Title: 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.