Cartesian product of regular expressions?
Gelade and Neven prove the following theorem in their paper Succinctness of the Complement and Intersection of
Regular Expressions:


  For every $n$ there exist two regular expressions $r_1,r_2$ of size $O(n^2)$ such that the smallest regular expression representing $L[r_1] \cap L[r_2]$ has size $\Omega(2^n)$.


This suggests that there is no analog of the product construction for regular expressions.