The ultimate rank of tropical matrices

Pierre Guillon, Zur Izhakian*, Jean Mairesse, Glenn Merlet

*Corresponding author for this work

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 languageEnglish
Pages (from-to)222-248
Number of pages27
JournalJournal of Algebra
Volume437
Early online date25 May 2015
DOIs
Publication statusPublished - 1 Sep 2015

Fingerprint

Semigroup
Finitely Generated
Semiring
Polynomial Algorithm
Polynomial time
Rank of a matrix

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

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

In: Journal of Algebra, Vol. 437, 01.09.2015, p. 222-248.

Research output: Contribution to journalArticle

Guillon, Pierre ; Izhakian, Zur ; Mairesse, Jean ; Merlet, Glenn. / The ultimate rank of tropical matrices. In: Journal of Algebra. 2015 ; Vol. 437. pp. 222-248.
@article{7e36eaf6fc2c4032a84d5db01f7673d7,
title = "The ultimate rank of tropical matrices",
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.",
keywords = "Matrix semigroups, Max-plus (tropical) algebra, Ranks of matrices, Strongly polynomial algorithm, Tropical matrices",
author = "Pierre Guillon and Zur Izhakian and Jean Mairesse and Glenn Merlet",
year = "2015",
month = "9",
day = "1",
doi = "10.1016/j.jalgebra.2015.02.026",
language = "English",
volume = "437",
pages = "222--248",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",

}

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

VL - 437

SP - 222

EP - 248

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -