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

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Abstract


Abstract. This paper is the rst of a two part series. In this paper, we rst
prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras
of level 1 satises the strongly blended _-De Morgan law introduced in [20].
Then, using this result and the results of [20], 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 RDQDStSH1 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 RDQDStSH1-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 RDQDStSH1. The Part II concludes with some open
problems for further investigation.

Keywords


Regular dually quasi-De Morgan semi-Heyting algebra of level 1, dually pseudocomple- mented semi-Heyting algebra, De Morgan semi-Heyting algebra, strongly blended dually quasi-De Morgan Stone semi-Heyting algebra, discriminator variety, simple, directly i

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