Simple image set of (max, +) linear mappings

Let us denote a b= max(a;b) and a b=a+b for a;b 2 R and extend this pair of operations to matrices and vectors in the same way as in conventional linear algebra, that is if A=(aij), B=(bij), C =(cij) are real matrices or vectors of compatible sizes then C =A B if cij = P k aik bkj for all i;j. If A is a real n n matrix then the mapping x 7! A x from R n to R n (n>1) is neither surjective 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-image. 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>n 1 or, equivalently, the set of xed points of f after normalization. ? 2000 Elsevier Science B.V. All rights reserved.