# Closure operators, frames, and neatest representations

## TL;DR

This paper characterizes when the lattice of closure-closed sets forms a frame, relates lattice properties to distributivity, and proves a poset has an $(,3)$-representation under certain conditions, answering a known question.

## Contribution

It provides a necessary and sufficient condition for the lattice of closure-closed sets to be a frame and links lattice distributivity to the failure of being a frame, also establishing a new representation result for certain posets.

## Key findings

- Lattice of closure-closed sets forms a frame under specific conditions.
- Failure to be a frame is equivalent to failure of $\sigma$-distributivity.
- Posets with a certain distributive property have an $(,3)$-representation.

## 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.