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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

REFERENCES

[1] Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, Columbia, Missouri, 1974.Google Scholar
[2] Berman, J., Congruence relations of pseudocomplemented distributive lattices, Algebra Universalis, vol. 3(1973), pp. 288–293.CrossRefGoogle Scholar
[3] Berman, J., Distributive lattices with an additional unary operation, Aequationes Mathematicae, vol. 16 (1977), pp. 165–171.CrossRefGoogle Scholar
[4] Birkhoff, G., Lattice theory, 3rd ed., American Mathematical Society, Providence, Rhode Island, 1967.Google Scholar
[5] Burris, S. and Sankappanavar, H. P., A course in universal algebra, Springer-Verlag, New York, 1981.Google Scholar
[6] Frink, O., Pseudocomplements in semilattices, Duke Mathematical Journal, vol. 29 (1962), pp. 505–514.Google Scholar
[7] Glivenko, V., Sur quelques points de la logique de M. Brouwer, Bulletin de l'Académie desSciences de Belgique, vol. 15 (1929), pp. 183–188.Google Scholar
[8] Grätzer, G., Lattice theory. First concepts and distributive lattices, Freeman, San Francisco, California, 1971.Google Scholar
[9] Grätzer, G. and Lakser, H., The structure of pseudocomplemented distributive lattices. II: Congruence extension and amalgamation, Transactions of the American Mathematical Society, vol. 156 (1971), pp. 343–358.Google Scholar
[10] Grätzer, G. and Lakser, H., The structure of pseudocomplemented distributive lattices. III: Injective andabsolute subretracts, Transactions of the American Mathematical Society, vol. 169 (1972), pp. 475–487.Google Scholar
[11] Jónsson, B., Algebras whose congruence lattices are distributive, Mathematica Scandinavica, vol. 21 (1967), pp. 110–121.CrossRefGoogle Scholar
[12] Kalman, J. A., Lattices with involution, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 485–491.CrossRefGoogle Scholar
[13] Lakser, H., The structure of pseudocomplemented distributive lattices. I: Subdirect decomposition, Transactions of the American Mathematical Society, vol. 156 (1971), pp. 335–342.Google Scholar
[14] Lakser, H., Principal congruences of pseudocomplemented distributive lattices, Proceedings of the American Mathematical Society, vol. 37 (1973), pp. 32–36.CrossRefGoogle Scholar
[15] Lee, K. B., Equational classes of distributive pseudocomplemented lattices, Canadian Journal of Mathematics, vol. 22 (1970), pp. 881–891.CrossRefGoogle Scholar
[16] Rasiowa, H., An algebraic approach to nonclassical logics, North-Holland, Amsterdam, 1974.Google Scholar
[17] Ribenboim, P., Characterization of the sup-complement in a distributive lattice with last element, Summa Brasiliensis Mathematicae, vol. 2 (1949), pp. 43–49.Google Scholar
[18] Sankappanavar, H. P., Congruence lattices of pseudocomplemented semilattices, Algebra Universalis, vol. 9 (1979), pp. 304–316.CrossRefGoogle Scholar
[19] Sankappanavar, H. P., A characterization of principal congruences of De Morgan algebras and its applications, Mathematical logic in Latin America, North-Holland, Amsterdam, 1980, pp. 341–349.Google Scholar
[20] Sankappanavar, H. P., Distributive lattices with a dual endomorphism, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 385–392.CrossRefGoogle Scholar
[21] Sankappanavar, H. P., Heyting algebras with dual pseudocomplementation, Pacific Journal of Mathematics, vol. 117 (1985), pp. 405–415.CrossRefGoogle Scholar
[22] Sankappanavar, H. P., Heyting algebras with a dual (lattice) endomorphism, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988) (to appear).Google Scholar
[23] Sankappanavar, H. P., Pseudocomplemented Ockham and De Morgan algebras, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 385–394.Google Scholar
[24] Sankappanavar, H. P., Demi-p-lattices: principal congruences and subdirect irreducibility (preprint).Google Scholar