Congruence Preserving Functions on Free Monoids
نویسندگان
چکیده
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 three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form x 7→ w0xw1 · · ·wn−1xwn for some finite sequence of words w0, . . . , wn. 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. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra.
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