  Profinite congruences on profinite algebras determining profinite quotients
are difficult to describe. In particular, no constructive description is known
of the least profinite congruence containing a given binary relation on the
algebra. On the other hand, closed congruences and fully invariant congruences
can be described constructively. In a previous paper, we conjectured that fully
invariant closed congruences on a relatively free profinite algebra are always
profinite. Here, we show that our conjecture fails for unary algebras and that
closed congruences on relatively free profinite semigroups are not necessarily
profinite. As part of our study of unary algebras, we establish an adjunction
between profinite unary algebras and profinite monoids. We also show that the
Polish representation of the free profinite unary algebra is faithful.

Group Theory;Logic in Computer Science