Cardinal representations for closures and preclosures

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 $|X \cap A|= |A|$ 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 $|\mathcal {A}| \leq k$ (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 $|\mathcal {A}| \leq k$ (Theorem 2.7).