Expansions of Semi-Heyting Algebras I: Discriminator Varieties

Abstract

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [48] and [50] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH. We also use it to prove that the variety DQSSH of dually quasi-Stone semiHeyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH. Finally, we apply these results to give bases for "small" subvarieties of BDQDSH, DQSSH, and DblSH.

Semi-Heyting Algebras and Identities of Associative Type

“…Indeed, it turned out to be the case. Conjecture 1 was settled later in [39], with the help of both semi-Heyting algebras and (a subvariety of) semi-De Morgan algebras.) Definition 1.1.…”

De Morgan Semi-Heyting and Heyting Algebras

“…The variety DQDSH of semi-Heyting algebras with a dually quasi-De Morgan negation was introduced and investigated in [12]. Several important subvarieties of DQDSH were also introduced in the same paper, some of which are the following: Subvarieties of DQD of level n, for n ∈ ω, the variety DMSH of De Morgan (symmetric) semi-Heyting algebra, and DmsStSH of dually ms Stone semi-Heyting algebras.…”

“…Several important subvarieties of DQDSH were also introduced in the same paper, some of which are the following: Subvarieties of DQD of level n, for n ∈ ω, the variety DMSH of De Morgan (symmetric) semi-Heyting algebra, and DmsStSH of dually ms Stone semi-Heyting algebras. The work of [12] was continued in [13], [14], [15], [16] and [17]. We also note that the variety DMSH is an equivalent algebraic semantics for the propositional logic, called "De Morgan semi-Heyting logic" which is recently introduced in [3].…”

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The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, for n ∈ ω, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH 1 -algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH 1 -algebras of level 1 satisfy Stone identity.

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A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions

“…In 2011, motivated by the similarities of the results and proofs in [San85] and [San87], the secondnamed author introduced in [San11] a more general variety of algebras called "dually hemimorphic semi-Heyting algebras" as expansions of semi-Heyting algebras by a dual hemimorphisma common generalization of De Morgan operation and the dual psedocomplementation. Definition 1.1 [San11] An algebra A = A; ∧, ∨, →, ′ , 0, 1 is a semi-Heyting algebra with a dual hemimorphism (or dually hemimorphic semi-Heyting algebra) if A satisfies the following conditions:…”