Subresiduated lattice ordered commutative monoids

TL;DR

This paper introduces subresiduated lattice ordered commutative monoids, proves they form a variety, and explores their algebraic structure and congruence lattices, extending previous algebraic frameworks.

Contribution

It establishes that srl-monoids form a variety and characterizes their congruence lattices and subalgebras, generalizing subresiduated and residuated lattices.

Findings

The class of srl-monoids forms a variety.

Congruence lattices are isomorphic to strongly convex subalgebras.

Provides an alternative equational basis for the variety.

Abstract

A subresiduated lattice ordered commutative monoid (or srl-monoid for short) is a pair $(\textbf{A},Q)$ where $\textbf{A}=(A,\wedge,\vee,\cdot,e)$ is an algebra of type $(2,2,2,0)$ such that $(A,\wedge,\vee)$ is a lattice, $(A,\cdot,e)$ is a commutative monoid, $(a\vee b)\cdot c = (a\cdot c) \vee (b\cdot c)$ for every $a,b,c\in A$ and $Q$ is a subalgebra of \textbf{A} such that for each $a,b\in A$ there exists $c\in Q$ with the property that for all $q\in Q$, $a\cdot q \leq b$ if and only if $q\leq c$. This $c$ is denoted by $a\rightarrow_Q b$, or simply by $a\rightarrow b$. The srl-monoids $(\textbf{A},Q)$ can be regarded as algebras $(A,\wedge,\vee,\cdot,\rightarrow, e)$ of type $(2,2,2,2,0)$. These algebras are a generalization of subresiduated lattices and commutative residuated lattices respectively. In this paper we prove that the class of srl-monoids forms a variety. We show that the lattice of congruences of any srl-monoid is isomorphic to the lattice of its strongly convex subalgebras and we also give a description of the strongly convex subalgebra generated by a subset of the negative cone of any srl-monoid. We apply both results in order to study the lattice of congruences of any srl-monoid by giving as application alternative equational basis for the variety of srl-monoids generated by its totally ordered members.