Independence Posets

TL;DR

This paper introduces independence posets, a new class of partially ordered sets generalizing distributive lattices through graph-theoretic involutions, and explores their properties, lattice conditions, and connections to algebraic structures.

Contribution

It defines independence posets based on involutions on independent sets of acyclic graphs, generalizing distributive and trim lattices, and characterizes when these posets form lattices.

Findings

Independence posets generalize distributive lattices.

A graph-theoretic condition characterizes when an independence poset is a lattice.

The paper extends rowmotion to independence posets and relates them to algebraic structures.

Abstract

Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph $G$. We characterize when an independence poset is a lattice with a graph-theoretic condition on $G$. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-simpleminded collections arising in the representation theory of certain acyclic finite-dimensional algebras.