What’s the difference between concatenation and cross product of any 2 languages in finite automata?

Question

I have come across many examples of concatenation and cross product but I still face difficulty in figuring out when to use either of them. Since both are combining the properties of any 2 languages I don’t know what makes both of them individually special. It’s confusing to pin point the clear differences between them. I hope if anyone can clarify this doubt of mine, I will be very grateful.

Answer 1

The operation of concatenation of strings (and by extension, languages) is well-known. I have never seen a definition of cross product of languages.

If talking about finite automata, one could call the "cross product" of two automata the automaton whose states are pairs of states, one of each of the inputs, and which follows the transitions of each automaton in the respective state. This construction is useful to prove that if $L_1$ and $L_2$ are regular (accepted by regular automata) then $L_1 \cap L_2$ is also regular (start in the respective start states, accept if both automata end up in accepting states). Similar constructions serve for all binary set operations (union, intersection, symmetric difference, ...).