In this paper we introduce the notion of generalized implication for lattices, as a binary function⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized im...

Inspired by classic work of Wallman and more recent Jung-Kegelmann-Moshier Vickers, we show how to encode general subbases stably locally compact spaces via certain entailment relations. We further build this up a categorical duality encompassing the Priestley-Stone its various extensions Shirota, De Vries, Hofmann-Lawson (in stable case), Jung-Sünderhauf, Hansoul-Poussart, Bezhanishvili-Jansan...

We carry out a detailed comparison of the two topological dualities for distributive meet-semilattices studied by Celani [3] and by Bezhanishvili and Jansana [2]. We carry out such comparison, that was already sketched in [2], by defining the functors involved in the equivalence of both dual categories of distributive meet-semilattices.

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of ...

A construction of canonical extensions of Stone algebras is presented that uses the natural duality based on the three-element generating algebra 3 rather than the Priestley duality based on 2 that is traditionally used to build the canonical extension. The new approach has the advantage that the canonical extension so constructed inherits its algebra structure pointwise from a power of the gen...

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploščica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass ...

The variety of RDP-algebras forms the algebraic semantics of RDPlogic, the many-valued propositional logic of the revised drastic product left-continuous triangular norm and its residual. We prove a Priestley duality for finite RDP-algebras, and obtain an explicit description of coproducts of finite RDP-algebras. In this light, we give a combinatorial representation of free finitely generated R...

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...

We investigate Heyting varieties determined by prohibition of systems of configuraations in Priestley duals; we characterize the configuration systems yielding such varieties. On the other hand, the question whether a given finitely generated Heyting variety is obtainable by such means is solved for the special case of systems of trees. Priestley duality provides a correspondence between bounde...

In their more general form, our equations are of the form u → v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman’s theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.)...