A generalization of Hiraguchi's: Inequality for posets

AbstractFor 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 | X | ⩾ 4, then Dim(X) ⩽ [| X |/2]. For each k ⩽ 2, we define Dimk(X) as the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains, each of length k. We then prove that if | X | ⩾ 5, Dim3(X) ⩽ {| X |/2} and if | X | ⩾ 6, then Dim4(X) ⩽ [| X |/2]