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A Condition for Equality of Cardinals of Minimal Generators Under Closure Operators

Published online by Cambridge University Press:  20 November 2018

Japheth Hall Jr
Japheth Hall Jr*
Affiliation:
Stillman College, Tuscaloosa, Alabama
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Let C be an operator on the subsets of a set X with values among the subsets of X. We assume that C is a closure operator in X, i.e. a monotone, idempotent and extensive operator in X (cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). If A ⊆ X and B ⊆ X, we say that A and B are C-equivalent if C(A) = C(B) (Bleicher- Marczewski [4, p. 210]).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 4 , December 1971 , pp. 569 - 570
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Schmidt, J., Einige grundlegende Begriffe und Satz aus der Theorie der Hullenoperatoren, Bericht uber mathematische Tagung, Berlin (1953), 21-48.Google Scholar
2. Schmidt, J., Mehrstufige Austauschstrukturen, Z. Math. Logik Grundlagen Math. 2 (1956), 233-249.Google Scholar
3. Birkhoff, G., Lattice theory, rev. Ed., Colloq. Publ., Amer. Math. Soc, New York, 1948.Google Scholar
4. Bleicher, M. N. and Marczewski, E., Remarks on dependence relations and closure operators, Colloquium Mathematicum IX (1962), 209-211.Google Scholar
5. Bleicher, M. N. and Preston, B. B., Abstract linear dependence relations, Publ. Math. Debrecen 8 (1961), 55-63.Google Scholar
6. Taylor, A. E., Introduction to Functional Analysis, Wiley, New York, 1964.Google Scholar