Hanamantagouda P. Sankappanavar

Semi-de Morgan algebras

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Laksers characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]). In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

Heyting Algebras with a Dual Lattice Endomorphism

In [14] we characterized non-regular subdirectly irreducible pseudocomplemented De Morgan algcbras, while the corresponding problem for regular algebras was left open. Our attempt to solve this problem led us naturally to consider Heyting algebras with a De Morgan operation. In this paper we examine Heyting algebras with a dual lattice endomorphism (OH-algebras, for short), with special emphasis on Heyting algebras with a De Morgan operation (i.e. De Morgan-Heyting algebras). Using the notion of a regular filter introduced in [14], i t is shown that the congruences on OHalgebras are determined by regular filters. This basic result is then applied to characterize the directly indecomposables, simples, finitely subdirectly irreducibles and subdirectly irreducibles in the variety of De Morgan-Heyting algebras. As consequences, we obtain a sequence of subvarieties which are discriminator varieties, and the #esult that finite simple De Morgan-Heyting algebras are quasiprimal. It turns out that the variety of De Morgan-Heyting algebras does not have equationally definable principal congruences. The functions on OH-algebras having congruence substitution property are also characterized. These results also hold for regular pseudocomplemented De Morgan algebras.

Expansions of Semi-Heyting Algebras I: Discriminator Varieties

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 semi- Heyting 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.

Quasi-Stone algebras

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras (QSAs). Among other things we characterize the class of simple QSAs and the class of subdirectly irreducible QSAs. It follows from this characterization that the subdirectly irreducible QSAs form an elementary class and that the variety of QSAs is locally finite. Furthermore we prove that the lattice of subvarieties of QSAs is an (ω + 1)-chain. MSC: 03G25, 06D16, 06E15.

Semisimple varieties of implication zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876–884, 1934) gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein’s theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21–50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties,

On implicator groupoids

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A Course in Universal Algebra

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