Lattice structure of Grid-Tamari orders

TL;DR

This paper proves that Grid-Tamari orders, a broad class of posets including Tamari and Cambrian lattices, are congruence-uniform lattices by analyzing biclosed sets of grid paths and their quotient structure.

Contribution

It establishes that all Grid-Tamari orders are congruence-uniform lattices, confirming a conjecture and generalizing known structures like Tamari and Cambrian lattices.

Findings

Grid-Tamari orders are congruence-uniform lattices.

Biclosed sets of paths form a congruence-uniform lattice.

Grid-Tamari order is a quotient of biclosed sets lattice.

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.