Algebra universalis

, Volume 78, Issue 3, pp 389–406 | Cite as

Congruence preserving functions on free monoids

  • Patrick Cégielski
  • Serge Grigorieff
  • Irène Guessarian


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.

2010 Mathematics Subject Classification

Primary: 08A30 Secondary: 08B20 

Key words and phrases

congruence preservation free monoid affine completeness 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Patrick Cégielski
    • 1
  • Serge Grigorieff
    • 2
  • Irène Guessarian
    • 2
    • 3
  1. 1.LACL, EA 4219, Université Paris-Est Créteil, IUTSénart-FontainebleauFrance
  2. 2.IRIF, UMR 8243, CNRS & Université Paris 7 ParisFrance
  3. 3.Emeritus at UPMC Université Paris 6ParisFrance

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