### Abstract

A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among others. In the present paper, we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix, which does not depend on the underlying notion of rank. Furthermore, we provide a simple formula for the ultimate rank of a matrix, which can therefore be computed in polynomial time. Then we turn our attention to finitely generated semigroups of matrices, for which our notion of ultimate rank is generalized naturally. We provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank. As a consequence, we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal.

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

Pages (from-to) | 222-248 |

Number of pages | 27 |

Journal | Journal of Algebra |

Volume | 437 |

Early online date | 25 May 2015 |

DOIs | |

Publication status | Published - 1 Sep 2015 |

### Fingerprint

### Keywords

- Matrix semigroups
- Max-plus (tropical) algebra
- Ranks of matrices
- Strongly polynomial algorithm
- Tropical matrices

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*437*, 222-248. https://doi.org/10.1016/j.jalgebra.2015.02.026

**The ultimate rank of tropical matrices.** / Guillon, Pierre; Izhakian, Zur; Mairesse, Jean; Merlet, Glenn.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 437, pp. 222-248. https://doi.org/10.1016/j.jalgebra.2015.02.026

}

TY - JOUR

T1 - The ultimate rank of tropical matrices

AU - Guillon, Pierre

AU - Izhakian, Zur

AU - Mairesse, Jean

AU - Merlet, Glenn

PY - 2015/9/1

Y1 - 2015/9/1

N2 - A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among others. In the present paper, we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix, which does not depend on the underlying notion of rank. Furthermore, we provide a simple formula for the ultimate rank of a matrix, which can therefore be computed in polynomial time. Then we turn our attention to finitely generated semigroups of matrices, for which our notion of ultimate rank is generalized naturally. We provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank. As a consequence, we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal.

AB - A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among others. In the present paper, we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix, which does not depend on the underlying notion of rank. Furthermore, we provide a simple formula for the ultimate rank of a matrix, which can therefore be computed in polynomial time. Then we turn our attention to finitely generated semigroups of matrices, for which our notion of ultimate rank is generalized naturally. We provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank. As a consequence, we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal.

KW - Matrix semigroups

KW - Max-plus (tropical) algebra

KW - Ranks of matrices

KW - Strongly polynomial algorithm

KW - Tropical matrices

UR - http://www.scopus.com/inward/record.url?scp=84929380131&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2015.02.026

DO - 10.1016/j.jalgebra.2015.02.026

M3 - Article

AN - SCOPUS:84929380131

VL - 437

SP - 222

EP - 248

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -