We present here theoretical results coming from the implementation of the package called AMULT (automata with multiplicities in several noncommutative variables). We show that classical formulas are "almost every time" optimal, characterize the dual laws preserving rationality and also relators that are compatible with these laws. boolean coefficients). In 1974, for the case of fields, Fliess [6] extended the proof of the equivalence of minimal linear representations, using Hankel matrices. All these results allow us to construct an algorithmic processing for this series and their associated operations. In fact, classical constructions of language theory have multiplicity analogues which can be used in every domain where linear recurrences between words are handled. All these operations can be found in the package over automata with multiplicities (called AMULT). This package is a component of the environment SEA (Symbolic Environment for Automata) under development at the University of Rouen.The structure of this paper is the following: In section 3 (the first section after introductory paragraphs), we recall the classical construction for simple rational laws (+, ., * , ×) and make some remarks concerning in particular the non-commutative case. The compositions are based on polynomial formulas which has an important consequence on composition of automata choosen "at random". In fact, this first result says that the classical formulas are "almost everywhere" optimal (which is clear from experimental tests at random).In section 4, we show that the three laws known to preserve rationality ( Hadamard, shuffle and infiltration products) are of the same nature: they arise by dualizing alphabetic morphisms. Moreover, they are, up to a deformation, the only ones of this kind, which of course, shows immediately in the implemented formulas.Section 5 is devoted to study the compatibility with relators. It was well known that, when coefficients are taken in a ring of characteristic 0, the only relators compatible with the shuffle were partial commutations ([3]). Here, we show that a similar result holds (up to the supplementary possibility of letters erasure) when K is a semiring which is not a ring. This implies the known case as a corollary. To end with, we give examples of some strange relators in characteristic 2.
PreambleLet K A be the set of noncommutative formal series with A a finite alphabet and K a semiring (commutative or not). A series denoted S = w∈A * S|w w is recognizable iff there exists a row vector λ ∈ K 1×n , a morphism of monoids µ : A * → K n×n and a column vector γ ∈ K n×1 , such that for all w ∈ A * , one has S|w = λµ(w)γ. Throughout the paper, we will denote by S : (λ, µ, γ) this property and say that (λ, µ, γ) is a linear repre-
