A Generalization of Hiraguchi's: Inequality for Posets

For a poset X, Dim(X) is the smallest positive integer t for which X is isomorphic to a subposet of the Cartesian product of t chains. Hiraguchi proved that if 1 X I > 4, then Dim(X) < [ X l/2]. For each k Q 2, we define Dim,(X) as the smallest positive integer t for which Xis isomorphic to a subposet of the Cartesian product oft chains, each of length k. We then prove that if 1 X I > 5, Dim,(X) & {I X1/2) and if / X ~ > 6, then Dim,(X) Q [i X1/2].