Rational series

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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\langle A \rangle.

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

\sum_{w \in A^*} c(w) w \ .

The set of formal series is denoted R\langle\langle A \rangle\rangle and becomes a semiring under the operations

c+d : w \mapsto c(w) + d(w) \
c\cdot d : w \mapsto \sum_{uv = w} c(u) \cdot 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

\sum_{w \in L} w \

corresponding to the characteristic function of L.

In R\langle\langle A \rangle\rangle one can define an operation of iteration expressed as

 S^* = \sum_{n \ge 0} S^n \

and formalised as

c^*(w) = \sum_{u_1 u_2 \cdots u_n = w} c(u_1)c(u_2) \cdots c(u_n) \ .

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\langle A \rangle.

See also[edit]


  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. Zbl 0819.11044. 

Further reading[edit]

  • Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. Part IV (where they are called \mathbb{K}-rational series). ISBN 978-0-521-84425-3. Zbl 1188.68177. 
  • 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