On the order dimension of convex polytopes (Q915134)

Let P be a convex polytope and L(P) be its face lattice. It is shown that \(\dim_ P(P)+1<\dim_ 0(L(P))\) where \(\dim_ P(P)\) is the affine dimension of P and \(\dim_ 0(L(P))\) is the order dimension of L(P). Although equality obtains for many standard examples, it does not do so in general. In fact, it is shown that for cyclic polytopes c(n,d) the order dimension can be arbitrarily large for fixed \(d>4\).