Stochastic matrices and majorization in max algebra

In this paper, we introduce and characterize max-doubly stochastic matrices within the framework of max algebra, where the operations are defined as $x \oplus y = \max(x, y)$ and $x \otimes y = xy$. We explore the fundamental properties of max-doubly stochastic matrices and their role in vector majorization. Specifically, we establish that for vectors $x$ and $y$ in max algebra, $x$ is majorized by $y$ if there exists a max-doubly stochastic matrix $D$ such that $x = D \otimes y$. This provides a new approach to majorization theory within tropical mathematics and enhances the understanding of vector relations in max algebra.