Bounds on the k-dimension of Products of Special Posets

Trotter conjectured that dim P × 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 lnn), the value 2 dimk(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 Sm and Sn of dimensions m and n, dimk(Sm × Sn) = m + n −min{2, k − 2}. For higher-order products of “standard” orders, dim2( ∏t i=1 Sni) = ∑ ni if each ni ≥ t.