Graph homomorphisms on rectangular matrices over division rings II

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 $\mathrm{rank}(A-B)=1$. By the adjacency, ${\mathbb{D}}^{m\times n}$ is a connected graph. Suppose $\mathbb{D},{\mathbb{D}}^{\prime }$ are division rings and $m,n,m^{\prime },n^{\prime }\ge 2$ are integers. We determine additive graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${{\mathbb{D}}^{\prime }}^{m^{\prime }\times n^{\prime }}$. When $|\mathbb{D}|\ge 4$, we characterize the graph homomorphism $\phi :{\mathbb{D}}^{n\times n}\to {{\mathbb{D}}^{\prime }}^{m^{\prime }\times n^{\prime }}$ if $\phi (0)=0$ and there exists $A_0\in {\mathbb{D}}^{n\times n}$ such that $\mathrm{rank}(\phi (A_0))=n$. We also discuss properties and ranges on degenerate graph homomorphisms. If $f:{\mathbb{D}}^{m\times n}\to {{\mathbb{D}}^{\prime }}^{m^{\prime }\times n^{\prime }}$ (where $\min \{m,n\}=2$) is a degenerate graph homomorphism, we prove that the image of $f$ is contained in a union of two maximal adjacent sets of different types. For the case of finite fields, we obtain two better results on degenerate graph homomorphisms.

33 pages