Semi-Heyting Algebras and Identities of Associative Type

JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR

BULLETIN OF THE SECTION OF LOGIC

University of Lodz

Año: 2019 vol. 48 p. 117 - 135

0138-0680

An algebra $mathbf A = langle A, lor, land, o, 0, 1angle$ is a semi-Heyting algebra if $langle A,lor, land ,0,1angle$ is a bounded lattice, and it satisfiesthe the identities :$x land (x o y) approx x land y$, $x land (y o z) approx x land [(x land y) o (x land z)]$, and$x o x approx 1$. $mathcal{SH}$ 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 containing three distinct variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of $mathcal{SH}$ 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 $mathcal{SH}$ of asociative type of length 3. Our main result shows that there are 3 such subvarities of $mathcal{SH}$.