If K is a subsemiring of a semiring L, each K-rational series is clearly L-rational. The main problem considered in this chapter is the converse: how to determine which of the L-rational series are rational over K. This leads to the study of semirings of a special type, and also shows the existence of remarkable families of rational series.

In the first section, we examine principal rings from this aspect. Fatou's Lemma is proved and the rings satisfying this lemma are characterized (Chabert's Theorem 1.5).

In the second section, Fatou extensions are introduced. We show in particular that ℚ+ is a Fatou extension of ℕ (Theorem 2.2 due to Fliess).

In the third section, we apply Shirshov's theorem on rings with polynomial identities to prove criteria for rationality of series and languages. This is then applied, in the last section, to Fatou ring extensions.

Rational series over a principal ring

Let K be a commutative principal ring and let F be its quotient field. Let S ϵ K⟪A⟫ be a formal series over A with coefficients in K. If S is a rational series over F, is it also rational over K? This question admits a positive answer, and there is even a stronger result, namely that S has a minimal linear representation with coefficients in K.

Theorem 1.1 (Fliess 1974a) Let S ϵ K⟪A⟫ be a series which is rational of rank n over F.