Group for Research in Decision Analysis

# The embedding theorem for tropical modules

## Edouard Wagneur

Tropical algebra is the algebra constructed over the tropical semifield $$R_{max}= (R\cup \{-\infty\},\max, +)$$. We show here that every $$m$$-dimensional tropical module $$M$$ over $$R_{max}$$, given by a $$n\times p$$ matrix $$A$$ can be embedded into $$R_{max}^n$$, iff $$n$$ of its rows are independent. This result yields a significant improvement to the Whitney embedding for tropical torsion modules published earlier.

, 10 pages