Poset Edge-Labellings and Left Modularity

Crosscut-simplicial Lattices

We call a lattice crosscut-simplicial if the crosscut complex of every atomic interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and Meszaros introduced SB-labellings and proved that if a lattice has an SB-labelling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labelling include the join-distributive lattices, the weak order of a Coxeter group, and the Tamari lattice. Generalizing these three examples, we prove that every meet-semidistributive lattice is …

Conditions forcing the existence of relative complements in lattices and posets

It is elementary and well-known that if an element x of a bounded modular lattice L has a complement in L then x has a relative complement in every interval [a,b] containing x. We show that the relatively strong assumption of modularity of L can be replaced by a weaker one formulated in the language of so-called modular triples. We further show that, in general, we need not suppose that x has a complement in L. By introducing the concept of modular triples in posets, we extend our results obtained for lattices to posets. It should be remarked that the notion of a complement can be introduced a…

Left-modular elements

Left-modularity is a concept that generalizes modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial of a lattice with such an element, one of which generalizes Stanley's Partial Factorization Theorem for a geometric lattice with a modular element. Both formulae provide us with inductive proofs of Blass and Sagan's Total Factorization Theorem for LL lattices. The characteristic polynomials and Mobius functions of non-crossing partition lattices and shuffle posets are computed as examples.

极大链上的预序：超平面排列、Cambrian 格与极大 green 序列

Preorders on maximal chains: hyperplane arrangements, Cambrian lattices, and maximal green sequences

本文研究多边形格及有限 Coxeter 排列中极大链等价类上的预序，证明商映射的保序、收缩与纤维连通性，并将其推广到 Cambrian 格、高阶 Bruhat/Stasheff–Tamari 序及极大 green 序列。

半分配格中的极值性

Extremality in semidistributive lattices

研究极值、左模、同余一致和半分配格的性质，给出左模格的边标号刻画及构造条件，证明同余一致格可壳性等价于极值性，并推广Dilworth定理到半分配极值格。