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We consider the four families of recognizable, synchronous,
deterministic rational and rational subsets of a direct product
of free monoids.
They form a strict hierarchy and we investigate the following
decision 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 unique
generator and fill some gaps in the general case.
In particular, adapting a proof of Stearns, we show that it is recursively decidable
whether...
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