The rank of a formal tree power series

Abstract

We prove that a tree series S: T_Σ → K (K a field) is recognizable iff dim V_s < ∞ iff dim F_s < ∞, where V_s (resp. F_s) is the subspace if K^P_Σ (resp. K^T_Σ) generated by the vectors t^-1 S=Σ_{τϵ PΣ} (S, tτ)τ, tϵ τ)t, τϵPΣ), and where T_Σ is the set of all trees over Σ and P_Σ is the free monoid of all pruned trees over Σ.

We finally give a Myhill-like criterion for tree-recognizability.