  For $P$ a poset, the dimension of $P$ is defined to be the least cardinal
$\kappa$ such that $P$ is embeddable in a direct product of $\kappa$ totally
ordered sets. We study the behavior of this function on finite-dimensional (not
necessarily finite) posets.
  In general, the dimension dim($P$ x $Q$) of a product of two posets can be
smaller than dim($P$) + dim($Q$), though no cases are known where the
discrepancy is greater than 2. We obtain a result that gives upper bounds on
the dimensions of certain products of posets, including cases where the
discrepancy 2 is achieved. But the paper is mainly devoted to stating
questions, old and new, about dimensions of product posets, noting implications
among their possible answers, and introducing some related concepts that might
be helpful in tackling these questions.
