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

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