Abstract
A tropical matrix is a matrix defined over the maxplus semiring. For such matrices, there exist several noncoinciding 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 (fromto)  222248 
Number of pages  27 
Journal  Journal of Algebra 
Volume  437 
Early online date  25 May 2015 
DOIs  
Publication status  Published  1 Sep 2015 
Keywords
 Matrix semigroups
 Maxplus (tropical) algebra
 Ranks of matrices
 Strongly polynomial algorithm
 Tropical matrices
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Profiles

Zur Izhakian
 School of Natural & Computing Sciences, Mathematical Science  Reader
 Mathematical Sciences (Research Theme)
Person: Academic