Congruence Preserving Functions on Free Monoids

Abstract

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 %nonmonogenic 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 $w_0,\dots ,w_n$. 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.

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