Equational Elements in Additive Algebras

Abstract.

Let K and Γ be a semiring and a finite ranked alphabet, respectively. We consider K -Γ -algebras with additive carriers and fully multilinear operations. The free objects in this category are the K -Γ -algebras $K\langle \langle T_{\Gamma} (X)\rangle \rangle$ of formal power series on trees with coefficients in K .

We study equational elements in such structures. More precisely we establish the following classical result of Mezei and Wright: the equational elements of $K\langle \langle T_{\Gamma} \rangle \rangle$ are the recognizable formal power series, whereas each equational element of an additive K -Γ -algebra $\cal A$ is the image via the unique morphism $H_{\cal A}: K \langle \langle T_{\Gamma} \rangle \rangle \rightarrow {\cal A}$ of a recognizable formal power series.

As a consequence we get closure of the family Eq( $\cal A$ ) under recognizable substitution. The following version of Kleene's theorem is also established: the recognizable series on trees form the smallest family including polynomials and closed under OI substitution and star.

A yield situation is a triple ( ${\cal A}, Y, {\cal M}$ ) formed by an additive algebra $\cal A$ (resp. semialgebra $\cal M$ ) and a fully linear function $Y: {\cal A} \rightarrow {\cal M}$ transforming all operations in $\cal A$ into products in $\cal M$ . It is shown that for each algebraic element m of a semialgebra $\cal M$ , there exist a yield situation ( ${\cal A}, Y, {\cal M}$ ) and an equational element q ∈ $\cal A$ such that m = Y(q) , thus generalizing corresponding well-known results in mathematical language theory. Several applications are given.