  We show that one can decide if a rational equivalence relation can be given
as the equivalence kernel of a sequential letter-to-letter transduction. This
problem comes from the setting of games with imperfect information. In [1, p.
6] the authors propose to model imperfect information by a rational equivalence
relation and leave open the problem of deciding if one can synthesize a
sequential letter-to-letter transducer (Mealy machine) which maps equivalent
histories to the same sequence of observations. We also show that knowing if an
equivalence relation can be given as the equivalence kernel of a sequential
transducer is undecidable, even if the relation is given as a letter-to-letter
transducer.
