Hemi-Nelson algebras

TL;DR

This paper introduces hemi-Nelson algebras, generalizing the connection between Heyting and Nelson algebras within hemi-implicative lattices, and characterizes their congruences and categorical equivalences.

Contribution

It defines hemi-Nelson algebras, characterizes their congruences via implicative filters, and establishes an equivalence with bounded distributive hemi-implicative lattices.

Findings

Hemi-Nelson algebras generalize Nelson algebras.

Lattice of congruences characterized by implicative filters.

Categorical equivalence established between hemi-implicative lattices and hemi-Nelson algebras.

Abstract

The aim of this paper is to generalize the link between Heyting algebras and Nelson algebras, established independently by Fidel and Vakarelov at the end of the 1970s, in the framework of bounded distributive hemi-implicative lattices. For this purpose, we introduce the variety of hemi-Nelson algebras. Moreover, we characterize the lattice of congruences of a hemi-Nelson algebra in terms of certain implicative filters. We also esta\-blish an equivalence between the algebraic category of bounded distributive hemi-implicative lattices and the one of centered hemi-Nelson algebras.