Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended lor-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety. In Part II, we prove, using the description of simples obtained in this Part, that the variety mathbf{RDQDStSH_1} of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element mathbf{RDQDStSH_1}-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of mathbf{RDQDStSH_1}. The Part II concludes with some open problems for further investigation.

Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity Keywords:

Regular dually , quasi-De Morgan , semi-Heyting algebra of level 1 , dually pseudocomplemented semi-Heyting algebra , De Morgan semi-Heyting algebra , strongly blended dually quasi-De Morgan Stone semi-Heyting algebra , discriminator variety , simple , directly indecomposable , subdirectly irreducible , equational base