Group for Research in Decision Analysis

G-2015-66

The dominant of a matrix - application to the classification of tropical modules

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.

, 18 pages