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 nonnegative linear algebra [G.F. Frobenius, Über Matrizen aus nicht negativen Elementen. Sitzungsber. Kön. Preuss. Akad. Wiss., 1912, in Ges. Abh., Vol. 3, Springer, 1968, pp. 546–557; D. Hershkowitz and H. Schneider, Solutions of Z-matrix equations, Linear Algebra Appl. 106 (1988), pp. 25–38; H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey, Linear Algebra Appl. 84 (1986), pp. 161–189], 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.