The dimension of the Cartesian product of partial orders

If P and Q are partial orders, then the dimension of the cartesian product P x Q does not exceed the sum of the dimensions of P and Q. There are several known sufficient conditions for this bound to be attained, on the other hand, the only known lower bound for the dimension of a cartesian product is the trivial inequality dim(P x Q) ~> max{dim P, dim O}. In partictdar, if P has dimension n, we know only that n ~~3, the crown S ° is an n-dimensional partial order for which dim(S°xS°)=2n-2. No example for which dim(Px Q) 3, la couronne S o est un ordre partiel de dimension n pour lequel dim(S°x S~)= 2n-2. On ne connait ancun exemple pour lequel dim(Px Q)<dimP+dim Q-2.