Closure operators, frames, and neatest representations

Abstract

Given a poset $P$ and a standard closure operator $\Gamma:\wp(P)\to\wp(P)$ we give a necessary and sufficient condition for the lattice of $\Gamma$-closed sets of $\wp(P)$ to be a frame in terms of the recursive construction of the $\Gamma$-closure of sets. We use this condition to show that given a set $\mathcal{U}$ of distinguished joins from $P$, the lattice of $\mathcal{U}$-ideals of $P$ fails to be a frame if and only if it fails to be $\sigma$-distributive, with $\sigma$ depending on the cardinalities of sets in $\mathcal{U}$. From this we deduce that if a poset has the property that whenever $a\wedge(b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(\omega,3)$-representation. This answers a question from the literature.