Tropical cones defined by max-linear inequalities

We consider a system of inequalities of the type A . x <= B . x over the idempotent semifield IRmax = (IR boolean OR {-infinity}, max, +), where A, B are matrices of size m x n with coefficients in IRmax, and try to determine the set of its solutions. For the case m = 1, we show that, for every k(0 <= k <= n), the set of solutions to a single inequality with A = (a(1),...,a(n)), and B = (b(1),...,b(n)) is an IRmax semi-module of dimension k(n + 1 - k), and determine its basis, where k is the number of a(i) <= b(i) (0 <= i <= n). We provide the necessary and sufficient conditions for the solution to be non trivial, and, in the case m = n = 3, determine all pairs (A, B) such that MA,B is non trivial. We also proceed to a detailed study of generators in the case n >= m = 2. We conclude the paper with two examples for m = 2, n = 7, and m = n = 3, respectively.