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. 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. 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 R ⟨ A ⟩ {displaystyle Rlangle A angle } . 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 R ⟨ ⟨ A ⟩ ⟩ {displaystyle Rlangle langle A angle angle } 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.