Varieties of recognizable tree series over fields

We introduce varieties of recognizable Σ-tree series (KΣ-VTS for short) over a field K and a ranked alphabet Σ. Our variety theorem establishes a bijective correspondence between these KΣ-VTSs and the varieties of finite-dimensional KΣ-algebras (KΣ-VFDA for short); a KΣ-algebra is a K-vector space equipped with multilinear Σ-operations. The link between KΣ-VTSs and KΣ-VFDAs is provided by the syntactic KΣ-algebras of tree series. The most immediate predecessors of this study are Berstel’s and Reutenauer’s (1982) [2], work on tree series over fields, Reutenauer’s (1980) [27] theory of varieties of string series, Bozapalidis’ and his associates (1983, 1989, 1991) [8,5,4] work on syntactic KΣ-algebras, Steinby’s (1979, 1992) [30,31] theory of varieties of tree languages, and our previous work (2009) on series of general algebras and their syntactic KΣ-algebras.