G-2015-66
The dominant of a matrix - application to the classification of tropical modules
BibTeX reference
Tropical algebra is the algebra constructed over the tropical semifield \(\mathbb{R}_{max}\)
. After revisiting the classification of 2-dimensional \(\mathbb{R}_{max}\)
semimodules, we define here the concept of dominant of a matrix and use it to show that every m-dimensional tropical module M over \(\mathbb{R}_{max}\)
with strongly independent basis can be embedded
into \(\mathbb{R}_{max}^m\)
. We also show that - up to matrix equivalence -
the right residuate of a matrix by itself characterises the isomorphy class of the semimodule generated by its columns.
The strong independence condition also yields a significant improvement to the Whitney embedding for tropical torsion modules published earlier.
We also show that the strong independence of the basis of M is equivalent to the unique representation of elements of M.
The results are illustrated with numerous examples.
Published July 2015 , 18 pages
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G1566.pdf (1 MB)