Lattice structure of Grassmann-Tamari orders

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. In this work, we consider a larger class of posets, the Grassmann-Tamari orders, which arise as an ordering on the facets of the non-crossing complex introduced by Pylyavskyy, Petersen, and Speyer. We prove that the Grassmann-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 inside a rectangle, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grassmann-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets.

Keywords

• biclosed sets
• tamari lattice
• congruence-uniform
• lattice quotient
• noncrossing complex
• grassmann-tamari associahedron
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]