  Our first motivation was the question: can a countable structure have an
automorphism group, which a free uncountable group? This is answered negatively
in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked
whether I am really speaking on Polish groups. We can prove this using a more
restrictive condition on the set of equations. Parallel theorems, hold for semi
groups and for metric algebras, e.g. with non-isolated unit. Here we do the
general case. For instance we show that there is no Polish group which as a
group is free and uncountable.
