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, ... , 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.