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 κ 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.

The Image of the Pop Operator on Various Lattices

“…In this paper, we settle four conjectures about the generating functions Pop(M; q) proposed by Defant and Williams [8]. In what follows, Weak(B n ) is the weak order of type B n , Tam(B n ) is the Tamari lattice of type B n , and J(Φ + An ) and J(Φ + Bn ) are the lattices of order ideals of the root posets of types A n and B n , respectively.…”

“…A descent of a permutation x = x 1 x 2 • • • x n ∈ S n is an adjacent pair of numbers x i and x i+1 such that x i > x i+1 . For x = 51763284, the descents of x are (5, 1), (7,6), (6,3), (3,2), and (8,4). We let |des(x)| represent the number of descents in x.…”

“…Definition 2.2 (Defant and Williams, [8]). Given a lattice M and an element x ∈ M, let Pop ↑ M (x) := ({y ∈ M : x ⋖ y} ∪ {x}).…”

“…Next we define the generating function that refines the count of the elements of M that belong to Pop M (M). Definition 2.3 (Defant and Williams, [8]). For a lattice M, let…”

“…In [8], Defant and Williams proved a theorem that related the image of Pop M with the image of the dual pop-stack sorting operator, the rowmotion operator, and independent dominating sets. Motivated by their theorem, Defant and Williams defined the following generating function that refines the count of elements of M that lie in the image of Pop M :…”