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The rank of a formal tree power series. (English) Zbl 0537.68053
A formal power series on trees is a mapping from a free algebra \(T_{\Sigma}\) into a field K; recall that \(\Sigma\) is a set of symbols of functions \(\Sigma =\Sigma_ 0\cup...\cup \Sigma_ n\cup...\) and that \(T_{\Sigma}\) (the set of trees) is defined inductively by: \(\Sigma_ 0\subset T_{\Sigma};\) if \(f\in \Sigma_ n\) and \(t_ 1,...,t_ n\in T_{\Sigma},\) then \(f(t_ 1,...,t_ n)\in T_{\Sigma}.\) A representation of \(T_{\Sigma}\) is a family of mappings \(\mu_ n:\Sigma_ n\to {\mathcal L}(V^ n,V),\) the space of all n-linear mappings \(V^ n\to V,\) where V is a finite-dimensional vector space over K. The mapping \(\mu_ 0:\Sigma_ 0\to V (\simeq {\mathcal L}(V^ 0,V))\) extends uniquely to a mapping \(T_{\Sigma}\to V\) through the formula \(\mu(f(t_ 1,...,t_ n))=\mu_ n(f)(\mu t_ 1,...,\mu t_ n).\) If \(\lambda:V\to K\) is linear, then the formal power series \(t\mapsto \lambda {\mathbb{O}}\mu(f)\) is called recognizable. The authors give a Hankel-like characterization of recognizable series. Let \(\kappa\) be a new variable and define \(\Sigma '\!_ 0=\Sigma_ 0\cup \kappa,\) T’\({}_{\Sigma}\) the resulting free algebra and \(P_{\Sigma}\) the subset of \(T'_{\Sigma}\) consisting of trees with exactly one occurrence of \(\kappa\). \(P_{\Sigma}\) acts naturally on \(T_{\Sigma}\) by substitution: \(T_{\Sigma}\times P_{\Sigma}\to T_{\Sigma}, (t,\tau)\mapsto t\tau.\) If S is a formal power series, \(S=\sum_{t\in T_{\Sigma}}(S,t)t,\) and if \(t\in T_{\Sigma}, \tau \in P_{\Sigma},\) define \(t^{-1}S=\sum_{\tau \in P_{\Sigma}}(S,t\tau)\tau, S\tau^{- 1}=\sum_{t\in T_{\Sigma}}(S,t\tau)t.\) Then S is recognizable iff the set of \(t^{-1}S (t\in T_{\Sigma})\) has finite rank in \(K^{P_{\Sigma}},\) iff these to \(S\tau^{-1}(\tau \in P_{\Sigma})\) has finite rank in \(K^{T_{\Sigma}}\). In this case, these ranks are equal.
Reviewer: Ch.Reutenauer

MSC:
68Q70 Algebraic theory of languages and automata
17A50 Free nonassociative algebras
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[1] Berstel, J.; Reutenauer, C., Recognizable formal power series on trees, Theoret. comput. sci., 18, 115-148, (1982) · Zbl 0485.68077
[2] Fliess, M., Matrices de Hankel, J. math. pures appl., 53, 197-222, (1974) · Zbl 0315.94051
[3] Salomaa, A.; Soittola, M., Automata-theoretic aspects of formal power series, (1978), Springer Berlin · Zbl 0377.68039
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