Note on Dilworth’s decomposition theorem for partially ordered sets

Abstract

Let P be a finite partially ordered set with elements 1, 2, , n and order relation denoted by >. A chain in P is a set of one or more elements ii, i2, *, ik with il >i2> . . .>ik. A decomposition of P is a partition of P into chains; a decomposition with the smallest number of chains is minimal. Two members i, j of P are unrelated if neither i>j nor j>i. Dilworth [2, Theorem 1.1 ] has proved that the number of chains in a minimal decomposition of P is equal to the maximal number of mutually unrelated elements of P. Recently Dantzig and Hoffman [I ] have formulated the problemn of finding a minimal decomposition of P as a transportation-type linear programming problem, and have shown that Dilworth's theorem follows from the duality theorem of linear inequality theory. Our aim here is to show that Dilworth's theorem can be deduced from the following theorem of Konig [4, p. 232 ]. Let L be a linear graph with node set N and suppose N is partitioned into fixed subsets N1, N2. An N1, N2 cut C of L is a subset of N having the property that every arc joining a node of N1 to a node of N2 has some node of C as end point, and no proper subset of C has this property. An N1, N2 join J of L is a set of arcs of L, each of which joins a node of N1 to a node of N2, and no two of which have a node in common. K5nig's theorem, applied to the given N1, N2 partition of L, asserts that maxr I JI = minc I C| (where |SI denotes the number of elements in set S), the maximum being taken over all N1, N2 joins J, the minimum over all N1, N2 cuts C. A proof of this theorem has also been given by Egervary [3]. We proceed to a deduction of Dilworth's theorem. Given the partially ordered set P = 1, 2, . . *, n }, let L be the linear graph consisting of 2n nodes, labeled a,, * , a, b1, . . . , b, and arcs defined from P by the rule: If i>j, then aibj is an arc of L; these are all the arcs of L. Let Nl1= {a,, . . , an}, N2= {bl, . .. , bn}. Henceforth all joins and cuts are relative to N1, N2.