A forbidden subposet characterization of an order — dimension inequality

Abstract

Dushnik and Miller defined the dimension of a partially ordered setX, DimX, as the minimum number of linear extensions ofX whose intersection is the partial ordering onX. The concept of dimension for a partially ordered set has applications to preference structures and the theory of measurement. Hiraguchi proved that DimX ≤ [|X|/2] when |X| ≥ 4. Bogart, Trotter, and Kimble gave a forbidden subposet characterization of Hiraguchi's inequality by constructing for eachn ≥ 2 the minimum collectionℬ _n of posets such that if [|X|/2] =n ≥ 2, then DimX < n unlessX contains one of the posets inℬ _n . Recently Trotter gave a simple proof of Hiraguchi's inequality based on the following theorem. IfA is an antichain ofX and |X − A| =n ≥ 2, then DimX ≤ n. In this paper we give a forbidden subposet characterization of this last inequality.