Rational series

In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring, and the indeterminates adjoined are not assumed to commute. They can be regarded as algebraic expressions of a formal language over a finite alphabet.
Definition.
Let "R" be a semiring and "A" a finite alphabet.
A "noncommutative polynomial" over "A" is a finite formal sum of words over "A". They form a semiring formula_1.
A "formal series" is a "R"-valued function "c", on the free monoid "A"*, which may be written as
The set of formal series is denoted formula_3 and becomes a semiring under the operations
A non-commutative polynomial thus corresponds to a function "c" on "A"* of finite support.
In the case when "R" is a ring, then this is the "Magnus ring" over "R".
If "L" is a language over "A", regarded as a subset of "A"* we can form the "characteristic series" of "L" as the formal series
corresponding to the characteristic function of "L".
In formula_3 one can define an operation of iteration expressed as
and formalised as
The "rational operations" are the addition and multiplication of formal series, together with iteration.
A rational series is a formal series obtained by rational operations from formula_1.