Semigroup identities of tropical matrices through matrix ranks

Zur Izhakian (1) , Glenn Merlet (2)

1 MD-UNIV. ABERDEEN - Mathematics Department of the University of Aberdeen
2 I2M - Institut de Mathématiques de Marseille

We prove the conjecture that, for any $n$, the monoid of all $n \times n$ 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.

Domaines: Anneaux et algèbres [math.RA]; Combinatoire [math.CO]

Zur Izhakian, Glenn Merlet. Semigroup identities of tropical matrices through matrix ranks. 2019. ⟨hal-02117595⟩