### G-2013-67

# Strong Independence and Injectivity in Tropical Modules

## Edouard Wagneur

Tropical algebra is the study of algebraic structures (tropical maps, idempotent semimodules, etc), constructed over the semifield `\(\underline{\mathbb{R}} = {\mathbb{R}} \cup \{-\infty\}\)`

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 `\(A : \underline{\mathbb{R}}^m \rightarrow \underline{\mathbb{R}}^n \)`

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 Im`\(A\)`

, we introduce the concept of injectivity domain, and study some of the properties of a matrix `\( \Gamma^A \)`

whose columns are given by the saturation of the equivalence class (modulo `\(A\)`

) of the canonical basis of `\( \underline{\mathbb{R}}^m\)`

. Numerous examples are provided.

Published **September 2013**
,
12 pages