The dimension of the cartesian product of posets

AbstractWe give a characterization of nonforced pairs in the cartesian product of two posets, and apply this to determine the dimension of P × Q, where P, Q are some subposets of 2n and 2m respectively. One of our results is dim Sn0×Sm0=n + m − 2 for n, m⩾3. This generalizes Trotter's result in [5], where he showed that dim Sn0 × Sn0=2n − 2. We also disprove the following conjecture [2]: If P, Q are two posets and 0,1 ∈ P, then dim P × Q ⩾dim P+ dim Q − 1