Recognizable subsets of some partially abelian monoids

AbstractWe show that, in a free partially abelian monoid generated by a finite alphabet A, the subset [X∗] of A∗ containing all the words equivalent to a product of words of X is rational if X is a finite set of words, each word containing at least one occurrence of any letter of A. We suppose that the graph the vertices of which are letters of A and the edges of which correspond to noncommuting pairs of letters is connected