Some non finitely generated monoids of repetition-free endomorphisms.
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 \(\text{Card} (A)\geqslant2\), 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 \(\text{Card} (A)\geqslant3)\), the monoid of \(k\)-power-free (uniform) endomorphisms on \(A\) (when \(\text{Card} (A)\geqslant2\) and \(k\geqslant3)\).
scientific article; zbMATH DE number 1856509