THE DIMENSION OF THE CARTESIAN PRODUCT OF POSETS

We give a characterization of nonforced pairs in the cartesian product of two posets, and apply this to determine the dimension of P x Q, wher P, Q are some subposets of 2n and 2m respectively. One of our results is dim S(n)0 x S(m)0 = n + m - 2 for n, m greater-than-or-equal-to 3. This generalizes Trotter's result in [5], where he showed that dim S(n)0 x S(n)0 = 2n - 2. We also disprove the following conjecture [2]: If P, Q are two posets and 0, 1 is-a-member-of P, then dim P x Q greater-than-or-equal-to dim P + dim Q - 1.