Classification of idempotent semi-modules with strongly independent basis

We show here that every \(m\)-dimensional semiring module \(M\) over an idempotent semiring \(S\) with strongly independent basis can be embedded into \(S^m\), and provide an algebraic invariant - the \(\Lambda\)-matrix - which characterises the isomorphy class of \(M\). The strong independence condition also yields a significant improvement to the Whitney embedding for tropical torsion modules published earlier [LAA 435, 1786-1795, 2011]. We also show that the strong independence of the basis of \(M\) is equivalent to the unique representation of elements of \(M\). Numerous examples illustrate our results, and a fast test for strong independence of the columns of a matrix is provided.