Unrestricted State Complexity of Binary Operations on Regular Languages

I study the state complexity of binary operations on regular languages over different alphabets. It is well known that if L′ m and Ln are languages over the same alphabet with m and n quotients, respectively, the state complexity of any binary boolean operation on L′ m and Ln is mn, and that of the product (concatenation) is (m − 1)2n + 2. In contrast to this, I show that if L′ m and Ln are over their own different alphabets, the state complexity of union and symmetric difference is mn + m + n + 1, that of intersection is mn + 1, that of difference is mn+m+ 1, and that of the product is m2n + 2.