Terms that define nuclei on residuated lattices: a case study of BL-algebras

TL;DR

This paper characterizes all terms defining nuclei on residuated lattices, especially BL-algebras, generalizing the double negation operation and providing a comprehensive description across subvarieties.

Contribution

It provides a complete description of all terms that define nuclei on residuated lattices and BL-algebras, extending the understanding of their algebraic structure.

Findings

Characterization of arbitrary nuclei on residuated lattices.

Identification of terms defining nuclei across subvarieties of BL-algebras.

Examples of nontrivial terms and their properties.

Abstract

A nucleus $\gamma$ on a (bounded commutative integral) residuated lattice $\mathbf{A}$ is a closure operator that satisfies the inequality $\gamma(a) \cdot \gamma(b) \leq \gamma(a \cdot b)$ for all $a,b \in A$. In this article, among several results, a description of an arbitrary nucleus on a residuated lattice is given. Special attention is given to terms that define a nucleus on every structure of a variety, as a means of generalizing the double negation operation. Some general results about these terms are presented, together with examples. The main result of this article consists of the description of all terms of this kind for every given subvariety of BL-algebras. We exhibit interesting nontrivial examples.