Generalization of Dilworth's Theorem on Minimal Chain Decomposition

The decomposition of a finite partially ordered set of elements as a union of chains was considered by Dilworth [2]. Dantzig and Hoffman [1] formulated this problem as a linear programming problem and obtained Dilworth's theorem from duality theory. For some practical applications and for a method to obtain a minimal decomposition see Ford and Fulkerson [3]. In this paper we generalize this problem to the case when the set is not necessarily partially ordered and obtain a method of finding a minimal chain decomposition of the set.