Subresiduated Nelson Algebras

TL;DR

This paper introduces subresiduated Nelson algebras, generalizing Nelson algebras within subresiduated lattices, and explores their algebraic properties, congruences, and categorical equivalences.

Contribution

It defines subresiduated Nelson algebras, characterizes their congruences, and establishes their algebraic and categorical properties, extending the theory of Nelson algebras.

Findings

Characterization of congruences via implicative filters

Existence of equationally definable principal congruences

Categorical equivalence between subresiduated lattices and centered subresiduated Nelson algebras

Abstract

In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.