Sur le monoide syntactique de L∗ lorsque L est un langage fini

Abstract

We prove that for every finite monoid M, there exists a finite language L so that M divides the syntactic monoid of L ∗. Moreover, one can choose for L a full finite prefix code. The same result for finite group has already been proved by Schützenberger. This result is crucial in the proof of the J.-F. Perrot's theorem that the only variety closed under star operation is the variety of all rational languages.