Semigroup identities of tropical matrices through matrix ranks

2018/06/28 by Zur Izhakian, Izhakian, Zur, Glenn Merlet +1
Computer Science · Mathematics · #20M05 #20M07 #20M30 #47D03 #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #math.CO #math.RA #msc:20M05 #msc:20M07 #msc:20M30 #msc:47D03 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1806.11028

13 pages

arxiv created 2018/06/28 · openalex publication_date 2018/06/28 · arxiv updated 2018/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the conjecture that, for any n, the monoid of all n × n tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.

Citations

• A semigroup identity for tropical 3x3 matrices
• Remarks on separating words

Related

• Semigroup identities in the monoid of two-by-two tropical matrices
• Identities of tropical matrices and plactic monoids
• Semigroup identities in the monoid of triangular tropical matrices
• Tropical representations and identities of plactic monoids
• A semigroup identity for tropical 3x3 matrices