Automorphisms of the MAX-matrix system (Q1823292)

Let \(F=R\cup \{-\infty \}\) where R is the set of the real numbers. A set of \(n\times n\) matrices \(A=(a_{ij})\) with \(a_{ij}\in F\) is called a system of MAX-matrices if the operations \(\oplus\) and \(\otimes\) are defined for any pair A, B in the following way: \(A\oplus B=C=(c_{ij})=(\max \{a_{ij},b_{ij}\}),\) \(A\otimes B=D=(d_{ij})=(\max_{k}\{a_{ik}+b_{kj}\}).\) The author shows that all automorphisms of a system of MAX-matrices are interior; in particular, if \(n\neq 6\), any automorphism \(\phi\) is given by the relation \(X^{\phi}=A\otimes X\otimes A^{-1}\cdot a\) for all matrices X of the system where A is an invertible MAX-matrix and a is a positive scalar (the dot denotes ordinary multiplication).