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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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##### References:
 [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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