Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T14:57:33.129Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal congruences in de Morgan algebras

Published online by Cambridge University Press:  20 January 2009

M. E. Adams
M. E. Adams
Affiliation:
State University of New York, New Paltz, New York 12561, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 3 , October 1987 , pp. 415 - 421
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Baker, K. A., Primitive satisfaction and equational problems for lattices and other algebras, Trans. Amer. Math. Soc. 190(1974), 125–150.Google Scholar
2.Balbes, R. and Dwinger, Ph., Distributive Lattices (University of Missouri Press, Columbia, Missouri, 1974).Google Scholar
3.Blyth, T. S. and Varlet, J. C., Congruences on A/S-algebras, Bull. Soc. Roy. Sci. Liege 53 (1984), 341–362.Google Scholar
4.Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer-Verlag, New York, 1981).CrossRefGoogle Scholar
5.Cornish, W. H. and Fowler, P. R., Coproducts of de Morgan algebras, Bull. Austral. Math. Soc. 16(1977), 1–13.CrossRefGoogle Scholar
6.Cornish, W. H. and Fowler, p. R., Coproducts of Kleene algebras, J. Austral. Math. Soc. Ser. A 27 (1979), 209–220.Google Scholar
7.Davey, B. A. and Duffus, D., Exponentiation and duality, in Ordered Sets (ed. Rival, I.), NATO Advanced Study Institutes Series (D. Reidel, Dordrecht, 1982), pp. 43–96.Google Scholar
8.Goldberg, M. S., Distributive p-algebras and Ockham Algebras; A Topological Approach (Ph.D. Thesis, La Trobe University, Australia, 1979).Google Scholar
9.Gratzer, G., Lattice Theory: First Concepts and Distributive Lattices (Freeman, San Francisco, California, 1971).Google Scholar
10.Kalman, J. A., Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), 485–491.CrossRefGoogle Scholar
11.Lakser, H., Principal congruences of pseudocomplemented distributive lattices, Proc. Amer. Math. Soc. 37 (1973), 32–36.CrossRefGoogle Scholar
12.Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.CrossRefGoogle Scholar
13.Priestley, H. A., Ordered sets and duality for distributive lattices, Ann. Discrete Math. 23 (1984), 39–60.Google Scholar
14.Sankappanavar, H. P., A characterization of principal congruences of de Morgan algebras and its applications, in Mathematical Logic in Latin America (ed. Arruda, A. I., Cuaqui, R. and da Costa, N. C. A.) (North-Holland (North-Holland, Amsterdam, 1980), pp. 341–349.Google Scholar