Group for Research in Decision Analysis

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.

, 12 pages