On a class of subreducts of the variety of integral srl-monoids and related logics

Abstract

An integral subresiduated lattice ordered commutative monoid (or integral srl-monoid for short) is a pair where is a lattice ordered commutative monoid, 1 is the greatest element of the lattice and Q is a subalgebra of A such that for each the set has maximum, which will be denoted by . The integral srl-monoids can be regarded as algebras of type (2, 2, 2, 2, 0). Furthermore, this class of algebras is a variety which properly contains the varieties of integral commutative residuated lattices and subresiduated lattices respectively. In this paper we study the quasivariety of -subreducts of integral srl-monoids, which will be denoted by . In particular, we show that is a variety. We also characterize simple and subdirectly irreducible algebras of respectively. Finally, through a Hilbert style system, we present a logic which has as algebraic semantics the variety and we apply this result in order to present an expansion of the previous logic which has as algebraic semantics the variety of integral srl-monoids.