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Equational Classes of Distributive Pseudo-Complemented Lattices

Published online by Cambridge University Press:  20 November 2018

K. B. Lee
K. B. Lee*
Affiliation:
McMaster University, Hamilton, Ontario
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A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 881 - 891
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Balbes, R. and Horn, A., Stone lattices (preprint copy).Google Scholar
2. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, rev. éd. (Amer. Math. Soc. (New York) Providence, R.I., 1960).Google Scholar
3. Bruns, G., Ideal representations of Stone lattices, Duke Math. J. 82 (1965), 555–556.Google Scholar
4. Frink, O., Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514.Google Scholar
5. Gràtzer, G., A generalization on Stone's representation theorem for Boolean algebras, Duke Math. J. 80 (1963), 469–474.Google Scholar
6. Gràtzer, G. and Schmidt, E. T., On a problem of M. H. Stone, Acta Math. Acad. Sci. Hungar. 8 (1957), 455–460.Google Scholar
7. Jönsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121.Google Scholar
8. Nachbin, L., Une propriété caractéristique des algèbres booléiennes, Portugal. Math. 6 (1947), 115–118.Google Scholar
9. Speed, T. P., On Stone lattices, J. Austral. Math. Soc. 9 (1969), 297–307.Google Scholar
10. Varlet, J., On the characterization of Stone lattices, Acta Sci. Math. Szeged 27, fasc. 1-2, (1966), 81–84.Google Scholar