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

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    We study Green's J-order and J-equivalence for the semigroup of all n× n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between certain 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≠J for the semigroup of all 3 × 3 matrices over the tropical semiring with -∞, but, in contrast, we show that D=J for all full matrix semigroups over the finitary tropical semiring. © 2012 Elsevier B.V.
    Original languageEnglish
    Pages (from-to)280-292
    Number of pages12
    JournalJournal of Pure and Applied Algebra
    Issue number2
    Publication statusPublished - Feb 2013


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