Lattice structure of Grid-Tamari orders

The Tamari order is a central object in algebraic combinatorics. 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 lattices, which settles a conjecture of Santos, Stump, and Welker on the Grassmann-Tamari order. To prove the conjecture, we define a closure operator on sets of paths in a shape λ, and prove that the biclosed sets of paths, ordered by inclusion, form a lattice. We then prove that the Grid-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets. This lattice of biclosed sets generalizes the weak order on permutations. The Tamari lattice and the weak order both possess additional structure: they are congruence-uniform lattices. We prove that the lattice of biclosed sets of paths is congruence-uniform and deduce that the Grid-Tamari lattices are congruence-uniform as well.