Priestley Duality Research Articles

Priestley's Duality from Stone's

Priestley's Duality from Stone's

A duality theory for bilattices

Recent studies of the algebraic properties of bilattices have provided insight into their internal strucutres, and have led to practical results, especially in reducing the computational complexity of bilattice-based multi-valued logic programs. In this paper the representation theorem for interlaced bilattices without negation found in [19] and extended to arbitrary interlaced bilattices without negation in [2] is presented. A natural equivalence is then established between the category of interlaced bilattices and the cartesian square of the category of bounded lattices. As a consequence a dual natural equivalence is obtained between the category of distributive bilattices and the coproduct of the category of bounded Priestley spaces with itself. Some applications of these equivalences are given. The subdirectly irreducible interlaced bilattices are characterized in terms of subdirectly irreducible lattices. A known characterization of the join-irreducible elements of the "knowledge" lattice of an interlaced bilattice is used to establish a natural equivalence between the category of finite, distributive bilattices and the category of posets of the form \( {\bf P} \oplus_\perp {\bf Q} \).

Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer from 1964

Let L be a bounded distributive lattice. For k⩾1, let Sk(L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean functions. The lattices that can arise as Sk(L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lattices L and M, it is shown that S1(L)≅S1(M) implies Sk(L)≅Sk(M). If L and M are finite, then Sk(L)≅Sk(M) implies L≅M. Some problems of Grätzer dating to 1964 are thus solved.

Extended ockham algebras

Here we initiate an investigation into the class eO of extended Ockham algebras, namely bounded distributive lattices endowed with a commuting pair of unary operations, one of which is an endomorphism and the other is a dual endomorphism. A consideration of the subdirectly irreducible algebras leads, via Priestley duality, to a description of all finite subdirectly irreducible eO-algebras. In particular, we show that there are precisely nine non-isomorphic subdirectly irreducible symmetric extended de Morgan algebras, all of which are simple. We also describe the corresponding lattice of varieties.

Priestley Duality for Quasi-Stone Algebras

In this paper we describe the Priestley space of a quasi-Stone algebra and use it to show that the class of finite quasi-Stone algebras has the amalgamation property. We also describe the Priestley space of the free quasi-Stone algebra over a finite set.

Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic

Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic

Distributive Lattices with a Negation Operator

AbstractIn this note we introduce and study algebras (L, V, Λ, ⌝, 0,1) of type (2, 2,1,1,1) such that (L, V, ⌝, 0,1) is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ (a V b) = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi‐Stone algebras.

Unification of Four Versions of Program Semantics

Abstract. Computing Science is a new subject, and we have not yet achieved the unification of theories that should support a proper understanding of its structure . CAR Hoare and He Jifeng, 1998. In this paper we use Priestley duality and Jónsson/Tarski duality to translate between four versions of program semantics: the relational model, predicate transformer semantics, information systems, and powerdomains. Our point of entry is the relational model, a kind of Kripke semantics, in which programs are thought of as input-output relations over a structured state space. Specifically, we present the state space as a certain kind of Priestley space, and programs as certain structure-preserving relations over such a space. We then derive, in circular fashion, a predicate transformer semantics from the relational model, an information system from the predicate transformer semantics, a powerdomain from the information system, and the original relational model back again from the powerdomain. The information system is also shown to be related to Hoare logic. To clarify the intuition behind this approach we present a case study, which is a ‘Priestley version’ of the so-called universal domain due to Plotkin, and we explicate various ideas about properties of programs and predicates in terms of this case study.

On Priestley Spaces of Lattice-Ordered Algebraic Structures

The laws defining many important varieties of lattice-ordered algebras, such as linear Heyting algebras, MV-algebras and l-groups, can be cast in a form which allows dual representations to be derived in a very direct, and semi-automatic, way. This is achieved by developing a new duality theory for implicative lattices, which encompass all the varieries above. The approach focuses on distinguished subsets of the prime lattice filters of an implicative lattice, ordered as usual by inclusion. A decomposition theorem is proved, and the extent to which the order on the prime lattice filters determines the implicative structure is thereby revealed.

Priestley duality for demi-p-lattices

Priestley spaces of demi-p-lattices and almost-p-lattices are described in terms of Priestley spaces of pseudo-complemented distributive lattices. Characterizations of subvarieties of demi-p-lattices generated by a single finite subdirectly irreducible demi-p-lattice are given.

Localic Priestley duality

Given the category of ordered Stone spaces (as introduced by Priestley, 1970) and the category of coherent spaces (= spectral spaces) we can construct a pair of functors CohSp ▪ OStoneSp between the categories. Priestley (1970) has shown, assuming the prime ideal theorem, that these define an equivalence. In this paper, we define ordered Stone locales. These are classically just the ordered Stone spaces. It is well known that the localic analogue of the coherent spaces is the category of coherent locales. We prove, entirely constructively, that the category of coherent locales is equivalent to the category of ordered Stone locales.

Principal congruences on distributive double p-algebras

We use Priestley's duality to characterize, via their dual space, the distributive double p-algebras on which all congruences are principal.

Extremal Prime Filters and Universality of Some Categories

Abstract Due to the existence of constants, classical topological categories cannot be universal in the sense of containing each concrete category as a full subcategory. In the point-free case, this obstruction vanishes and the question of universality makes sense again. The main problem, namely that as to whether the category of locales and localic morphisms is universal is still open; we prove, however, the universality of the following categories: - pairs (locale, sublocale) with the localic morphisms preserving the distinguished sublocales, - frames with frame homomorphisms reflecting the maximal prime ideals, - Priestley spaces with f-maps preserving the maximal elements.

Priestley duality for some subalgebra lattices

Priestley duality can be used to study subalgebras of Heyting algebras and related structures. The dual concept is that of congruence on the dual space and the congruence lattice of a Heyting space is dually isomorphic to the subalgebra lattice of the dual algebra. In this paper we continue our investigation of the congruence lattice of a Heyting space that was undertaken in [10], [8] and [12]. Our main result is a characterization of the modularity of this lattice (Theorem 2.12). Partial results about its complementedness are also given, and among other things a characterization of those finite Heyting algebras with a complemented subalgebra lattice (Theorem 3.5).

Optimal natural dualities for varieties of Heyting algebras

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.

Semi-DeMorgan algebras

Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of “partial diagrams” is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest variety of semi-DeMorgan algebras with the congruence extension property.

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice Lσ of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-Cech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; XY denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of LP is determined, where P is a poset and L a bounded distributive lattice.

Duality for algebras of relevant logics

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.

A simplified duality for implicative lattices and l-groups

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them.

Distributive lattices with an operator

It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].