Extension of Brzozowski’s derivation calculus of rational expressions to series over the free partially commutative monoids

We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called ϕ-derivations) are shown to satisfy the commutation relations.Our main result states that for every so-called rigid rational expression, there exists a stable finitely generated submodule containing it. Moreover, this submodule is generated by what we call Words, that is by products of letters and of pure stars.Consequently this submodule is free and it follows that every rigid rational expression represents a recognizable series in K《A/C》. This generalizes the previously known property where the star was restricted to mono-alphabetic and connected series.