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

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.