Geodesic
Semi-Heyting Algebras and Identities of Associative Type
Bulletin of the Section of Logic, Tome 48 (2019) no. 2.

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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. ℋ 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 ℋ 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 ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of ℋ.
Keywords: semi-Heyting algebra, Heyting algebra, identity of associative type, subvariety of associative type 
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     author = {Cornejo, Juan M. and Sankappanavar, Hanamantagouda P.},
     title = {Semi-Heyting {Algebras} and {Identities} of {Associative} {Type}},
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Cornejo, Juan M.; Sankappanavar, Hanamantagouda P. Semi-Heyting Algebras and Identities of Associative Type. Bulletin of the Section of Logic, Tome 48 (2019) no. 2. http://geodesic.mathdoc.fr/item/BSL_2019_48_2_a3/