Formal Series of General Algebras over a Field and Their Syntactic Algebras

Any mapping S : C → K, where K is a field and C = (C,Σ) is a Σ-algebra, is called a KC-series. These series are natural generalizations of both formal series on strings over a field and the tree series introduced by Berstel and Reutenauer [1]. We consider various operations on KC-series and their effects on the linear extensions of the series. We also study some algebraic aspects of theKΣ-algebra formed by the KC-polynomials (i.e., the KC-series with finitely many non-zero coefficients); a KΣ-algebra is a Σ-algebra based on a K-vector space in which all the Σ-operations are multilinear. The syntactic congruence of a KC-series S is a congruence on this KΣ-algebra of KC-polynomials, and the syntactic KΣ-algebra SA(S) of S is the corresponding quotient algebra. These syntactic algebras generalize Reutenauer’s syntactic K-algebras of string series [13] and the syntactic KΣ-algebras of tree series studied by Bozapalidis et al. [5, 4, 3]. It is shown that SA(S) is finite-dimensional iff the series S is recognizable. We also characterize the subdirectly irreducible KΣ-algebras and show that all of them are syntactic. Furthermore, we show how various operations on KC-series relate to the syntactic KΣ-algebras.