Graph homomorphisms on rectangular matrices over divisionrings I

Abstract

Let ${\mathbb{D}}^{m\times~n}$ be the set of $m\times~n$ matrices over a division ring $\mathbb{D}$.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 $\mathbb{D},~\mathbb{D}'$ are division rings with $|\mathbb{D}|\geq~4$ and $m,n,m',n'\geq2$ are integers.Using the geometric method, we characterize every non-degenerate graph homomorphism$\varphi$ from ${\mathbb{D}}^{m\times~n}$ to ${\mathbb{D}'}^{m'\times~n'}$ if $\varphi(0)=0$ and $\varphi$ preserves the dimensions of two standard maximal adjacent setsof different types in ${\mathbb{D}}^{m\times~n}$. As a 推论, when $\mathbb{D}$ is an EAS (every endomorphism to be automatically surjective) division ring, we get algebraic formulas of every non-degenerate graph homomorphismfrom ${\mathbb{D}}^{m\times~n}$ to $\mathbb{D}^{m'\times~n'}$.