Lattice structure of Grid-Tamari orders
Thomas McConville
Abstract
The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. We consider a larger class of posets, the Grid-Tamari orders, which arise as an ordering on the facets of the non-kissing complex introduced by Pylyavskyy, Petersen, and Speyer. In addition to Tamari orders, some interesting examples of Grid-Tamari orders include the Type A Cambrian lattices and Grassmann-Tamari orders. We prove that the Grid-Tamari orders are congruence-uniform lattices, which resolves a conjecture of Santos, Stump, and Welker. Towards this goal, we define a closure operator on sets of paths in a square grid, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grid-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets.
中文速览
Tamari序（Tamari order）是代数组合学中一个核心对象，本质上是结合律的传递闭包，已知它具有"同余一致格"（congruence-uniform lattice）这一精细的代数结构；但将其推广到更大的格-Tamari序（Grid-Tamari order）后，这一性质是否仍然成立一直是个悬而未决的猜想。本文通过在方格网路径上定义一种闭包算子，构造出"双闭集"（biclosed sets）的包含序格，并证明它是同余一致格；进一步证明格-Tamari序恰好是该格的商格，从而确认格-Tamari序也是同余一致格，解决了Santos、Stump和Welker提出的猜想。这一结果统一涵盖了经典Tamari格、A型Cambrian格和Grassmann-Tamari序等多个重要例子，为研究三角剖分与路径组合背后的格论结构提供了系统性框架。
原文 arXiv:1504.05213;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1504.05213v2