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 three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form \({x \mapsto w_{0}xw_{1} \cdots w_{n-1}xw_n }\) 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. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra.
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Presented by M. Jackson.
This work was partially supported by TARMAC ANR agreement 12 BS02 007 01.
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Cégielski, P., Grigorieff, S. & Guessarian, I. Congruence preserving functions on free monoids. Algebra Univers. 78, 389–406 (2017). https://doi.org/10.1007/s00012-017-0464-x
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DOI: https://doi.org/10.1007/s00012-017-0464-x