### Abstract

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

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) |

Publisher | Springer Verlag |

Pages | 42-56 |

Number of pages | 15 |

Volume | 2030 |

ISBN (Print) | 3540418644 |

Publication status | Published - 2001 |

Event | 4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001 - Genova, Italy Duration: Apr 2 2001 → Apr 6 2001 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2030 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001 |
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Country | Italy |

City | Genova |

Period | 4/2/01 → 4/6/01 |

### Fingerprint

### Keywords

- Commutative idempotent weak semirings
- Complete axiomatizations
- Convexity
- Equational logic
- Exponential time complexity
- Relative axiomatizations
- Tropical semirings
- Varieties

### 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. 2030, pp. 42-56). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2030). Springer Verlag.

**Axiomatizing tropical semirings.** / Aceto, Luca; Ésik, Z.; Ingólfsdóttir, Anna.

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. 2030, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2030, Springer Verlag, pp. 42-56, 4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001, Genova, Italy, 4/2/01.

}

TY - GEN

T1 - Axiomatizing tropical semirings

AU - Aceto, Luca

AU - Ésik, Z.

AU - Ingólfsdóttir, Anna

PY - 2001

Y1 - 2001

N2 - This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

AB - This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

KW - Commutative idempotent weak semirings

KW - Complete axiomatizations

KW - Convexity

KW - Equational logic

KW - Exponential time complexity

KW - Relative axiomatizations

KW - Tropical semirings

KW - Varieties

UR - http://www.scopus.com/inward/record.url?scp=23044526057&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044526057&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:23044526057

SN - 3540418644

VL - 2030

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

SP - 42

EP - 56

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

PB - Springer Verlag

ER -