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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
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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 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

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