Simple image set of (max,+) linear mappings

Let us denote a + b = max(a,b) and a x b = a + b for a,b is an element of R and extend this pair of operations to matrices and vectors in the same way as in conventional linear algebra, that is if A = (a(ij)), B = (h(ij)), C = (C-ij) are real matrices or vectors of compatible sizes then C = A x B if c(ij) = Sigma(k)(+) a(ik) x b(kj) for all i,j. If A is a real n x n matrix then the mapping x --> A x x from R-n to R-n (n > 1) is neither subjective nor injective. However, for some of such mappings (called strongly regular) there is a nonempty subset (called the simple image set) of the range, each element of which has a unique pre-imagic. We present a description of simple image sets, from which criteria for strong regularity follow. We also prove that the closure of the simple image set of a strongly regular mapping f is the image of the kth iterate of f after normalization for any k greater than or equal to n - 1 or, equivalently, the set of fixed points of f after normalization. (C) 2000 Elsevier Science B.V. All rights reserved.