Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T01:05:53.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations for Prime Semilattices

Published online by Cambridge University Press:  20 November 2018

K. P. Shum ,
M. W. Chan ,
C. K. Lai  and
K. Y. So
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
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.

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 a ∧ b, and the supremum, if it exists, by a ∨ b. A prime semilattice is a meet semilattice such that the infimum distributes over all existing finite suprema, in the sense that if x1 ∨ x2 … ∨ xn exists then (x ∧ x1) ∨ (x ∧ x2) … ∨ (x ∧ xn) exists for any x and equals x ∧ (x1 ∨ x2 … ∨ 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 x ∧ y ∊ F if and only if x ∊ F and y ∊ F.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1059 - 1073
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Balbes, R., A representation theory for prime and implicative semilattices, Trans. Amer. Math. Soc. 136 (1969), 261–267.Google Scholar
2. Cornish, W. C. and Hickman, R. C., Weakly distributive semilattices, Acta. Math. Acad. Sci. Hungar. 32 (1978), 5–16.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), 35–45.Google Scholar
5. Mandelker, M., Relative annihilators in lattices, Duke Math. J. 37 (1970), 377–386.Google Scholar
6. Pawar, Y. S. and Thakare, N. K., On prime semilattices, Canad. Math. Bull. 23 (1980), 291–298.Google Scholar
7. Schein, B. M., On the definition of distributive semilattices, Alg. Universalis 2 (1972), 1–2.Google Scholar
8. Shum, K. P. and Hoo, C. S., O-distributive and P-uniform semilattices, Canad. Math. Bull. 25 (1982), 317–323.Google Scholar
9. Varlet, J. C., On separation properties in semilattices, Semigroup Forum 10 (1975), 220–228.Google Scholar
10. Varlet, J. C., Relative annihilators in semilattices, Bull. Austral. Math. Soc. 9 (1973), 169–185.Google Scholar