The embedding theorem for tropical modules

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.