : : :; ng. Proof : The proof proceeds as the previous ones; we choose the set ? of formulae as in (1), and then note that in XL (hence also in X1 ) the following sentence holds: 8v(:Y (v) ) 9v1 ; : : : ; 9vn (Y 1 (v1 ) ^ : : : ^ Y n (vn ) ^ Q0 (v1 ; : : : ; vn ; v))): 22

Theorem 19 Let K be a class of ordered relational structures closed under ultraproducts. Let fXi j i 2 I g P be a family of elements in K , and let Y be an ultrapower of i2I Xi . Then O(Y ) 2 ISP (K + ), where K + = fO(X ) j X 2 K g. Proof : The proof extends the P proof of Lemma 3.5 in [Gol88] to the richer structure of the relational spaces considered here. Let X = i2I Xi and let Y = X J =U for some index set J and some ultra lter U  P (J ). For every t 2 Y let ft 2 X J be such that t = [ft ]U . For every j 2 J there exists ij 2 I with ft(j ) 2 Xij . Q Construct Yt = ( j2J Xij )=U . We know that for every j 2 J , Xij 2 K , hence, since K is closed under ultraproducts, Yt 2 K for every t 2 Y . Thus, O(Yt ) 2 K + for every t 2 Y . Q J U Q Note that j 2J Xij  X . We now closely analyze Yt . An element of Yt is of the form g = fh 2 j 2J Xij j g U hg  [g ]U , where the last equivalence class is in Y (the two equivalence classes in Yt and Y are notQequal in general). Let t : ( j2J Xij )=U ! X J =U be de ned by t (gU ) = [g]U . QIt is easy to see that t is wellde ned and that t (ftU ) = [ft ]U = t. Note that for every g; h 2 j2J Xij gU = hU if and only if . (In what follows we will fj j g(j ) = h(j ) 2 Yij g 2 U if and only Q if g U h if and only if [g ]U = [h]UQ denote the functions and relations on ( j2J Xij )=U by indexing them with ( j2J Xij )=U , and those on X J =U by indexing them with U .) Claim 1. t is an order-embedding. Q Proof : For every g; h 2 j2J Xij  X J , gU  hU if and only if fj 2 J j g(j )  h(j )g 2 U if and only if [g]U  [h]U . Claim 2. t is a morphism. Proof : We show that t preserves the operations and relations. Q

(1) For every H 2 Lh[La, and every g 2 j2J Xij , t (HQj2J Xij =U (gU )) = t ((HQj2J Xij (g))U ) = [HQj2J Xij (g)]U = [HX J (g)]U = HU ([g]U ) = HU (t (gU )). Q (2) Let R 2 Jh [ Mh [ Ja [ Ma and g1 ; : : : ; gn ; gn+1 2 j2J Xij . If RQ Xij =U (g1U ; : : : ; gnU+1 ) then fj j RXij (g1 (j ); : : : ; gn+1 (j ))g 2 U , and since gk (j ) 2 Xij for every j 2 J , by the de nition of RX it follows that fj j RX (g1 (j ); : : : ; gn+1 (j ))g 2 U , hence RU ([g1 ]U ; : : : ; [gn+1 ]U ). Claim 3. t is a bounded morphism. Q Proof : Assume that in X J =U we have RU ([f1 ]U ; : : : ; [fn ]U ; t (gU )) with g 2 j2J Xij and fk 2 X J for every k. Then F = fj j RX (f1 (j ); : : : ; fn (j ); g(j ))g 2 U , where g(j ) 2 Xij . Since U is an ultra lter, F 6= ;. Let j be an arbitrary element of F . By the de nition of RX , since g(j ) 2 Xij , it follows that for every k = 1; : : : ; n, fk (j ) 2 XiQ j and RXij (f1 (j ); : : : ; fn (j ); g (j )). For every k 2 f1; : : : ; ng let hk be an arbitrary member of j2J Xij which agrees with fk on F . Then hk U fk , hence t (hUk ) = [fk ]U ; and F  fj j RXij (h1 (j ); : : : ; hn (j ); g(j ))g. Since UQis a lter it follows that fj j RXij (h1 (j ); : : : ; hn (j ); g(j ))g 2 U . Thus, there exist hU1 ; : : : ; hUn 2 ( j2J Xij )=U such that R(Q Xij )=U (hU1 ; : : : ; hUn ; gU ) and t (hUk ) = [fk ]U .

This shows that for every t 2 Y , t is a bounded morphism from Yt to Y . P Let  : t2Y Yt ! Y , such that for every t 2 Y , jYt = t . It can be seen that  is well-de ned and it is a bounded morphism. Also,  is surjective because for every t 2 Y , t = t (ftU ) = (ftU ). It follows that there exists an injective homomorphism

O(Y ) ! O(

X

t2Y

Yt ) '

Y

t2Y

O(Yt ) 2 IP (K +):

Thus, O(Y ) 2 ISP (K +): 2 Corollary 20 Let fD(Aj ) j j 2 J g be a nite set, where for every j 2 J , Aj is nite, such that fAj j j 2 J g = fB1 ; : : : ; Bmg. Let Y be an ultrapower of Pj2J D(Aj ). Then O(Y ) 2 ISP (B1 ; : : : ; Bm ). Proof : From Lemma 17 we know that K = fD(Aj ) j j 2 J g is closed under ultraproducts. Thus, the conclusion follows from Theorem 19. 2 23

Corollary 21 A 2 HSP (A1; : : : ; An), Ai nite for every i = 1; : : : ; n. Then O(D(A)) 2 HSP (A1; : : : ; An). Proof : The result follows immediately from Lemma 15, Theorem 18, Theorem 19, and Corollary 20.

2

An even more general result can be proved, that extends Theorem 3.6 in [Gol88]. The proof proceeds as the one given by Goldblatt. Corollary 22 Let K be a class of ordered -relational structures which is closed under ultraproducts, and let K + = fO(X ) j X 2 K g. Then HSP (K +) is closed under canonical embedding algebras. Proof : We rst prove that ISP (K +) is closed under canonical embedding algebras. Let B 2 + ISP P that B is isomorphic to a subalgebra of Qa family fXi j i 2 I g  K such Q (K ). Then there exists O ( X ) is isomorphic to O ( O ( X ). By Lemma 13, i i i2I Xi ). It follows that O(DP(B )) is isomori2P I i2I X ))). By Theorem 18, there exists an ultrapower ( i2I Xi )J =U of phic to a subalgebra of O ( D ( O ( i i2IP P P J i2I Xi , and a bounded P morphism ( i2I Xi ) =U ! D(O( i2I Xi )) which is onto. Therefore, O(D(B )) is a subalgebra of O( i2I Xi )J =U ), which, by Theorem 19, is an element in ISP (K +). Thus, we showed that ISP (K +) is closed under canonical embedding algebras. Let now A 2 HSP (K +). Then A is a homomorphic image of some B 2 ISP (K +), and hence, O(D(A)) is a homomorphic image of O(D(B )). We proved that ISP (K +) is closed under canonical embedding algebras, so O(D(B )) 2 ISP (K +), and, therefore, O(D(A)) 2 HSP (K +). 2

5 Conclusions In this paper we presented an extension of the Priestley duality to distributive lattices with operators in the following classes: lattice homomorphisms and antimorphisms; join- and meet-hemimorphisms; and join- and meet-hemiantimorphisms. These results can be seen as an extension of the results presented in [Gol89]. We then used these results in order to study the closure under canonical embedding algebras of certain varieties of distributive lattices with operators. Our results extend both the results in [Gol89] (we consider distributive lattices with operators instead of Boolean algebras with operators) and those in [GJ94] (we also consider certain classes of antimorphisms). It turns out that this type of results can be applied to the study of non-classical logics. In the second part of this paper [SS98] we will present a way of de ning classes of Kripke-style models for certain logics for which an algebraic semantics is known, and we will show that, under certain circumstances, the soundness and completeness of a logic L with respect to a variety VL of algebras implies its soundness and completeness with respect to a class of relational Priestley spaces, VL Sp, regarded as (non-topological) frames, or, alternatively, to the class of all ordered relational structures X of a certain similarity type with the property that O(X ) (endowed with operators corresponding to the relations on X ) is an element in VL . An extension to more general classes of operators that are hemimorphisms in some of their arguments and hemiantimorphisms in the others is not completely immediate; in future work we would like to also study these classes of operators.

Acknowledgments

This work was partially supported by MEDLAR II (ESPRIT Basic Research Project 6471; nanced for Austria by FWF) and partially by a postdoctoral scholarship at MPII. I also gratefully acknowledge the support from COST Action 15. I thank J. Pfalzgraf for directing my research to the area of decompositions of logics; L. Iturrioz, E. Orlowska, and H. Priestley for very interesting and inspiring discussions; E. SanJuan for bringing to my knowledge the work of Gehrke and Jonsson on closure under canonical embedding algebras for distributive lattices with additive operators; and H. Ganzinger for helpful advices concerning the structure of the paper. Many thanks to the anonymous referee for references to related work, for the very helpful suggestions that led to an improvement of the structure of the paper, for pointing out two errors, and for suggesting a shorter proof of Lemma 17.

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