Semi-Heyting Algebras and Identities of Associative Type

Abstract

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras.  They share several important properties with Heyting algebras.  An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of 𝒮ℋ.