  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.
