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 $({\textbf {A}},Q)$ where ${\textbf {A}}=(A,\wedge ,\vee ,\cdot ,1)$ is a lattice ordered commutative monoid, 1 is the greatest element of the lattice $(A,\wedge ,\vee )$ and Q is a subalgebra of A such that for each $a,b\in A$ the set $\{q \in Q: a \cdot q \le b\}$ has maximum, which will be denoted by $a\rightarrow b$. The integral srl-monoids can be regarded as algebras $(A,\wedge ,\vee ,\cdot ,\rightarrow ,1)$ 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 $\{\wedge ,\cdot ,\rightarrow ,1\}$-subreducts of integral srl-monoids, which will be denoted by $\mathsf {SR^s}$. In particular, we show that $\mathsf {SR^s}$ is a variety. We also characterize simple and subdirectly irreducible algebras of $\mathsf {SR^s}$ respectively. Finally, through a Hilbert style system, we present a logic which has as algebraic semantics the variety $\mathsf {SR^s}$ 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.