Order embeddings of real matrix domains

Abstract

Let $n$ be a positive integer, $n\ne 1$, and $S_n$ the set of all $n\times n$ real symmetric matrices. A nonempty subset $\U \subset S_n$ is called a matrix domain if it is open and connected and a map $ϕ: \U \to S_n$ is said to be an order emebedding if for every pair $X,Y \in \U$ we have $X\le Y⟺\varphi (X)\le \varphi (Y)$. We describe the general form of such maps.