Cardinal representations for closures and preclosures

Authors:
F. Galvin, E. C. Milner and M. Pouzet

Journal:
Trans. Amer. Math. Soc. **328** (1991), 667-693

MSC:
Primary 06A06; Secondary 04A20, 06A15, 54A05

DOI:
https://doi.org/10.1090/S0002-9947-1991-1016806-3

MathSciNet review:
1016806

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Abstract | References | Similar Articles | Additional Information

Abstract: A cardinal representation of a preclosure on a set is a family such that for any set holds if and only if for every . We show, for example (Theorem 2.3) that any topological closure has such a representation, but there are closures which have no cardinal representation (Theorem 11.2). We prove that, if is finite and a closure has no independent set of size , then it has a cardinal representation, , of size (Theorem 2.4). This result is used to give a new proof of a theorem of D. Duffus and M. Pouzet [4] about gaps in a lattice of finite breadth. We do not know if a closure which has no infinite independent set necessarily has a cardinal representation, but we do prove this is so for the special case of a closure on a countable set (Theorem 2.5). Even in this special case, nothing can be said about the size of the cardinal representation; however, if the closure is algebraic, then there is a finite cardinal representation (Theorem 2.6). These results do not hold for preclosures in general, but if a preclosure on a countable set has no independent set of size ( finite), then it has a cardinal representation of size (Theorem 2.7).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1016806-3

Keywords:
Cardinal number,
directed graph,
quasi-order,
algebraic and topological closures,
lattice,
breadth,
independent set

Article copyright:
© Copyright 1991
American Mathematical Society