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Semi-de Morgan algebras

Published online by Cambridge University Press:  12 March 2014

Hanamantagouda P. Sankappanavar
Hanamantagouda P. Sankappanavar*
Affiliation:
Department of Mathematics and Computer Science, State University of New York at New Paltz, New Paltz, New York 12561
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Extract

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 Lakser's 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 712 - 724
Copyright
Copyright © Association for Symbolic Logic 1987

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References

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