Embedding finite posets in cubes

AbstractIn this paper we define the n-cube Qn as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim2 X as the smallest positive integer n such that X can be embedded as a subposet of Qn. For any poset X we then have log2 |X| ⩽ dim2 X ⩽ |X|. For the distributive lattice L = 2 X, dim2 L = |X| and for the crown Skn, dim2 (Skn) = n + k. For each k ⩾ 2, there exist positive constants c1 and c2 so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c1 log2 n < dim2(X) < c2 log2 n holds for all n with k < n. A poset is called Q-critical if dim2 (X − x) < dim2(X) for every x ϵ X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection M ⊆ Q consisting of all posets X for which dim2 (X) = |X|

Weak embeddings of posets to the Boolean lattice

The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs and of Patkos. As an equivalent reformulation of one of these problems, we also derive that it is NP-complete to decide whether a given graph can be embedded to the two middle levels of some hypercube

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]