Subresiduated lattice ordered commutative monoids

Abstract

A subresiduated lattice ordered commutative monoid (or srl-monoid for short) is a pair $(\text{A},Q)$ where $\text{A}=(A,\wedge ,\vee ,\cdot ,e)$ 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 $q\in Q$, $a\cdot q\le b$ if and only if $q\le c$. This $c$ is denoted by $a{\to }_{Q}b$, or simply by $a\to b$. The srl-monoids $(\text{A},Q)$ can be regarded as algebras $(A,\wedge ,\vee ,\cdot ,\to ,e)$ 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.

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