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

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

STUDIA LOGICA

Springer

Año: 2023

0039-3215

An integral subresiduated lattice ordered commutative monoid (or integralsrl-monoid for short) is a pair (A;Q) where A = (A,wedge,ee, prod, 1) is a lattice ordered commutative monoid, 1 is the greatest element of the lattice (Awedge,ee) and Q is a subalgebra of A such that for each a, b in A the set {qin Q : a prod q leq b} has maximum, which will be denoted by a ightarrow b.The integral srl-monoids can be regarded as algebras (A,wedge ,prod,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 infimum-product-implicative-1 subreducts of integral srl-monoids, which will be denoted by SRs. In particular, we show that SRs is a variety. We also characterize simple and subdirectly irreducible algebras of SRs respectively. Finally, through a Hilbert style system, we present a logic which has as algebraic semantics the variety SRs 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.