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.

Source : oai:HAL:hal-01337806v1

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)

Section: Proceedings

Published on: January 1, 2015

Submitted on: November 21, 2016

Keywords: biclosed sets,Tamari lattice,congruence-uniform,lattice quotient,noncrossing complex,Grassmann-Tamari associahedron,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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