Bounds on the k-dimension of Products of Special Posets

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.