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
Full-text PDF
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