On a variety of hemi-implicative semilattices

Abstract

A hemi-implicative semilattice is an algebra (A, ∧ , → , 1) of type (2, 2, 0) such that (A, ∧ , 1) is a bounded semilattice and the 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 the variety of hemi-implicative semilattices, ShIS, which also properly contains some varieties of interest for algebraic logic. Our main goal is to show a representation theorem for ShIS. More precisely, we prove that every algebra of ShIS is isomorphic to a subalgebra of a member of ShIS whose underlying bounded semilattice is the bounded semilattice of upsets of a poset.