We show that the variety generated by the three-element Heyting algebra admits a meet dense, regular completion even though it is not closed under MacNeille completions.

A relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting...

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and of Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion ̂ H of a Heyting algebra H, and characterize the dual sp...

In this paper a simple proof of the completeness theorem of the intuitionistic predicate calculus with respect to the topological semantics is shown. From a technical point of view the proof of the completeness theorem is based on a Rasiowa-Sikirski-like theorem for the countable Heyting algebras which allows to embadd any countable Heyting algebra into a suitable topology in a such way that a ...

In this paper we survey some recent developments in duality for lattices with additional operations paying special attention to Heyting algebras and the connections to Esakia’s work in this area. In the process we analyse the Heyting implication in the setting of canonical extensions both as a property of the lattice and as an additional operation. We describe Stone duality as derived from cano...

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation τ(a) ≤ b ∨ (b → a), for all a, b ∈ A. These operators were studied from an algebraic, logical an...

In [3] we have claimed that finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor, and in particular all finite chains generate a proper subvariety of the latter. As Xavier Caicedo made us notice, this claim is not true. He proved, using techniques of Kripke models, that the intuitionistic calculus with S has finite model propert...

We present a fuzzy version of the notion relational Galois connection between transitive directed graphs (fuzzy T-digraphs) on specific setting in which underlying algebra truth values is complete Heyting algebra. The components such are relations satisfying certain reasonable properties expressed terms so-called full powering. Moreover, we provide necessary and sufficient condition under it po...

In this paper we introduce a notion of dimension and codimension for every element of a distributive bounded lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is prec...

We answer a question [2] by Vasco Brattka and Guido Gherardi by proving that the Weihrauch lattice is not a Brouwer algebra. The computable Weihrauch lattice is also not a Heyting algebra, but the continuous Weihrauch lattice is. We further investigate embeddings of the Medvedev degrees into the Weihrauch degrees.