On principal congruences in distributive lattices with a commutative monoidal operation and an implication

TL;DR

This paper introduces a new class of algebras extending known structures like residuated lattices and Heyting algebras, providing a characterization of principal congruences and analyzing compatible functions within this framework.

Contribution

It defines a broader algebraic variety and offers a novel characterization of principal congruences, advancing the understanding of these algebraic structures.

Findings

Characterization of principal congruences in the new variety

Application to compatible functions analysis

Extension of existing algebraic frameworks

Abstract

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.