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Characterizations for Prime Semilattices

Published online by Cambridge University Press:  20 November 2018

K. P. Shum
Affiliation:
The Chinese University of Hong Kong, Shatin, Hong Kong
M. W. Chan
Affiliation:
The Chinese University of Hong Kong, Shatin, Hong Kong
C. K. Lai
Affiliation:
The Chinese University of Hong Kong, Shatin, Hong Kong
K. Y. So
Affiliation:
The Chinese University of Hong Kong, Shatin, Hong Kong
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Throughout this paper when we refer to a semilattice S we shall mean that S is a meet semilattice. We shall denote the infimum of two elements a, b of S by ab, and the supremum, if it exists, by ab. A prime semilattice is a meet semilattice such that the infimum distributes over all existing finite suprema, in the sense that if x 1x 2 … ∨ xn exists then (xx 1) ∨ (xx 2) … ∨ (xxn ) exists for any x and equals x ∧ (x 1x 2 … ∨ xn ). Such semilattices were first studied by Balbes [1] and we use his terminology.

A non-empty subset F of S is a filter provided that xyF if and only if xF and yF.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Balbes, R., A representation theory for prime and implicative semilattices, Trans. Amer. Math. Soc. 136 (1969), 261267.Google Scholar
2. Cornish, W. C. and Hickman, R. C., Weakly distributive semilattices, Acta. Math. Acad. Sci. Hungar. 32 (1978), 516.Google Scholar
3. Grätzer, G., Lattice theory: First concepts and distributive lattices (W. H. Freeman, San Francisco, 1971).Google Scholar
4. Hickman, R. C., Distributivity in semilattices, Acta. Math. Acad. Hungarica 32 (1978), 3545.Google Scholar
5. Mandelker, M., Relative annihilators in lattices, Duke Math. J. 37 (1970), 377386.Google Scholar
6. Pawar, Y. S. and Thakare, N. K., On prime semilattices, Canad. Math. Bull. 23 (1980), 291298.Google Scholar
7. Schein, B. M., On the definition of distributive semilattices, Alg. Universalis 2 (1972), 12.Google Scholar
8. Shum, K. P. and Hoo, C. S., O-distributive and P-uniform semilattices, Canad. Math. Bull. 25 (1982), 317323.Google Scholar
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10. Varlet, J. C., Relative annihilators in semilattices, Bull. Austral. Math. Soc. 9 (1973), 169185.Google Scholar
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