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.

## Contents

* 1 Definition
* 2 See also
* 3 References
* 4 Further reading

## Definition

Let R be a semiring and A a finite alphabet.

A non-commutative polynomial over A is a finite formal sum of words over A. They form a semiring R⟨A⟩.

A formal series is a R-valued function c, on the free monoid A*, which may be written as

∑w∈A*c(w)w.

The set of formal series is denoted R⟨⟨A⟩⟩ and becomes a semiring under the operations

c+d:w↦c(w)+d(w)
c⋅d:w↦∑uv=wc(u)⋅d(v)

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.[1]

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

∑w∈Lw

corresponding to the characteristic function of L.

In R⟨⟨A⟩⟩ one can define an operation of iteration expressed as

S*=∑n≥0Sn

and formalised as

c*(w)=∑u1u2⋯un=wc(u1)c(u2)⋯c(un).

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 R⟨A⟩.

## See also

* Formal power series
* Rational language
* Rational set
* Hahn series (Malcev–Neumann series)
* Weighted automaton

## References

1. ↑ Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci.. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 167. ISBN 3-540-63003-1.

* Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0.

## Further reading

* Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. Part IV (where they are called 𝕂-rational series). ISBN 978-0-521-84425-3.
* Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1
* Sakarovitch, J. Rational and Recognisable Power Series. Handbook of Weighted Automata, 105–174 (2009). doi:10.1007/978-3-642-01492-5_4
* W. Kuich. Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, Chapter 9, pages 609–677. Springer, Berlin, 1997

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