The minimum number of chains in a noncrossing partition of a poset

TL;DR

This paper introduces noncrossing partitions for posets, focusing on the minimal number of chains needed, and provides a characterization of this quantity for general posets, extending classical results.

Contribution

It defines noncrossing partitions for posets and characterizes the minimum number of chains in such partitions for arbitrary posets.

Findings

Minimum chains in noncrossing partitions reflect poset complexity

For the chain poset [n], the minimum number is one

The paper provides a characterization for the minimum number in general posets

Abstract

The notion of noncrossing partitions of a partially ordered set (poset) is introduced here. When the poset in question is $[n]=\{1,2,\dots, n\}$ with the complete order of natural numbers, conventional noncrossing partitions arise. The minimum possible number of chains contained in a noncrossing partition of a poset clearly reflects the structural complexity of the poset. For the poset $[n]$, this number is just one. However, for a generic poset, it is a challenging task to determine the minimum number. Our main result in the paper is some characterization of this quantity.