Geodesic
Lattice structure of Grassmann-Tamari orders
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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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. 
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     author = {McConville, Thomas},
     title = {Lattice structure of {Grassmann-Tamari} orders},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2460},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2460/}
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McConville, Thomas. Lattice structure of Grassmann-Tamari orders. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2460. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2460/

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