Semidistrim Lattices

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.

Homomesy via toggleability statistics

“…Rowmotion, as we have defined it, acts on a distributive lattice. There has been recent interest [Bar19,DW23,TW19a,TW19b] in extending rowmotion to act on other classes of lattices (or even other classes of posets). In [Hop19], the toggleability statistics technique is used to establish some homomesies for rowmotion acting on certain semidistributive lattices.…”