Linear extensions and order-preserving poset partitions

TL;DR

This paper explores the topological structure of the lattice of order-preserving partitions of finite posets, revealing a homotopy equivalence to a wedge of spheres and linking the number of spheres to linear and cyclic extensions.

Contribution

It establishes a homotopy equivalence between the order complex of the lattice of order congruences and a wedge of spheres, connecting combinatorial properties of posets to topological invariants.

Findings

Order complex is homotopy equivalent to a wedge of spheres of dimension n-3.

Number of spheres equals the number of linear extensions for connected posets.

Number of spheres equals the number of cyclic extensions in general.

Abstract

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$.