JI-distributive, dually quasi-De Morgan semi-Heyting and Heyting algebras

TL;DR

This paper introduces new subvarieties within semi-Heyting algebras, characterizes their structure, and explores their algebraic properties, including discriminator levels, subdirect irreducibility, and lattice structures.

Contribution

It defines and investigates the JI-distributive, dually quasi-De Morgan semi-Heyting algebras and their subvarieties, providing structural characterizations and algebraic property analyses.

Findings

DSt and JID are discriminator varieties of levels 1 and 2.

JID1 is the join of DSt and De Morgan Boolean semi-Heyting algebras.

JIDL1 is the join of DStHC and a 4-element De Morgan Boolean Heyting algebra.

Abstract

The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in a series of four papers. In this paper we define and investigate a new subvariety JID of DQD, called JI-distributive, dually quasi-De Morgan semi-Heyting algebras, as well as the variety DSt of dually Stone semi-Heyting algebras. We first prove that DSt and JID are discriminator varieties of level 1 and level 2, respectively. Secondly, we give a characterization of subdirectly irreducible algebras of the subvariety JID1 of level 1 of JID. As a first application of it, we derive that the variety JID1 is the join of the variety DSt and the variety of De Morgan Boolean semi-Heyting algebras. As a second application, we give a concrete description of the subdirectly irreducible algebras in the subvariety JIDL1 of JID1 defined by the linear identity, and deduce that the variety JIDL1 is the join of the variety DStHC generated by the dually Stone Heyting chains and the variety generated by the 4-element De Morgan Boolean Heyting algebra. Several applications of this result are also given, including a description of the lattice of subvarieties of JIDL1, equational bases of all subvarieties of JIDL1, and the amalgamation property of all subvarieties of DStHC.