Some non finitely generated monoids of repetition-free endomorphisms
Résumé
We answer a question raised by Mitrana in Information Processing Letters 64 about primitive morphisms, that is, morphisms that preserve primitiveness of words. Given an alphabet A with Card(A) >= 2, the monoid of primitive endomorphisms on A and the monoid of primitive uniform endomorphisms on A are not finitely generated. Moreover we show that it is also the case for the following monoids: the monoid of overlap-free (uniform) endomorphisms on A (when Card(A) >= 3), the monoid of k-power-free (uniform) endomorphisms on A (when Card(A) >= 2 and k >= 3).