### 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 |
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Pages (from-to) | 793-811 |

Number of pages | 19 |

Journal | Information and Computation |

Volume | 207 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 2009 |

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### ASJC Scopus subject areas

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

### Cite this

*Information and Computation*,

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

**Axiomatizing rational power series over natural numbers.** / Bloom, S. L.; Ésik, Z.

Research output: Contribution to journal › Article

*Information and Computation*, vol. 207, no. 7, pp. 793-811. https://doi.org/10.1016/j.ic.2009.02.003

}

TY - JOUR

T1 - Axiomatizing rational power series over natural numbers

AU - Bloom, S. L.

AU - Ésik, Z.

PY - 2009/7

Y1 - 2009/7

N2 - Iteration semi-rings are Conway semi-rings satisfying Conway's group identities. We show that the semi-rings Nrat {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.

AB - Iteration semi-rings are Conway semi-rings satisfying Conway's group identities. We show that the semi-rings Nrat {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.

UR - http://www.scopus.com/inward/record.url?scp=67349165374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349165374&partnerID=8YFLogxK

U2 - 10.1016/j.ic.2009.02.003

DO - 10.1016/j.ic.2009.02.003

M3 - Article

AN - SCOPUS:67349165374

VL - 207

SP - 793

EP - 811

JO - Information and Computation

JF - Information and Computation

SN - 0890-5401

IS - 7

ER -