Poset topology of $s$-weak order via SB-labelings

TL;DR

This paper studies the topological structure of certain lattice intervals derived from generalized weak orders on permutations, revealing their homotopy types using SB-labelings.

Contribution

It introduces SB-labelings for $s$-weak order and $s$-Tamari lattices, characterizing their homotopy types as balls or spheres and determining sphere dimensions.

Findings

Intervals are homotopy equivalent to balls or spheres.

Characterization of intervals homotopy equivalent to spheres.

Determination of sphere dimensions for certain intervals.

Abstract

Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to spheres and which are homotopy equivalent to balls; we also determine the dimension of the spheres for the intervals yielding spheres.