Graph homomorphisms on rectangular matrices over division rings I

Abstract: Let $\mathbb{D}$ be a division ring, and let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency, ${\mathbb{D}}^{m\times n}$ is a connected graph. Suppose that $m,n,m',n'\geq2$ are integers and $\mathbb{D}'$ is a division ring. Using the weighted semi-affine map and algebraic method, we characterize graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${\mathbb{D}'}^{m'\times n'}$ (where $|\mathbb{D}|\geq 4$) under some weaker conditions.