Z-matrix equations in max algebra, nonnegative linear algebra and other   semirings
  We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to that in non-negative linear algebra, characterizing the solvability in terms of supports and access relations. We give a description of the solution set as combination of the least solution and the eigenspace of the matrix, and provide a general algebraic setting in which this result holds. 
Peter Butkovic, Hans Schneider, Sergei Sergeev
math.RA