Let $(B,\leq)$ be a dense, linearly ordered set with maximum and minimum element and $(\oplus,\otimes)=(\max, \min)$. We say that an $(m,n)$ matrix $A=(a_1,a_2,\ldots,a_n)$ has: (i) weakly linearly independent $(WLI)$ columns if for each vector $b$ the system $A\otimes x=b$ has at most one solution; (ii) regularly linearly independent columns $(RLI)$ if for each vector $b$ the system $A\otimes x = b$ is uniquely solvable; (iii) strongly linearly independent columns $(SLI)$ if there exist vectors $d_1,d_2,\ldots,d_r$, $r\geq0$ such that for each vector $b$ the system $(a_1,\ldots, a_n, d_1,\ldots,d_r)\otimes x = b$ is uniquely solvable. For these linear independences we derive necessary and sufficient conditions which can be checked by polynomial algorithms as well as their coherences with definition of matroids.