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Relative annihilators in semilattices

Published online by Cambridge University Press:  17 April 2009

J.C. Varlet
J.C. Varlet
Affiliation:
Institut de Mathématique, Université de Liège, B-4000, Liège, Belgique.
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Abstract

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An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 2 , October 1973 , pp. 169 - 185
Copyright
Copyright © Australian Mathematical Society 1973

References

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