Semidistrim Lattices

TL;DR

This paper introduces semidistrim lattices, a new class generalizing semidistributive and trim lattices, and explores their properties, graph representations, and dynamic operators like rowmotion and pop-stack sorting.

Contribution

The paper defines semidistrim lattices, proves their key properties, and establishes connections between rowmotion, pop-stack sorting, and Galois graph structures.

Findings

Elements correspond to independent sets in the Galois graph

Products and intervals of semidistrim lattices are semidistrim

Order complex is contractible or homotopy equivalent to a sphere

Abstract

We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim, and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere. Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard's $\overline\kappa$ map on semidistributive lattices as well as Thomas and the second author's rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element $x$ to the meet of the elements covered by $x$. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.