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
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
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
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