The ultimate rank of tropical matrices

Pierre Guillon, Zur Izhakian, Jean Mairesse, Glenn Merlet

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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
StatePublished - 1 Sep 2015
Externally publishedYes


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


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