TY - JOUR
T1 - The ultimate rank of tropical matrices
AU - Guillon, Pierre
AU - Izhakian, Zur
AU - Mairesse, Jean
AU - Merlet, Glenn
N1 - Publisher Copyright:
© 2015 Elsevier Inc.
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
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AN - SCOPUS:84929380131
SN - 0021-8693
VL - 437
SP - 222
EP - 248
JO - Journal of Algebra
JF - Journal of Algebra
ER -