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A Representation Theorem for Relatively Complemented Distributive Lattices

Published online by Cambridge University Press:  20 November 2018

Philip Nanzetta
Philip Nanzetta*
Affiliation:
Case Western Reserve University, Cleveland, Ohio
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In this note, we are concerned with the following generalization of a wellknown theorem of M. H. Stone; see (2, 8.2).

Theorem 1. Let L be a relatively complemented distributive lattice.

(I) If L has no least element, then L is isomorphic to the lattice of non-empty compact-open subsets of an anti-Hausdorff, nearly-Hausdorff, T1-space with a base of open sets consisting of compact-open sets.

(II) (3, Theorem 1) If L has a least element, then L is isomorphic to the lattice of all compact-open subsets of a locally compact totally disconnected space. Moreover, the spaces of (I) and (II) are compact if and only if L has a greatest element.

The space in question is the space of prime ideals of L with the hull-kernel topology.

The author is indebted to M. G. Stanley for several conversations concerning this note.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 756 - 758
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This research was supported by NSF grant GP 5301.

References

1. de Groot, J., An isomorphism principle in general topology, Bull. Amer. Math. Soc, 73 (1967), 465–467.Google Scholar
2. Sikorski, R., Boolean algebras, 2nd ed. (Springer-Verlag, New York, 1964).Google Scholar
3. Stone, M. H., Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc, 41 (1937), 375–481.Google Scholar