Acyclic reorientation lattices and their lattice quotients

TL;DR

This paper characterizes when the acyclic reorientation poset of a DAG forms a lattice and explores its algebraic properties, introducing combinatorial models and geometric constructions for its congruences and quotients.

Contribution

It provides a complete characterization of the lattice structure of acyclic reorientation posets and introduces the concept of ropes to analyze their algebraic and geometric properties.

Findings

The acyclic reorientation poset is a lattice iff the transitive reduction of any induced subgraph is a forest.

The lattice is always congruence normal and semidistributive if and only if the graph is filled.

The lattice is distributive iff the graph is a forest.

Abstract

We prove that the acyclic reorientation poset of a directed acyclic graph $D$ is a lattice if and only if the transitive reduction of any induced subgraph of $D$ is a forest. We then show that the acyclic reorientation lattice is always congruence normal, semidistributive (thus congruence uniform) if and only if $D$ is filled, and distributive if and only if $D$ is a forest. When the acyclic reorientation lattice is semidistributive, we introduce the ropes of $D$ that encode the join irreducibles acyclic reorientations and exploit this combinatorial model in three directions. First, we describe the canonical join and meet representations of acyclic reorientations in terms of non-crossing rope diagrams. Second, we describe the congruences of the acyclic reorientation lattice in terms of lower ideals of a natural subrope order. Third, we use Minkowski sums of shard polytopes of ropes to construct a quotientope for any congruence of the acyclic reorientation lattice.