Congruence Preserving Functions on Free Monoids
Résumé
A function on an algebra is congruence preserving
if, for any congruence, it maps congruent elements to congruent elements.
We show that, on a free monoid generated by at least 3 letters,
a function from the free monoid into itself is congruence preserving
if and only if it is of the form $x \mapsto w_0 x w_1 \cdots w_{n-1} x w_n$
for some finite sequence of words $w_0,\ldots,w_n$.
We generalize this result to functions of arbitrary arity.
This shows that a free monoid with at least three generators is a (noncommutative)
affine complete algebra. Up to our knowledge,
it is the first (nontrivial) case of a noncommutative
affine complete algebra.