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The least extension of a closure operator on a sublattice

Published online by Cambridge University Press:  24 October 2008

Roy O. Davies
Roy O. Davies
Affiliation:
University of Leicester
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Extract

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditions

for all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have

.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 1 , January 1967 , pp. 9 - 10
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

(1)Birkhoff, G.Lattice theory, revised edition (American Mathematical Society, New York, 1948).Google Scholar
(2)Everett, C. J.Closure operators and Galois theory of lattices. Trans. Amer. Math. Soc. 55 (1944), 514–525.CrossRefGoogle Scholar
(3)Ward, Morgan.The closure operators of a lattice. Ann. of Math. (2) 43 (1942), 191–196.CrossRefGoogle Scholar