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

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.