Congruence preserving functions on free monoids |
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Authors: | Patrick Cégielski Serge Grigorieff Irène Guessarian |
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Institution: | 1.LACL, EA 4219, Université Paris-Est Créteil, IUT,Sénart-Fontainebleau,France;2.IRIF, UMR 8243, CNRS & Université Paris 7, Paris,France;3.Emeritus at UPMC Université Paris 6,Paris,France |
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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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