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Pré-Publication, Document De Travail Année : 2016

Congruence Preserving Functions on Free Monoids

Patrick Cegielski
  • Fonction : Auteur
Grigorieff Serge
  • Fonction : Auteur

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.
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Dates et versions

hal-01360242 , version 1 (05-09-2016)

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  • HAL Id : hal-01360242 , version 1

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Irene Guessarian, Patrick Cegielski, Grigorieff Serge. Congruence Preserving Functions on Free Monoids. 2016. ⟨hal-01360242⟩
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