Is there a well-defined division operation on finite automata? Background:

Given two deterministic finite automata A and B, we form the product C by letting the states in C be the cartesian product of states in A and states in B. Then, we choose transitions, initial state, and final states so the language accepted by C is the intersection of the languages for A and B.

Questions:

(1) Can we "divide" C by B to find A? Is A even unique, up to isomorphism? We care about the state diagrams, not the languages here and below. Thus, we don't allow compressing the state diagrams to reduce the number of states.

(2) If A is unique, is there an efficient algorithm to find it?

(3) Does every deterministic finite automaton have a unique factorization into "primes". A prime here means an automaton that cannot be factored, that is, written as a product of 2 smaller automata.


Work with @MichaelWehar
Take a look at this MFCS 2013 paper, which studies compositionality in automata. Perhaps it will help.