Sur un Théorème de Shamir

enShamir has proved that any algebraic power series over a free monoïd X*, with coefficients in an abelian semiring A, may be calculated by a representation of X* into the A-ring A〈H〉 of a half-free-group H. The same result has been proved by Nivat, with the A-ring A〈G〉 of a free group G.We state here the converse of this result: Any power series defined by such a representation is algebraic. In order to prove that, we use the following material: a particular class of transductions, which we call “regulated rational transductions”. They seem to be the good generalization for power series of the “faithfully rational transductions” for languages