Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-04T21:32:11.883Z Has data issue: false hasContentIssue false

On rings of sets II. Zero-sets

Published online by Cambridge University Press:  09 April 2009

T. P. Speed
Affiliation:
Department of Probability of Statistics University Sheffield, S3 7RH, England
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.

In an earlier paper [11] we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illustrations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison [5] for the simple proof that Z(X) is a (2, σ)-ring of subsets of X, where we denote aleph-zero by σ.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Birkhoff, G., Lattice Theory Colloq. Publ. No. XXV, 3rd Ed., (Amer. Math. Soc. Providence, Rhode Island, 1967).Google Scholar
[2]Bourbaki, N., Elements of mathematics, General topology, Part 1 (Hermann, Paris: Addison Wesley, Reading Massachussetts, Palo Alto, London, Don Mills Ontario; 1966).Google Scholar
[3]Čech, E., Topological Spaces (Interscience Publishers, John Wiley and Sons, London, New York; Sydney, 1966).Google Scholar
[4]Cornish, W. H., ‘Normal lattices’, J. Austral Math. Soc. 14 (1972), 200215.CrossRefGoogle Scholar
[5]Gillman, L. and Jerison, M., Rings of continuous functions (D. Van Nostrand Princeton, New Jersey, Toronto, London, New York, 1960).CrossRefGoogle Scholar
[6]Gordon, H., ‘Rings of functions determined by zero sets’, Pacific. J. Math. 36 (1971), 133157.Google Scholar
[7]Kesttan, J., ‘Eine Charakterisierung der vollständig regulären Raume’, Math. Nachr. 17 (1958/1959), 2746.Google Scholar
[8]Kist, J., ‘Minimal prime ideals in commutative semigroups’, Proc. London Math. Soc. (3) 13 (1963), 3150.Google Scholar
[9]Lorch, E. R., ‘Compactification, Baire functions, and Daniell integration’, Acta. Sci. Math. Szeged, 24 (1963), 204218.Google Scholar
[10]Mandelker, M., ‘Relative annihilators in lattices’, Duke Math. J. 37 (1970), 377386.CrossRefGoogle Scholar
[11]Speed, T. P., ‘On rings of sets’, J. Austral. Math. Soc. 8 (1968), 723730.CrossRefGoogle Scholar