Subresiduated lattice ordered commutative monoids

Abstract

A subresiduated lattice ordered commutative monoid (or srl-monoid for short) is a lattice ordered commutative monoid with a particular subalgebra which contains a residuated implication. The srl-monoids can be regarded as algebras that generalize 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. The above mentioned variety contains the variety of prelinear integral srl-monoids, whose expansion with bottom can be seen as a generalization of that of MTL-algebras.