On a Variety of Hemi-Implicative Semilattices

JOSÉ LUIS CASTIGLIONI; VICTOR FERNÁNDEZ; FEDERICO MALLEA; HERNAN J. SAN MARTIN

SOFT COMPUTING - (Print)

SPRINGER

Año: 2022 vol. 26 p. 3187 - 3195

1472-7643

A hemi-implicative semilattice is an algebra (A,∧,→, 1) of type (2, 2, 0) such that (A,∧, 1) is a bounded semilattice andthe following conditions are satisfied:1. for every a, b, c ∈ A, if a ≤ b → c then a ∧ b ≤ c and2. for every a ∈ A, a → a = 1.The class of hemi-implicative semilattices forms a variety. In this paper we introduce and study a proper subvariety of thevariety of hemi-implicative semilattices, ShIS, which also properly contains some varieties of interest for algebraic logic. Ourmain goal is to show a representation theorem for ShIS. More precisely, we prove that every algebra of ShIS is isomorphic toa subalgebra of a member of ShIS whose underlying bounded semilattice is the bounded semilattice of upsets of a poset.