Simple image set of linear mappings in a max–min algebra

For a given linear mapping, determined by a square matrix AA in a max–min algebra, the set SASA consisting of all vectors with a unique pre-image (in short: the simple image set of AA) is considered. It is shown that if the matrix AA is generally trapezoidal, then the closure of SASA is a subset of the set of all eigenvectors of AA. In the general case, there is a permutation ππ, such that the closure of SASA is a subset of the set of all eigenvectors permuted by ππ. The simple image set of the matrix square and the topological aspects of the problem are also described.