Bounds on the k-dimension of Products of Special Posets

摘要

Trotter conjectured that dim P × Q ≥ dim P ＋ dim Q － 2 for all posets P and Q. To shed light on this, we study the /c-dimension of products of finite orders. For k ∈ o(ln n), the value 2dim_k(P) - dim_k(P × P) is unbounded when P is an n-element antichain, and 2dim_2(mP) - dim_2(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, dim_k(S_m × S_n) = m ＋ n － min{2, k － 2}. For higher-order products of "standard" orders, dim_2(∏_i~l=_1S_(ni)) = ∑_(ni) if each ni ≥ t.