Аннотация:We investigate different notions of linear independence
and of matrix rank that are relevant for max-plus or tropical semirings.
The factor rank and tropical rank have already received attention,
we compare them with the ranks defined in terms of signed tropical
determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra,
establishing properties of linear spaces, linear systems,
and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.