On a Kind of Fatou Property of Context-Free Groups

In [4] Berstel and Sakarovitch pointed out that from a proof given in [2] one can deduce that a rational set of a group G which is included in a subgroup H of G is already rational in H (an explicit proof is given in [12]). They called this “a kind of Fatou property of groups” according to similar properties of certain formal power series [14]. Another kind of Fatou property was shown by Eilenberg and Schtitzenberger for rational sets and commutative monoids in [7]. Naturally, the question arises whether these results remain true for other than rational sets. In this paper we restrict to context-free groups and show that in these groups the above-cited result remains true for those sets which are defined by homomorphic images of languages which belong to a cone closed under union. Especially for algebraic sets, i.e. sets which are homomorphic images of context-free languages, we obtain the following corollary: