  A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector
$x$ belonging to the interval $X=\{x\colon \underline{x}\leq
x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in
$X$. The main result of this paper is a combinatorial characterization of such
matrices in the linear algebra over max-min (fuzzy) semiring.
  The characterized property is related to and motivated by the general
development of tropical linear algebra and interval analysis, as well as the
notions of simple image set and weak robustness (or weak stability) that have
been studied in max-min and max-plus algebras.
