### Abstract

A Conway semiring is a semiring S equipped with a unary operation ^{*}:S∈→∈S, called "star", satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ^{ω} : S∈→∈V, called "omega power", subject to the sum omega and product omega identities. A Kleene type theorem holds in all Conway semiring-semimodule pairs that can be instantiated to give the equivalence of Büchi automata and regular languages over ω-words. However, sometimes the star and omega power operations cannot be defined in an appropriate manner on the whole semiring S. To handle this situation, we introduce partial Conway semiring-semimodule pairs and develop their basic theory in connection with automata. We prove a Kleene theorem, applicable to all partial Conway semiring-semimodule pairs.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 56-71 |

Number of pages | 16 |

Volume | 7020 LNCS |

DOIs | |

Publication status | Published - 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 7020 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 7020 LNCS, pp. 56-71). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7020 LNCS). https://doi.org/10.1007/978-3-642-24897-9_3

**Partial conway and iteration semiring-semimodule pairs.** / Ésik, Z.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 7020 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7020 LNCS, pp. 56-71. https://doi.org/10.1007/978-3-642-24897-9_3

}

TY - GEN

T1 - Partial conway and iteration semiring-semimodule pairs

AU - Ésik, Z.

PY - 2011

Y1 - 2011

N2 - A Conway semiring is a semiring S equipped with a unary operation *:S∈→∈S, called "star", satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ω : S∈→∈V, called "omega power", subject to the sum omega and product omega identities. A Kleene type theorem holds in all Conway semiring-semimodule pairs that can be instantiated to give the equivalence of Büchi automata and regular languages over ω-words. However, sometimes the star and omega power operations cannot be defined in an appropriate manner on the whole semiring S. To handle this situation, we introduce partial Conway semiring-semimodule pairs and develop their basic theory in connection with automata. We prove a Kleene theorem, applicable to all partial Conway semiring-semimodule pairs.

AB - A Conway semiring is a semiring S equipped with a unary operation *:S∈→∈S, called "star", satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ω : S∈→∈V, called "omega power", subject to the sum omega and product omega identities. A Kleene type theorem holds in all Conway semiring-semimodule pairs that can be instantiated to give the equivalence of Büchi automata and regular languages over ω-words. However, sometimes the star and omega power operations cannot be defined in an appropriate manner on the whole semiring S. To handle this situation, we introduce partial Conway semiring-semimodule pairs and develop their basic theory in connection with automata. We prove a Kleene theorem, applicable to all partial Conway semiring-semimodule pairs.

UR - http://www.scopus.com/inward/record.url?scp=84856621044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856621044&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-24897-9_3

DO - 10.1007/978-3-642-24897-9_3

M3 - Conference contribution

AN - SCOPUS:84856621044

SN - 9783642248962

VL - 7020 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 56

EP - 71

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -