Strong Independence and Injectivity in Tropical Modules

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.