JI-distributive dually quasi-De Morgan linear semi-Heyting algebras

The main purpose of this paper is to axiomatize the join of the variety DPCSHC of dually pseudocomplemented semi-Heyting algebras generated by chains and the variety generated by D2, the De Morgan expansion of the four element Boolean Heyting algebra. Toward this end, we first introduce the variety DQDLNSH of dually quasi-De Morgan linear semi-Heyting algebras defined by the linearity axiom and the variety JIDSH of JI-distributive dually quasi-De Morgan semiHeyting algebras, and present some properties thereof. We then give an explicit description of simple (= subdirectly irreducible) algebras in the variety JIDLNSH1 of JI-distributive dually quasi-De Morgan linear semi-Heyting algebras of level 1 by applying the results of [24] and [25], from which we deduce our main theorem that says that JIDLNSH1 is the join of the variety DPCSHC and the variety V(D2), solving the problem mentioned above. We give some applications of this theorem. First, it is shown that the lattice of nontrivial subvarieties of JIDLNSH1 is isomorphic to the direct product of an infinite chain with a top element and the 2-element chain. We, then, present (small) bases for all of the subvarieties of JIDLNSH1. Finally, we show that all subvarieties of DPCSHC have the amalgamation property. The paper concludes with some open problems for further investigation.