Bounds on the k-dimension of Products of Special Posets

Abstract

Trotter conjectured that \(\dim\, P\times Q\ge \dim P+\dim Q-2\) 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 \(2{\dim_k}(P)-{\dim_k}(P\times P)\) is unbounded when P is an n-element antichain, and \(2{\dim_2}(mP)-{\dim_2}(mP\times mP)\) 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, \(\dim_k(S_m\times S_n)=m+n-\min\{2,k-2\}\). For higher-order products of “standard” orders, \({\dim_2}(\prod_{i=1}^t S_{n_i}) = \sum n_i\) if each n _i  ≥ t.