The dimension of the Cartesian product of partial orders (Q1061151)

Let P and Q be posets. The maximum of the dimensions of the posets is a lower bound of the dimension of the cartesian product \(P\times Q\) and the sum of the dimensions of the posets is an upper bound. The author is interested in problems of the lower bound. He shows, that for each \(n\geq 3\), the crown \(S^ 0_ n\) satisfies the equality \(\dim (S^ 0_ n\times S^ 0_ n)=2n-2\) and he reminds, that there is no example known for which \(\dim (P\times Q)<\dim P+\dim Q-2\).