Embedding finite posets in cubes

ln this paper we defme the n-cube Qn as the poset obtained by t&kmg the Cartesian product sf n chains each zonsjsting of two points. For a finite poset X, we then define dims A as tY;e smallest positive integer n such that X can be embedded as a subpos:: t of QC For any poset X we then have Ilog2 WI 6 dim2 X G IXr. For the distributive Mice f, = z*, dim2 L 5 iXt and for the crown $, dim2(3$) =: n + k. For each k 3 2, there exist positive constants ~1 8%nd c2 so that for the poset X consisting of all one element and k-element subsets of an nelement set, the inequality cl log2 n < dim&Q < ca togs 1'1 holds for all n with k < n. iA goset is called Q-arritical if dimz(X-x) 6 dim2(x) fcr every x E X. We define a join operation @ on posets under which the collection Q of 311 Q-critical powts which are not ehai;ls forms a semigroup in which unique factorizatdool holds. We then completely determine the subcollecticln C Q consisting of all poszts X for which dimzCX) = 1x1. A partially ordered set or poset is a pair (A', P) where x' is a set and P is a reflexive, antisymmetric, and transitive relation on X'. The notations (x, V)E P, x dc;y in P, and x Py are (used interc:hangeably. If neither (x, JJ) nor @, x) is in P, we say x and y are inczornparable and write