Linear Independences In Bottleneck Algebra And Their Coherences With Matroids

Let (B; ) be a dense, linearly ordered set with maximum and minimum element and (\Phi;\Omega ) = (max; min). We say that an (m; n) matrix A = (a 1 ; a 2 ; : : : ; an ) has: (i) weakly linearly independent (WLI) columns if for each vector b the system A\Omega x = b has at most one solution; (ii) regularly linearly independent columns (RLI) if for each vector b the system A\Omega x = b is uniquely solvable; (iii) strongly linearly independent columns (SLI) if there exist vectors d 1 ; d2 ; : : : ; dr , r 0 such that for each vector b the system (a1 ; : : : ; an ; d1 ; : : : ; dr )\Omega 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