We consider the four families of recognizable, synchronous,deterministic rational and rational subsets of a direct productof free monoids.They form a strict hierarchy and we investigate the followingdecision problem: given a relation in one of the families,does it belong to a smaller family?We settle the problem entirely when all monoids have a uniquegenerator and fill some gaps in the general case.In particular, adapting a proof of Stearns, we show that it is recursively decidablewhether or not a deterministic subset of an arbitrarynumber of free monoids is recognizable.Also we exhibit a single exponential algorithmfor determining if a synchronous relation is recognizable.
