Semidistrim Lattices
Colin Defant, Nathan F. Williams · Forum of Mathematics Sigma · 2023
Abstract We introducesemidistrim 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 theGalois 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 naturalrowmotionoperator, 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 associatedpop-stack sortingoperator that sends an elementxto the meet of the elements covered byx. 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.