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Transactions of the American Mathematical Society

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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
MathSciNet review: 1016806
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Abstract: A cardinal representation of a preclosure $ \varphi $ on a set $ E$ is a family $ \mathcal{A} \subseteq \mathcal{P}(E)$ such that for any set $ X \subseteq \cup \mathcal{A},\varphi (X) = E$ holds if and only if $ \vert X \cap A\vert= \vert A\vert$ for every $ A \in \mathcal{A}$. 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 $ k$ is finite and a closure has no independent set of size $ k + 1$, then it has a cardinal representation, $ \mathcal{A}$, of size $ \vert\mathcal{A}\vert \leq k$ (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 $ k + 1$ ($ k$ finite), then it has a cardinal representation $ \mathcal{A}$ of size $ \vert\mathcal{A}\vert \leq k$ (Theorem 2.7).

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Keywords: Cardinal number, directed graph, quasi-order, algebraic and topological closures, lattice, breadth, independent set
Article copyright: © Copyright 1991 American Mathematical Society

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