Extending the scalars of minimizations

In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of Σ∗ where Σ is a set of symbols (or the alphabet). Now, given a semiring (K,+, .), one can construct K-subsets of Σ∗ in the sense of Eilenberg [5], that are alternatively called noncommutative formal power series [2, 13] for which a framework very similar to language theory has been constructed (see [11, 12] and [2]). This extension has applications in many domains. Let us cite, for example, enumeration (non-commutative as used by instance for alignment of genomic sequences), image processing [3], task-ressource problems [8] and real-time applications where multiplicities are used to prove the modularity of the validation method by means of the Hadamard product of two integer valued automata (see the contribution by Geniet and Dubernard [9]). Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or K-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of Kautomata already exist for K being: a) a field [2], b) a noncommutative field [7], c) a PID [6]. When K is the bolean semiring, such a minimization process (with isomor-