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arXiv · 1102.2707

Green's J-order and the rank of tropical matrices

Marianne Johnson·Mark Kambites
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Abstract

We study Green's J-order and J-equivalence for the semigroup of all n-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that $D \neq J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with $-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups over the finitary tropical semiring.

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Marianne Johnson, Mark Kambites. 2012-01-04. Green's J-order and the rank of tropical matrices. https://arxiv.org/abs/1102.2707

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