Bounds on the k-dimension of Products of Special Posets

Abstract

Trotter conjectured that dimP×Q≥dimP+dimQ−2 for all posets P and Q. To shed light on this, we study the k-dimension of products of finite orders. For k ∈ o(ln n), the value 2dimk(P)−dimk(P×P) is unbounded when P is an n-element antichain, and 2dim2(mP)−dim2(mP×mP) is unbounded when P is a fixed poset with unique maximum and minimum. For products of the “standard” orders S m and S n of dimensions m and n, dimk(Sm×Sn)=m+n−min{2,k−2} . For higher-order products of “standard” orders, dim2(∏ti=1Sni)=∑ni if each n i ≥ t.