G-2013-67
Strong Independence and Injectivity in Tropical Modules
BibTeX reference
Tropical algebra is the study of algebraic structures (tropical maps, idempotent semimodules, etc), constructed over the semifield R_=R∪{−∞}
with the max operator (resp. usual addition) as 1st (resp.2nd) composition law.
After showing first that the concept of injectivity module of a tropical map A:R_m→R_n
introduced in [Wagneur, E., The Whitney embedding theorem for tropical torsion modules. Classification of tropical modules, Linear Algebra and its Applications, 435, 2011, 1786-1795] fails to be sufficient for the characterisation of ImA
, we introduce the concept of injectivity domain, and study some of the properties of a matrix ΓA
whose columns are given by the saturation of the equivalence class (modulo A
) of the canonical basis of R_m
. Numerous examples are provided.
Published September 2013 , 12 pages