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

CORNEJO, JUAN MANUEL; SAN MARTÍN, HERNÁN JAVIER; SÍGAL, VALERIA ANAHÍ

STUDIA LOGICA

Springer Netherlands

Año: 2023

0039-3215

An integral subresiduated lattice ordered commutative monoid (or integral srl-monoid for short) is a pair$(\textbf{A},Q)$ where $\textbf{A}=(A,\we,\vee,\cdot,1)$is a lattice ordered commutative monoid, $1$ is the greatest element of the lattice $(A,\wedge,\vee)$ and $Q$ is a subalgebra of\textbf{A} such that for each $a,b\in A$ the set $\{q \in Q: a \cdot q \leq b\}$ has maximum, which will be denoted by $a\ra b$.The integral srl-monoids can be regarded as algebras $(A,\we,\vee,\cdot,\ra,1)$ of type $(2,2,2,2,0)$.Furthermore, this class of algebras is a variety which properly contains the varieties of integral commutativeresiduated lattices and subresiduated lattices respectively.In this paper we study the quasivariety of $\{\we,\cdot,\ra,1\}$-subreducts of integral srl-monoids, whichwill be denoted by $\nV$. In particular, we show that $\nV$ is a variety. We also characterize simple and subdirectlyirreducible algebras of $\nV$ respectively. Finally, through a Hilbert style system, we present a logic which has asalgebraic semantics the variety $\nV$ and we apply this result in order to present an expansion of the previous logic which has asalgebraic semantics the variety of integral srl-monoids.