The Core Label Order of a Congruence-Uniform Lattice

TL;DR

This paper studies the core label order of congruence-uniform lattices, characterizing when it forms a lattice, and explores its properties and connections to other lattice structures.

Contribution

It provides an equivalent characterization for the core label order to be a lattice and shows this property is inherited by lattice quotients.

Findings

Core label order is always a lattice for lattices from simplicial hyperplane arrangements.

The paper characterizes when the core label order forms a lattice in general.

It connects core label lattices to Boolean lattices and biclosed set lattices.

Abstract

We investigate the alternate order on a congruence-uniform lattice $\mathcal{L}$ as introduced by N. Reading, which we dub the core label order of $\mathcal{L}$. When $\mathcal{L}$ can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general $\mathcal{L}$, however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets.