### Abstract

Iteration semi-rings are Conway semi-rings satisfying Conway's group identities. We show that the semi-rings N^{rat} {left double angle bracket} Σ^{*} {right double angle bracket} of rational power series with coefficients in the semi-ring N of natural numbers are the free partial iteration semi-rings. Moreover, we characterize the semi-rings N_{∞rat} {left double angle bracket} Σ^{*} {right double angle bracket} as the free semi-rings in the variety of iteration semi-rings defined by three additional simple identities, where N_{∞} is the completion of N obtained by adding a point of infinity. We also show that this latter variety coincides with the variety generated by the complete, or continuous semirings. As a consequence of these results, we obtain that the semi-rings N_{∞}^{rat} {left double angle bracket} Σ^{*} {right double angle bracket}, equipped with the sum order, are free in the class of symmetric inductive ^{*}-semi-rings. This characterization corresponds to Kozen's axiomatization of regular languages.

Original language | English |
---|---|

Pages (from-to) | 793-811 |

Number of pages | 19 |

Journal | Information and Computation |

Volume | 207 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 2009 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Information Systems
- Computer Science Applications
- Computational Theory and Mathematics

## Fingerprint Dive into the research topics of 'Axiomatizing rational power series over natural numbers'. Together they form a unique fingerprint.

## Cite this

*Information and Computation*,

*207*(7), 793-811. https://doi.org/10.1016/j.ic.2009.02.003