Varieties of recognizable tree series over fields
For a field \(K\) and a (finite) ranked alphabet \(\Sigma\) and a leaf alphabet (i.e., a finite set of variables) \(X\), a tree series (or a \(K\Sigma X\)-series) is a mapping from the term algebra \(T_\Sigma(X)\) to \(K\). To each tree series, a special \(K\Sigma\)-algebra (called its syntactic \(K\Sigma\)-algebra) is associated; a \(\Sigma K\)-algebra is a \(K\)-vector space equipped with multi-linear \(\Sigma\)-operations (i.e., a \(\Sigma\)-algebra whose every \(\Sigma\)-operation is linear in all components). A tree series is recognizable if (and only if) its syntactic algebra is finite dimensional. A variety of tree series is a family \({\mathcal V}=\{{\mathcal V}(X)\}\), where \(X\) ranges over all leaf alphabets, such that for each \(X\), the class \({\mathcal V}(X)\) is a \(K\)-vector space of recognizable \(K\Sigma X\)-series closed under inverse translations and inverse homomorphisms. A variety of finite dimensional \(K\Sigma\)-algebras is a class closed under subalgebras, homomorphic images and direct products of finite families of algebras. The main result of the paper is a variety theorem that establishes a bijection between varieties of tree (\(K\Sigma X\)-)series and varieties of finite dimensional \(K\Sigma\)-algebras. Some interesting examples are presented at the end. The authors note that another theory of varieties of tree series has been proposed in [\textit{S. Bozapalidis} and \textit{S. Ioulidis}, ``Varieties of formal series on trees and Eilenberg's theorem'', Inf. Process. Lett. 29, No. 4, 171--175 (1988; Zbl 0658.08002)] with no (concrete) examples, and since that approach is quite different from the authors', the two theories have not been compared in the present paper.
scientific article; zbMATH DE number 5853807
scientific article