Graph homomorphisms on rectangular matrices over division rings II

TL;DR

This paper characterizes additive graph homomorphisms between matrix spaces over division rings, revealing their structure, properties, and ranges, especially for finite fields and degenerate cases.

Contribution

It provides a complete characterization of additive graph homomorphisms on matrices over division rings, including degenerate cases and finite field improvements.

Findings

Characterized additive graph homomorphisms for matrices over division rings.

Described properties and ranges of degenerate graph homomorphisms.

Obtained improved results for finite fields.

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 $\mathbb{D}, \mathbb{D}'$ are division rings and $m,n,m',n'\geq2$ are integers. We determine additive graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${\mathbb{D}'}^{m'\times n'}$. When $|\mathbb{D}|\geq 4$, we characterize the graph homomorphism $\varphi: {\mathbb{D}}^{n\times n}\rightarrow {\mathbb{D}'}^{m'\times n'}$ if $\varphi(0)=0$ and there exists $A_0\in {\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_0))=n$. We also discuss properties and ranges on degenerate graph homomorphisms. If $f:{\mathbb{D}}^{m\times n}\rightarrow {\mathbb{D}'}^{m'\times n'}$ (where ${\rm min}\{m,n\}=2$) 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.

Equations

$$d(\phi (A),\phi (B))\le d(A,B)$$

$${\mathcal{M}}_i=\left\{\sum _{j=1}^n{x}_j{E}_{ij}:x_j\in \mathbb{D}\right\},{\mathcal{N}}_j=\left\{\sum _{i=1}^m{y}_i{E}_{ij}:y_i\in \mathbb{D}\right\}\subset {\mathbb{D}}^{m\times n};$$

$${\mathcal{M}}_i^{\prime }=\left\{\sum _{j=1}^{n^{\prime }}{x}_j{E}_{ij}^{\prime }:x_j\in {\mathbb{D}}^{\prime }\right\},{\mathcal{N}}_j^{\prime }=\left\{\sum _{i=1}^{m^{\prime }}{y}_i{E}_{ij}^{\prime }:y_i\in {\mathbb{D}}^{\prime }\right\}\subset {{\mathbb{D}}^{\prime }}^{m^{\prime }\times n^{\prime }}.$$

$${\mathbb{D}}_{\le k}^{m\times n}=\left\{X\in {\mathbb{D}}^{m\times n}:\mathrm{rank}(X)\le k\right\},$$

$${\mathbb{B}}_A={\mathbb{D}}_{\le 1}^{m\times n}+A,$$

$${\mathbb{N}}_A={\mathbb{D}}_1^{m\times n}+A\subset {\mathbb{B}}_A.$$

$$\ell=P\left(\begin{array}[]{cc}\mathbb{D}\alpha+\beta\\ 0\\ \end{array}\right)+A$$

$$\dim (\phi (\mathcal{S}))\le \dim (\mathcal{S}).$$

$${\rm rank}\left(\begin{array}[]{cc}a_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right)={\rm rank}\left(\begin{array}[]{cc}a_{11}-1&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right)={\rm rank}\left(\begin{array}[]{cc}a_{11}-b&A_{12}-\beta\\ A_{21}&A_{22}\\ \end{array}\right)=1.$$

$$\varphi (X)=P{X}^{\tau }Q,X\in {\mathbb{D}}^{m\times n};$$

$$\varphi (X)=P{}^t{X}^{\sigma }Q,X\in {\mathbb{D}}^{m\times n};$$

$$\varphi (E_{ii})=E_{ii}^{\prime },i=1,2.$$

$$\varphi (-E_{11}-E_{12}-E_{22})=-E_{11}^{\prime }+(-1{)}^{\ast }{E}_{12}^{\prime }-E_{22}^{\prime }.$$

$$\mathrm{rank}\left(\varphi (E_{21})-(-E_{11}^{\prime }+(-1{)}^{\ast }{E}_{12}^{\prime }-E_{22}^{\prime })\right)=2,$$

$$\phi (T_3^{-1}{\mathcal{M}}_i{T}_4^{-1})\subseteq T_1{\mathcal{M}}_i^{\prime }{T}_2$$

$$\phi (T_3^{-1}{\mathcal{M}}_i{T}_4^{-1})\subseteq T_1{\mathcal{N}}_i^{\prime }{T}_2$$

$$\mathcal{M}=\left\{\left(\begin{array}[]{c}a_{1}x\\ a_{2}x\\ \vdots\\ a_{m}x\\ \end{array}\right):x\in\mathbb{D}^{n}\right\}$$

$$\phi (T_3^{-1}{\mathcal{M}}_i)\subseteq T_1{\mathcal{M}}_i^{\prime }$$

$$\phi (T_3^{-1}{\mathcal{M}}_i{T}_4^{-1})\subseteq T_1{\mathcal{M}}_i^{\prime }{T}_2$$

$$\phi(X)=P\left(\begin{array}[]{cc}X^{\tau}&0\\ 0&0\\ \end{array}\right)Q,\ \,X\in{\mathbb{D}}^{2\times 2};$$

$$\phi(X)=P\left(\begin{array}[]{cc}{{}^{t}X^{\sigma}}&0\\ 0&0\\ \end{array}\right)Q,\ \,X\in{\mathbb{D}}^{2\times 2}.$$

$$\varphi ({\mathcal{M}}_i)\subseteq {\mathcal{M}}_i^{\prime }$$

$$\phi\left(\begin{array}[]{cc}x_{11}&x_{12}\\ x_{21}&x_{22}\\ \end{array}\right)=\left(\begin{array}[]{cc}\left(\begin{array}[]{cc}x_{11}^{\sigma_{11}}&x_{12}^{\sigma_{12}}\\ x_{21}^{\sigma_{21}}&x_{22}^{\sigma_{22}}\\ \end{array}\right)&0\\ 0&0\\ \end{array}\right),\ x_{ij}\in\mathbb{D}.$$

$$X⟼\mathrm{diag}(1,(1^{{\sigma }_{21}}{)}^{-1},1,\dots ,1)\varphi (X)\mathrm{diag}(1,(1^{{\sigma }_{12}}{)}^{-1},1,\dots ,1),$$

$$\varphi (X)=P{X}^{\tau }Q,X\in {\mathbb{D}}^{m\times n};$$

$$\varphi (X)=P{}^t{X}^{\sigma }Q,X\in {\mathbb{D}}^{m\times n}.$$

$$\varphi ({\mathcal{M}}_i)\subseteq {\mathcal{M}}_i^{\prime }$$

$$\phi\left(\begin{array}[]{cc}X&0\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}X^{\tau}&0\\ 0&0\\ \end{array}\right),\ \,X\in{\mathbb{D}}^{2\times 2},$$

$$\psi_{ij}\left(\begin{array}[]{cc}x_{1i}&x_{1j}\\ x_{2i}&x_{2j}\\ \end{array}\right)=\phi\left(x_{1i}E_{1i}+x_{1j}E_{1j}+x_{2i}E_{2i}+x_{2j}E_{2j}\right),\ \ x_{1i},x_{1j},x_{2i},x_{2j}\in\mathbb{D}.$$

$$\psi_{kj}\left(\begin{array}[]{cc}x_{1k}&x_{1j}\\ x_{2k}&x_{2j}\\ \end{array}\right)=\phi\left(x_{1k}E_{1k}+x_{1j}E_{1j}+x_{2k}E_{2k}+x_{2j}E_{2j}\right)=P_{k}\left(\begin{array}[]{cc}\left(\begin{array}[]{cc}x_{1k}^{\mu}&x_{1j}^{\mu}\\ x_{2k}^{\mu}&x_{2j}^{\mu}\\ \end{array}\right)&0\\ 0&0\\ \end{array}\right)Q_{j},$$

Full text

Keywords:   graph homomorphism, matrix, division ring, additive graph homomorphism, degenerate graph homomorphism, geometry of matrices

2010 AMS Classification: 05C50, 15B33, 05C60, 51A10, 51D20

1 Introduction

The study of the geometry of matrices was initiated by Hua in the mid 1940s [8, 26]. The fundamental problem in the geometry of matrices is to characterize the transformation group of matrices by as few geometric invariants as possible. In the view of equivalence, the basic problem of the geometry of matrices is also to study graph isomorphisms on matrices. In 1951, Hua [8] proved the fundamental theorem of the geometry of rectangular matrices over division rings. Hua’s theorem also characterized the graph isomorphism on rectangular matrices, and his work was continued by many scholars (cf. [3], [9]-[19], [21], [23]-[28]).

In the algebraic graph theory, the research of graph homomorphisms is an important subject [6, 7]. Thus, it is of significance to determine graph homomorphisms on matrices. Recently, literatures [24, 17, 23, 13, 14] discussed the graph homomorphisms on matrices over a division ring, and these work is interesting for the geometry of matrices, the algebraic graph theory and the preserver problems. This paper is continuation of literature [14], and our goal is further to characterize the graph homomorphism on rectangular matrices. All mathematical symbols and definitions that are not explained, see [14].

Throughout this paper, we assume that $\mathbb{D},\mathbb{D}^{\prime}$ are division rings, ${\mathbb{D}}^{*}=\mathbb{D}\setminus\{0\}$, and $m,n,m^{\prime},n^{\prime}$ are positive integers. Denote by $\mathbb{F}_{q}$ the finite field with $q$ elements where $q$ is a power of a prime. Let $|X|$ be the cardinality of a set $X$. Let ${\mathbb{D}}^{n}$ [resp. ${}^{n}{\mathbb{D}}$] denote the left [resp. right] vector space over $\mathbb{D}$ whose elements are $n$-dimensional row [resp. column] vectors over $\mathbb{D}$. On the basic properties of matrices over a division ring, one may see literatures [20, 26].

Let ${\mathbb{D}}^{m\times n}$ and ${\mathbb{D}}^{m\times n}_{r}$ denote the sets of $m\times n$ matrices and $m\times n$ matrices of rank $r$ over $\mathbb{D}$, respectively. A matrix in ${\mathbb{D}}^{m\times n}$ is also called a point. Denote the set of $n\times n$ invertible matrices over $\mathbb{D}$ by $GL_{n}(\mathbb{D})$. Let $I_{r}$ ($I$ for short) be the $r\times r$ identity matrix, $0_{m,n}$ the $m\times n$ zero matrix ([math] for short) and $0_{n}=0_{n,n}$. Let $E^{m\times n}_{ij}$ ($E_{ij}$ for short) denote the $m\times n$ matrix whose $(i,j)$-entry is $1$ and all other entries are [math]’s. Denote by ${}^{t}A$ the transpose matrix of a matrix $A$. If $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is a map and $A=(a_{ij})\in{\mathbb{D}}^{m\times n}$, we write $A^{\sigma}=(a_{ij}^{\sigma})$ and ${}^{t}A^{\sigma}=\,^{t}(A^{\sigma})$.

Let $\Gamma(\mathbb{D}^{m\times n})$ be the graph whose vertex set is ${\mathbb{D}}^{m\times n}$ and two vertices $A,B\in{\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. We write $A\sim B$ if ${\rm rank}(A-B)=1$. The $\Gamma(\mathbb{D}^{m\times n})$ is called the graph on $m\times n$ matrices over $\mathbb{D}$. $\Gamma(\mathbb{D}^{m\times n})$ is a connected distance transitive graph [6]. When $\mathbb{D}=\mathbb{F}_{q}$, $\Gamma(\mathbb{F}_{q}^{m\times n})$ is also called a bilinear forms graph [2]. For $A,B\in{\mathbb{D}}^{m\times n}$, $d(A,B):={\rm rank}(A-B)$ is the distance between $A$ and $B$ in $\Gamma(\mathbb{D}^{m\times n})$ (cf. [20, 26]). The diameter of a subgraph $G$ of $\Gamma(\mathbb{D}^{m\times n})$, denoted by ${\rm diam}(G)$, is the maximum distance between two distinct vertices in $G$.

Let $\varphi:\mathbb{D}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a map. The $\varphi$ is called a graph homomorphism if $A\sim B$ implies that $\varphi(A)\sim\varphi(B)$. The $\varphi$ is called a distance preserving map if $d(A,B)=d(\varphi(A),\varphi(B))$ for all $A,B\in{\mathbb{D}}^{m\times n}$. The $\varphi$ is called a distance $k$ preserving map if $d(A,B)=k$ implies that $d(\varphi(A),\varphi(B))=k$ for some fixed $k$. In the geometry of matrices, a graph homomorphism [resp. graph isomorphism] is also called an adjacency preserving map [resp. adjacency preserving bijection in both directions]. If $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism, then

[TABLE]

A nonempty subset $\cal S$ of ${\mathbb{D}}^{m\times n}$ is called an adjacent set if any two distinct vertices (matrices) in $\cal S$ are adjacent. An adjacent set $\cal M$ in ${\mathbb{D}}^{m\times n}$ is called a maximal adjacent set (maximal set for short), if there is no adjacent set in ${\mathbb{D}}^{m\times n}$ which properly contains $\cal M$ as a subset. In graph theory, a maximal set is also called a maximal clique [2, 6]. In ${\mathbb{D}}^{m\times n}$, every adjacent set can be extended to a maximal set, and there are only two type of maximal sets. For convenience, we think that a maximal set and its vertex set are equal.

Suppose that $\mathbb{D}$ and $\mathbb{D}^{\prime}$ are division rings and $m,n,m^{\prime},n^{\prime}\geq 2$ are integers. Write $E_{ij}=E_{ij}^{m\times n}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. For $1\leq i\leq m$ and $1\leq j\leq n$ (or $1\leq i\leq m^{\prime}$ and $1\leq j\leq n^{\prime}$), we let

[TABLE]

[TABLE]

Lemma 1.1

(cf. [26, Prpopsition 3.8])*  Every maximal set ${\cal M}$ of ${\mathbb{D}}^{m\times n}$ is one of the following forms.*

Type one.   ${\cal M}=P{\cal M}_{1}Q+A=P{\cal M}_{1}+A$, where $P\in GL_{m}(\mathbb{D})$, $Q\in GL_{n}(\mathbb{D})$ and $A\in{\mathbb{D}}^{m\times n}$.

Type two.  ${\cal M}=P{\cal N}_{1}Q+A={\cal N}_{1}Q+A$, where $P\in GL_{m}(\mathbb{D})$, $Q\in GL_{n}(\mathbb{D})$ and $A\in{\mathbb{D}}^{m\times n}$.

For $1\leq k\leq{\rm min}\{m,n\}$ and $A\in{\mathbb{D}}^{m\times n}$, we let

[TABLE]

[TABLE]

[TABLE]

The $\mathbb{B}_{A}$ is called the unit ball with a central point $A$, and the $\mathbb{N}_{A}$ is called the neighbourhood of $A$. If $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism, then $\varphi(\mathbb{N}_{A})\subseteq\mathbb{N}_{\varphi(A)}$ and $\varphi(\mathbb{B}_{A})\subseteq\mathbb{B}_{\varphi(A)}$ for every $A\in{\mathbb{D}}^{m\times n}$.

Let $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a graph homomorphism. The homomorphism $\varphi$ is called degenerate, if there exists a matrix $A\in\mathbb{D}^{m\times n}_{\leq 1}$ and there are two maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi\left(\mathbb{B}_{A}\right)\subseteq\mathcal{M}\cup\mathcal{N}$ with $\varphi(A)\in\mathcal{M}\cap\mathcal{N}$. The homomorphism $\varphi$ is called non-degenerate if it is not degenerate. The homomorphism $\varphi$ is called a (vertex) colouring if $\varphi(\mathbb{D}^{m\times n})$ is an adjacent set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$.

Every (vertex) colouring is a degenerate graph homomorphism. It is easy to see that $\varphi$ is a (vertex) colouring if and only if ${\rm diam}(\varphi(\mathbb{D}^{m\times n}))=1$. The word “colouring” is derived from graph theory [6]. Any distance preserving map is a non-degenerate graph homomorphism but not vice versa.

This paper is organized as follows. In Section 2, we introduce some Lemmas on maximal sets. In Section 3, We determine additive graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. In Section 4, we will characterize the graph homomorphism $\varphi$ from ${\mathbb{D}}^{n\times n}$ to ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ (where $|\mathbb{D}|\geq 4$) if there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$, this result extends [24, Theorem 4.1] to the cases of two division rings and $n=2$. In Section 5, we discuss the degenerate graph homomorphisms. Let $f:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ (where $|\mathbb{D}|\geq 4$) be a degenerate graph homomorphism. We prove that $f\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ and $f(\mathbb{B}_{A_{0}})$ are two adjacent sets, if there exists $A_{0}\in{\mathbb{D}}^{m\times n}$ such that ${\rm rank}(f(A_{0}))={\rm min}\{m,n\}$. Moreover, when ${\rm min}\{m,n\}=2$, the image of $f$ is contained in a union of two maximal sets of different types. For the case of finite fields, we obtain two better results on the degenerate graph homomorphisms.

2 Lemmas on maximal sets

In this section, we will introduce maximal sets on ${\mathbb{D}}^{m\times n}$ and their affine geometries.

Lemma 2.1

(cf. [9, Lemma 3.2])*  Let $m,n,r,s$ be integers with $1\leq r,s<{\rm min}\{m,n\}$. Assume that $\alpha=\{i_{1},\ldots,i_{r}\}$, $\beta=\{j_{1},\ldots,j_{s}\}$, where $1\leq i_{1}<\cdots<i_{r}\leq m$ and $1\leq j_{1}<\cdots<j_{s}\leq n$. Let $A=(a_{ij})\in\mathbb{D}^{m\times n}$, $B_{i}=\sum_{t=1}^{r}\sum_{k=1}^{s}b^{(i)}_{i_{t}j_{k}}E_{i_{t}j_{k}}\in\mathbb{D}^{m\times n}$ (where $b^{(i)}_{i_{t}j_{k}}\in\mathbb{D}$), $i=1,2$, and $B_{1}\neq B_{2}$. If $A\sim B_{i}$, $i=1,2$, then either $a_{ij}=0$ for all $i\notin\alpha$, or $a_{ij}=0$ for all $j\notin\beta$.*

Using Lemmas 2.1 and 1.1, we can prove the following results.

Corollary 2.2

(cf. [26, Corollary 3.10])* Let $A$ and $B$ be two adjacent points in ${\mathbb{D}}^{m\times n}$. Then there are exactly two maximal sets $\mathcal{M}$ and $\mathcal{M}^{\prime}$ containing $A$ and $B$. Moreover, $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are of different types.*

Corollary 2.3

(cf. [9, 12, 26])* If $\mathcal{M}$ and $\mathcal{N}$ are two distinct maximal sets of the same type $[$resp. different types$]$ in ${\mathbb{D}}^{m\times n}$ with $\mathcal{M}\cap\mathcal{N}\neq\emptyset$, then $|\mathcal{M}\cap\mathcal{N}|=1$ $[$resp. |\mathcal{M}\cap\mathcal{N}|\geq 2$$].*

Lemma 2.4

(cf. [14, Lemma 3.4])* Suppose that $\mathcal{M}$, $\mathcal{N}$ are two distinct maximal sets in ${\mathbb{D}}^{m\times n}$ with $\mathcal{M}\cap\mathcal{N}\neq\emptyset$. Then:*

(i)*   if $\mathcal{M}$, $\mathcal{N}$ are of different types, then for any $A\in\mathcal{M}\cap\mathcal{N}$, there are $P\in GL_{m}(\mathbb{D})$ and $Q\in GL_{n}(\mathbb{D})$ such that ${\cal M}=P{\cal M}_{1}Q+A=P{\cal M}_{1}+A$ and ${\cal N}=P{\cal N}_{1}Q+A={\cal N}_{1}Q+A$;*

(ii)*   if both $\mathcal{M}$ and $\mathcal{N}$ are of type one $[$resp. type two$]$, then there exists an invertible matrix $P$ $[$resp. Q$$] such that ${\cal M}=P{\cal M}_{1}+A$ and ${\cal N}=P{\cal M}_{2}+A$ $[$resp. ${\cal M}={\cal N}_{1}Q+A$ and {\cal N}={\cal N}_{2}Q+A$$], where $\mathcal{M}\cap\mathcal{N}=\{A\}$.*

There is an axiomatic definition of the affine geometry (cf. [1, 5]). Let $V$ be an $r$-dimensional left vector subspace of ${\mathbb{D}}^{n}$ and $a\in\mathbb{D}^{n}$. Then $V+a$ is called an $r$-dimensional left affine flat (flat for short) over $\mathbb{D}$. When $r\geq 2$, the set of all flats in $V+a$ is called the left affine geometry on $V+a$, which is denoted by $AG(V+a)$. The dimension of $AG(V+a)$ is $r$, denoted by ${\rm dim}(AG(V+a))=r$. The flats of dimensions $0,1,2$ are called points, lines, planes in $AG(V+a)$. Similarly, we have the right affine geometry on a right affine flat over $\mathbb{D}$.

Let ${\cal M}=P{\cal M}_{1}+A$ be a maximal set of type one in $\mathbb{D}^{m\times n}$, where $P\in GL_{m}(\mathbb{D})$ and $A\in\mathbb{D}^{m\times n}$. Then we have a left affine geometry $AG(P{\cal M}_{1}+A)$ such that $AG(P{\cal M}_{1}+A)$ and $AG({\mathbb{D}}^{n})$ are affine isomorphic. Similarly, we have a right affine geometry $AG({\cal N}_{1}Q+A)$ such that $AG({\cal N}_{1}Q+A)$ and $AG(^{m}{\mathbb{D}})$ are affine isomorphic, where $Q\in GL_{n}(\mathbb{D})$ (cf. [14]). In $AG(P{\cal M}_{1}+A)$, the parametric equation of a line $\ell$ is

[TABLE]

Lemma 2.5

(cf. [26, p.95])*  Let ${\cal M}$ be a maximal set in $\mathbb{D}^{m\times n}$. Then $\ell$ is a line in $AG({\cal M})$ if and only if $\ell=\mathcal{M}\cap\mathcal{N}$, where $\mathcal{M}$ and $\mathcal{N}$ are two maximal sets of different types with $\mathcal{M}\cap\mathcal{N}\neq\emptyset$. Moreover, $|\ell|=|\mathcal{M}\cap\mathcal{N}|=|\mathbb{D}|$.*

Lemma 2.6

(see [14, Lemma 4.6])*  Let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers. Suppose that $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a non-degenerate graph homomorphism with $\varphi(0)=0$. Then*

• (a)

if ${\cal M}$ is a maximal set of type one [resp. type two] containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi({\cal M})\subseteq{\cal M}^{\prime}$, where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then for any $A\in{\cal M}$, there exists a maximal set ${\cal R}$ of type one [resp. type two] in ${\mathbb{D}}^{m\times n}$, such that $\varphi({\cal R})\subseteq{\cal R}^{\prime}$ and ${\cal R}\cap{\cal M}=\{A\}$, where ${\cal R}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, ${\cal R}^{\prime}$ and ${\cal M}^{\prime}$ are of the same type with ${\cal R}^{\prime}\neq{\cal M}^{\prime}$;

• (b)

if ${\cal M}$ and ${\cal N}$ are two distinct maximal sets of the same type [resp. different types] in ${\mathbb{D}}^{m\times n}$ such that $0\in\mathcal{M}$ and ${\cal M}\cap{\cal N}\neq\emptyset$, then $\varphi({\cal M})\subseteq{\cal M}^{\prime}$ and $\varphi({\cal N})\subseteq{\cal N}^{\prime}$, where ${\cal M}^{\prime}$ and ${\cal N}^{\prime}$ are two maximal sets of the same type [resp. different types] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$;

• (c)

if ${\cal M}$ is a maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then ${\cal M}^{\prime}$ is the unique maximal set containing $\varphi({\cal M})$, and $\varphi({\cal M})$ is not contained in any line of $AG({\cal M}^{\prime})$.

Let $\mathcal{S}$ be an adjacent set in ${\mathbb{D}}^{m\times n}$ with $0\in\mathcal{S}$ and $|\mathcal{S}|\geq 2$. Then there exists a maximal set $\mathcal{M}$ containing $\mathcal{S}$. By Lemma 1.1, $\mathcal{M}=P\mathcal{M}_{1}$ or $\mathcal{M}=\mathcal{N}_{1}Q$, where $P$ and $Q$ are invertible matrices. Hence $P^{-1}\mathcal{S}\subseteq\mathcal{M}_{1}$ or $\mathcal{S}Q^{-1}\subseteq\mathcal{N}_{1}$. When $P^{-1}\mathcal{S}\subseteq\mathcal{M}_{1}$ [resp. $\mathcal{S}Q^{-1}\subseteq\mathcal{N}_{1}$], the dimension of $\mathcal{S}$, denoted by ${\rm dim}(\mathcal{S})$, is the number of matrices in a maximal left [resp. right] linear independent subset of $P^{-1}\mathcal{S}$ [resp. $\mathcal{S}Q^{-1}$]. The ${\rm dim}(\mathcal{S})$ is uniquely determined by $\mathcal{S}$ and ${\rm dim}(\mathcal{S})\leq{\rm max}\{m,n\}$.

Lemma 2.7

(see [14, Lemma 4.13])*  Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$. Suppose $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a non-degenerate graph homomorphism with $\varphi(0)=0$. Let ${\cal M}$ be a maximal set containing [math] in ${\mathbb{D}}^{m\times n}$. If $\varphi({\cal M})\subseteq{\cal M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then the restriction map $\varphi\mid_{\mathcal{M}}:\mathcal{M}\rightarrow\mathcal{M}^{\prime}$ is an injective weighted semi-affine map. Moreover, if $\mathcal{S}$ is an adjacent set in ${\mathbb{D}}^{m\times n}$ with $0\in\mathcal{S}$ and $|\mathcal{S}|\geq 2$, then*

[TABLE]

Lemma 2.8

Let $\mathcal{M}$ be a maximal set in $\mathbb{D}^{m\times n}$ $(m,n\geq 2)$ and $A\in\mathbb{D}^{m\times n}$. Then $A\in\mathcal{M}$ if and only if $A$ is adjacent with three noncollinear points in $AG(\mathcal{M})$.

*Proof. *If $A\in\mathcal{M}$, it is clear that $A$ is adjacent with three noncollinear points in $AG(\mathcal{M})$. Now, suppose that $A$ is adjacent with three noncollinear points $B_{1},B_{2},B_{3}$ in $AG(\mathcal{M})$. Then, there are $P\in GL_{m}(\mathbb{D})$ and $Q\in GL_{n}(\mathbb{D})$ such that $B_{2}-B_{1}=PE_{11}Q$. After by the map $X\mapsto P^{-1}(X-B_{1})Q^{-1}$, we may assume with no loss of generality that $B_{1}=0$ and $B_{2}=E_{11}$. By Corollary 2.2, we have either $\mathcal{M}=\mathcal{M}_{1}$ or $\mathcal{M}=\mathcal{N}_{1}$. Without loss of generality we assume that $\mathcal{M}=\mathcal{M}_{1}$. Clearly, $\ell=\{xE_{11}:x\in\mathbb{D}\}$ is a line in $AG(\mathcal{M}_{1})$ containing [math] and $E_{11}$. Since $0,E_{11},B_{3}$ are noncollinear points in $AG(\mathcal{M}_{1})$, $B_{3}$ is of the form \scriptsize B_{3}=\left(\begin{array}[]{cc}b&\beta\\ 0&0\\ \end{array}\right) where $0\neq\beta\in\mathbb{D}^{n-1}$. Write \scriptsize A=\left(\begin{array}[]{cc}a_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $a_{11}\in\mathbb{D}$. By $A\sim 0$, $A\sim E_{11}$ and \scriptsize A\sim\left(\begin{array}[]{cc}b&\beta\\ 0&0\\ \end{array}\right), we get

[TABLE]

It follows from Lemma 2.1 that $(A_{21},A_{22})=0$, and hence $A\in\mathcal{M}_{1}$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

3 Additive graph homomorphisms on rectangular matrices

Let $\phi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a map. The $\phi$ is called an additive map if $\phi(A+B)=\phi(A)+\phi(B)$ for all $A,B\in\mathbb{D}^{m\times n}$. The $\phi$ is called an additive graph homomorphism if $\phi$ is a graph homomorphism and an additive map. If $\phi$ is an additive graph homomorphism, then $\phi(0)=0$ and $\phi(A-B)=\phi(A)-\phi(B)$ for all $A,B\in\mathbb{D}^{m\times n}$.

The additive preserver problems on matrices is an active research area in linear algebra and the theory of matrices [29]. In the additive preserver problems, an additive graph homomorphism is also called an additive rank-$1$ preserving map. When $\mathbb{D}=\mathbb{D}^{\prime}$ is a field, additive rank-$1$ preserving maps from ${\mathbb{D}}^{m\times n}$ to ${\mathbb{D}}^{m^{\prime}\times n^{\prime}}$ is determined by [29]. In this section, we discuss the case of division rings, and our main result is the following theorem.

Theorem 3.1

Suppose $\mathbb{D},\mathbb{D}^{\prime}$ are division rings and $m,n,m^{\prime},n^{\prime}\geq 2$ are integers. Let $\phi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be an additive graph homomorphism. Then either there exist matrices $P\in{\mathbb{D}^{\prime}}^{m^{\prime}\times m}$ and $Q\in{\mathbb{D}^{\prime}}^{n\times n^{\prime}}$ with their ranks $\geq 2$, and a nonzero ring homomorphism $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

or there exist matrices $P\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n}$ and $Q\in{\mathbb{D}^{\prime}}^{m\times n^{\prime}}$ with their ranks $\geq 2$, and a nonzero ring anti-homomorphism $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

or $\phi$ is a (vertex) colouring.

In order to prove Theorem 3.1, we need the following results.

Theorem 3.2

Suppose $\mathbb{D},\mathbb{D}^{\prime}$ are division rings and $m,n,m^{\prime},n^{\prime}\geq 2$ are integers. Let $\phi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be an additive graph homomorphism. If $\phi$ is degenerate, then $\phi$ is a (vertex) colouring.

*Proof. *Assume that $\phi$ is degenerate. Since $\phi$ is additive, there are two maximal sets $\mathcal{M},\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}\cup\mathcal{N}$ with $0\in\mathcal{M}\cap\mathcal{N}$. We show that $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set by contradiction as follows.

Suppose $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set. Then there are matrices $A,B\in\mathbb{D}^{m\times n}_{1}$ such that ${\rm rank}(A-B)={\rm rank}(\phi(A)-\phi(B))=2$. Let $E_{ij}=E_{ij}^{m\times n}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. There are invertible matrices $P_{1},Q_{1}$ over $\mathbb{D}$ such that $A=P_{1}E_{11}Q_{1}$ and $B=P_{1}E_{22}Q_{1}$. Also, there are invertible matrices $P_{2},Q_{2}$ over $\mathbb{D}^{\prime}$ such that $P_{2}\phi(A)Q_{2}=E_{11}^{\prime}$ and $P_{2}\phi(B)Q_{2}=E_{22}^{\prime}$. Replacing $\phi$ by the map $X\longmapsto P_{2}\phi(P_{1}XQ_{1})Q_{2}$, we have

[TABLE]

By Corollary 2.2, there are exactly two maximal sets containing $E_{ii}$ and [math] [resp. $E_{ii}^{\prime}$ and [math]], they are $\mathcal{M}_{i}$ and $\mathcal{N}_{i}$ [resp. $\mathcal{M}_{i}^{\prime}$ and $\mathcal{N}_{i}^{\prime}$] ($i=1,2$). Thus, by (6) and $\phi(0)=0$, we get that $\phi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$ or $\phi(\mathcal{M}_{1})\subseteq\mathcal{N}_{1}^{\prime}$. Without loss of generality, we assume $\phi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. From the above discussion we have $\phi(\mathcal{N}_{2})\subseteq\mathcal{N}_{2}^{\prime}$ or $\phi(\mathcal{N}_{2})\subseteq\mathcal{M}_{2}^{\prime}$. Since $\mathcal{M}_{1}\cap\mathcal{N}_{2}=\mathbb{D}E_{12}$ and $\mathcal{M}_{1}^{\prime}\cap\mathcal{M}_{2}^{\prime}=\{0\}$, we must have $\phi(\mathcal{N}_{2})\subseteq\mathcal{N}_{2}^{\prime}$. By $\mathcal{M}_{1}^{\prime}\cap\mathcal{N}_{2}^{\prime}=\mathbb{D}^{\prime}E_{12}^{\prime}$, we obtain $\phi(\lambda E_{12})=\lambda^{*}E_{12}^{\prime}$ for all $\lambda\in\mathbb{D}$. Since $\phi$ is additive,

[TABLE]

Recall that $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}\cup\mathcal{N}$. Using Corollary 2.2, it is clear that $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}\cup\mathcal{N}_{2}^{\prime}$. Since $E_{21}\sim(-E_{11}-E_{12}-E_{22})$, $\phi(E_{21})\sim(-E_{11}^{\prime}+(-1)^{*}E_{12}^{\prime}-E_{22}^{\prime})$. On the other hand, by $\phi(E_{21})\in\mathcal{M}_{1}^{\prime}\cup\mathcal{N}_{2}^{\prime}$ and $\phi(E_{21})\sim E_{ii}^{\prime}$ ($i=1,2$), we must have $\phi(E_{21})=bE_{12}^{\prime}$ where $b\in{\mathbb{D}^{\prime}}^{*}$. Hence

[TABLE]

a contradiction.

Thus, $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set. Then there is a fixed maximal set $\mathcal{M}^{\prime}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\phi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}^{\prime}$. Note that $Y_{1},Y_{2}\in\mathcal{M}^{\prime}$ implies that $Y_{1}+Y_{2}\in\mathcal{M}^{\prime}$. Since $\phi$ is additive, it is clear that $\phi\left(\mathbb{D}^{m\times n}\right)\subseteq\mathcal{M}^{\prime}$, and hence $\phi$ is a (vertex) colouring. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 3.3

Suppose $\mathbb{D},\mathbb{D}^{\prime}$ are division rings and $m,n,m^{\prime},n^{\prime}\geq 2$ are integers. Let $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a non-degenerate graph homomorphism with $\varphi(0)=0$. Then there exist two invertible matrices $T_{1}$ and $T_{2}$ over $\mathbb{D}^{\prime}$, and two invertible matrices \scriptsize T_{3}^{-1}=\left(\begin{array}[]{cc}1&*\\ 0&*\\ \end{array}\right)\in GL_{m}(\mathbb{D}) and \scriptsize T_{4}^{-1}=\left(\begin{array}[]{cc}1&0\\ &*\\ \end{array}\right)\in GL_{n}(\mathbb{D}), such that either

[TABLE]

or

[TABLE]

Proof. Case 1.  $\varphi(\mathcal{M}_{1})$ is contained in a maximal set of type one. Then there exists an invertible matrix $T_{0}$ over $\mathbb{D}^{\prime}$ such that $\varphi(\mathcal{M}_{1})\subseteq T_{0}\mathcal{M}_{1}^{\prime}$. By Lemma 2.6(a), there exists a type one maximal set ${\cal M}$ in ${\mathbb{D}}^{m\times n}$ containing [math], such that ${\cal M}\neq{\cal M}_{1}$ and $\varphi({\cal M})\subseteq{\cal M}^{\prime}$, where ${\cal M}^{\prime}$ is a type one maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ containing [math] and ${\cal M}^{\prime}\neq T_{0}{\cal M}_{1}^{\prime}$. Using Lemma 2.4, there exists $T_{1}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ such that $T_{0}\mathcal{M}_{1}^{\prime}=T_{1}\mathcal{M}_{1}^{\prime}$ and $\mathcal{M}^{\prime}=T_{1}\mathcal{M}_{2}^{\prime}$. Thus, $\varphi(\mathcal{M}_{1})\subseteq T_{1}\mathcal{M}_{1}^{\prime}$ and $\varphi(\mathcal{M})\subseteq T_{1}\mathcal{M}_{2}^{\prime}$. Since

[TABLE]

there is \scriptsize T_{3}^{-1}=\left(\begin{array}[]{cc}1&*\\ 0&*\\ \end{array}\right)\in GL_{m}(\mathbb{D}) such that $\mathcal{M}=T_{3}^{-1}\mathcal{M}_{2}$ and $\mathcal{M}_{1}=T_{3}^{-1}\mathcal{M}_{1}$. Thus

[TABLE]

Similarly, there exists a type two maximal set ${\cal N}$ in ${\mathbb{D}}^{m\times n}$ containing [math], such that $\mathcal{N}\neq\mathcal{N}_{1}$, $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}T_{2}=T_{1}\mathcal{N}_{1}^{\prime}T_{2}$ and $\varphi(\mathcal{N})\subseteq\mathcal{N}_{2}^{\prime}T_{2}=T_{1}\mathcal{N}_{2}^{\prime}T_{2}$, where $T_{2}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$. Moreover, there is \scriptsize T_{4}^{-1}=\left(\begin{array}[]{cc}1&0\\ &*\\ \end{array}\right)\in GL_{n}(\mathbb{D}) such that $\mathcal{N}=\mathcal{N}_{2}T_{4}^{-1}=T_{3}^{-1}\mathcal{N}_{2}T_{4}^{-1}$ and $\mathcal{N}_{1}=\mathcal{N}_{1}T_{4}^{-1}=T_{3}^{-1}\mathcal{N}_{1}T_{4}^{-1}$. It follows that $\varphi(T_{3}^{-1}\mathcal{N}_{i}T_{4}^{-1})\subseteq T_{1}\mathcal{N}_{i}^{\prime}T_{2}$, $i=1,2$. Clearly, $T_{3}^{-1}\mathcal{M}_{i}=T_{3}^{-1}\mathcal{M}_{i}T_{4}^{-1}$ and $T_{1}\mathcal{M}_{i}^{\prime}=T_{1}\mathcal{M}_{i}^{\prime}T_{2}$, $i=1,2$. Therefore, we obtain

[TABLE]

Case 2.   $\varphi(\mathcal{M}_{1})$ is contained in a maximal set of type two. Similarly, there exist two invertible matrices $T_{1}$ and $T_{2}$ over $\mathbb{D}^{\prime}$, and two invertible matrices \scriptsize T_{3}^{-1}=\left(\begin{array}[]{cc}1&*\\ 0&*\\ \end{array}\right)\in GL_{m}(\mathbb{D}) and \scriptsize T_{4}^{-1}=\left(\begin{array}[]{cc}1&0\\ &*\\ \end{array}\right)\in GL_{n}(\mathbb{D}), such that $\varphi(T_{3}^{-1}\mathcal{M}_{i}T_{4}^{-1})\subseteq T_{1}\mathcal{N}_{i}^{\prime}T_{2}$ with $\varphi(T_{3}^{-1}\mathcal{N}_{i}T_{4}^{-1})\subseteq T_{1}\mathcal{M}_{i}^{\prime}T_{2}$, $i=1,2$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Theorem 3.4

Suppose $\mathbb{D},\mathbb{D}^{\prime}$ are division rings and $m^{\prime},n^{\prime}\geq 2$ are integers. Let $\phi:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a non-degenerate additive graph homomorphism. Then either there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, and a nonzero ring homomorphism $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

or there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, and a nonzero ring anti-homomorphism $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

*Proof. *By Lemma 3.3, we may assume with no loss of generality either $\phi(\mathcal{M}_{i})\subseteq\mathcal{M}_{i}^{\prime}$ with $\phi(\mathcal{N}_{i})\subseteq\mathcal{N}_{i}^{\prime}$ $(i=1,2)$, or $\phi(\mathcal{M}_{i})\subseteq\mathcal{N}_{i}^{\prime}$ with $\phi(\mathcal{N}_{i})\subseteq\mathcal{M}_{i}^{\prime}$ $(i=1,2)$. We prove this result only for the first case; the second case is similar. From now on we assume that

[TABLE]

We show that $\phi$ is of the form (7) as follows. (For the second case, we can prove similarly $\phi$ is of the form (8).)

Let $E_{ij}=E_{ij}^{2\times 2}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. By (9) we have that $\phi(xE_{ij})=x^{\sigma_{ij}}E_{ij}^{\prime}$ for all $x\in\mathbb{D}$, $i,j=1,2$, where $\sigma_{ij}:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is an additive map with $0^{\sigma_{ij}}=0$. Since $\phi$ is additive, we get

[TABLE]

Without loss of generality, we assume that $1^{\sigma_{11}}=1^{\sigma_{22}}=1$.

Since \scriptsize\left(\begin{array}[]{cc}x&x\\ 1&1\\ \end{array}\right)\sim 0 and \scriptsize\left(\begin{array}[]{cc}1&x\\ 1&x\\ \end{array}\right)\sim 0 for all $x\in\mathbb{D}$, \scriptsize\left(\begin{array}[]{cc}x^{\sigma_{11}}&x^{\sigma_{12}}\\ 1^{\sigma_{21}}&1\\ \end{array}\right)\sim 0 and \scriptsize\left(\begin{array}[]{cc}1&x^{\sigma_{12}}\\ 1^{\sigma_{21}}&x^{\sigma_{22}}\\ \end{array}\right)\sim 0 for all $x\in\mathbb{D}$. It follows that $x^{\sigma_{11}}=x^{\sigma_{12}}1^{\sigma_{21}}$ and $x^{\sigma_{22}}=1^{\sigma_{21}}x^{\sigma_{12}}$ for all $x\in\mathbb{D}$. Therefore, $1=1^{\sigma_{21}}1^{\sigma_{12}}=1^{\sigma_{12}}1^{\sigma_{21}}$ and hence $1^{\sigma_{12}}=(1^{\sigma_{21}})^{-1}$. Replacing the map $\phi$ by the map

[TABLE]

we have that $1^{\sigma_{ij}}=1$ for $i,j=1,2$. Then $x^{\sigma_{11}}=x^{\sigma_{22}}=x^{\sigma_{12}}$ for all $x\in\mathbb{D}$. Write $\tau=\sigma_{11}$. Since \scriptsize\left(\begin{array}[]{cc}x&x\\ x&x\\ \end{array}\right)\sim 0 for all $x\in\mathbb{D}^{*}$, \scriptsize\left(\begin{array}[]{cc}x^{\tau}&x^{\tau}\\ x^{\sigma_{21}}&x^{\tau}\\ \end{array}\right)\sim 0 for all $x\in\mathbb{D}^{*}$, which implies that $\sigma_{21}=\tau$. Since \scriptsize\left(\begin{array}[]{cc}xy&x\\ y&1\\ \end{array}\right)\sim 0 for all $x,y\in\mathbb{D}$, \scriptsize\left(\begin{array}[]{cc}(xy)^{\tau}&x^{\tau}\\ y^{\tau}&1\\ \end{array}\right)\sim 0 for all $x,y\in\mathbb{D}$. Consequently $(xy)^{\tau}=x^{\tau}y^{\tau}$ for all $x,y\in\mathbb{D}$. Thus $\tau$ is a nonzero ring homomorphism, and hence (7) holds. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Corollary 3.5

If $\phi:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ $(m^{\prime},n^{\prime}\geq 2)$ is a non-degenerate additive graph homomorphism, then $\phi$ is a distance preserving map.

Theorem 3.6

Suppose $\mathbb{D},\mathbb{D}^{\prime}$ are division rings and $m,n,m^{\prime},n^{\prime}\geq 2$ are integers. Let $\phi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a non-degenerate additive graph homomorphism. Then either there exist matrices $P\in{\mathbb{D}^{\prime}}^{m^{\prime}\times m}$ and $Q\in{\mathbb{D}^{\prime}}^{n\times n^{\prime}}$ with their ranks $\geq 2$, and a nonzero ring homomorphism $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

or there exist matrices $P\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n}$ and $Q\in{\mathbb{D}^{\prime}}^{m\times n^{\prime}}$ with their ranks $\geq 2$, and a nonzero ring anti-homomorphism $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, such that

[TABLE]

*Proof. *By Lemma 3.3, we may assume with no loss of generality either $\phi(\mathcal{M}_{i})\subseteq\mathcal{M}_{i}^{\prime}$ with $\phi(\mathcal{N}_{i})\subseteq\mathcal{N}_{i}^{\prime}$ $(i=1,2)$, or $\phi(\mathcal{M}_{i})\subseteq\mathcal{N}_{i}^{\prime}$ with $\phi(\mathcal{N}_{i})\subseteq\mathcal{M}_{i}^{\prime}$ $(i=1,2)$. If the first case [resp. second case] happens, we can prove that $\phi$ is of the form (10) [resp. (11)]. We prove this theorem only for the first case; the second case is similar. From now on we assume that

[TABLE]

By Theorem 3.4 and its proof, we may assume with no loss of generality that

[TABLE]

where $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is a nonzero ring homomorphism.

Write $E_{ij}=E_{ij}^{m\times n}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. For $i\in\{1,2\}$ and $j\in\{3,\ldots,n\}$, we let

[TABLE]

Then $\psi_{ij}:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is an additive graph homomorphism. For $j\in\{3,\ldots,n\}$, there exists some $k\in\{1,2\}$ such that $\psi_{kj}$ is non-degenerate. Otherwise, both $\psi_{1j}$ and $\psi_{2j}$ are degenerate, and Theorem 3.2 implies that $\psi_{1j}({\mathbb{D}}^{2\times 2})$ and $\psi_{2j}({\mathbb{D}}^{2\times 2})$ are two adjacent sets. By Corollary 2.3, (13) and Lemma 2.5, it is easy to see that $\psi_{1j}({\mathbb{D}}^{2\times 2})\subseteq\mathcal{N}_{1}^{\prime}$ and $\psi_{2j}({\mathbb{D}}^{2\times 2})\subseteq\mathcal{N}_{2}^{\prime}$. It follows that $\phi\left(x_{1j}E_{1j}+x_{2j}E_{2j}\right)\in\mathcal{N}_{1}^{\prime}\cap\mathcal{N}_{2}^{\prime}=\{0\}$ for all $x_{1j},x_{2j}\in\mathbb{D}$, a contradiction. Thus $\psi_{kj}$ is non-degenerate for some $k\in\{1,2\}$.

By Theorem 3.4 and (13), we can assume that

[TABLE]

for all $x_{1k},x_{1j},x_{2k},x_{2j}\in\mathbb{D}$, where $P_{k}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ is a diagonal matrix, $Q_{j}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ and $\mu:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is a nonzero ring homomorphism. Let $q_{ji}=(q_{ji}^{(1)},\ldots,q_{ji}^{(n^{\prime})})$ be the $i$-th row of $Q_{j}$ and let $P_{k}={\rm diag}(p_{k1},\ldots,p_{km^{\prime}})$. Then (14) and (13) imply that

[TABLE]

[TABLE]

By (15), we have $x^{\tau}=p_{k1}x^{\mu}q_{j1}^{(k)}$ and $x^{\tau}=p_{k2}x^{\mu}q_{j1}^{(k)}$ for all $x\in\mathbb{D}$. Thus $p_{k1}=p_{k2}=(q_{j1}^{(k)})^{-1}$.

Let $e_{i}$ be the $i$-th row of $I_{m^{\prime}}$, and let $e_{i}^{\prime}$ be the $i$-th row of $I_{n^{\prime}}$. Write $q_{j2}^{\prime}=(q_{j1}^{(k)})^{-1}q_{j2}$. Then (16) can be written as

[TABLE]

For $s\in\{1,2\}$ and $i\in\{3,\ldots,n\}$, we define

[TABLE]

Then $\psi_{si}^{\prime}:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is an additive graph homomorphism. Similarly, there exists some $s\in\{1,2\}$ such that $\psi_{si}^{\prime}$ is non-degenerate. Similar to the proof of (17), there is $0\neq p_{i2}^{\prime}\in{\mathbb{D}^{\prime}}^{m^{\prime}\times 1}$ such that

[TABLE]

Recall that $\psi_{kj}$ is non-degenerate for some $k\in\{1,2\}$. Put $i\in\{3,\ldots,m\}$, $j\in\{3,\ldots,n\}$ and $r\in\{1,2\}$. We let

[TABLE]

Then $\theta_{ij}^{(r)}:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is an additive graph homomorphism. There exists some $r\in\{1,2\}$ such that $\theta_{ij}^{(r)}$ is non-degenerate. Otherwise, both $\theta_{ij}^{(1)}$ and $\theta_{ij}^{(2)}$ are degenerate. By Theorem 3.2, we have $\theta_{ij}^{(1)}({\mathbb{D}}^{2\times 2})\subseteq\mathcal{M}$, where $\mathcal{M}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ containing [math]. From (13) we get \scriptsize\theta_{ij}^{(1)}\left(\begin{array}[]{cc}x_{1k}&0\\ 0&0\\ \end{array}\right)=x_{1k}^{\tau}E_{1k}^{\prime} for all $x_{1k}\in\mathbb{D}$. Thus, Corollaries 2.2 and 2.3 imply that $\mathcal{M}=\mathcal{M}_{1}^{\prime}$ or $\mathcal{M}=\mathcal{N}_{k}^{\prime}$. Suppose that $\mathcal{M}=\mathcal{N}_{k}^{\prime}$. Then (12) implies that \scriptsize\theta_{ij}^{(1)}\left(\begin{array}[]{cc}x_{1k}&x_{1j}\\ 0&0\\ \end{array}\right)\in\mathcal{N}_{k}^{\prime}\cap\mathcal{M}_{1}^{\prime}=\mathbb{D}^{\prime}E_{1k}^{\prime}, and hence $\phi(E_{1j})=a_{1}E_{1k}^{\prime}$ where $a_{1}\in{\mathbb{D}^{\prime}}^{*}$. By (15) and (16), it is easy to see that $q_{j1}$ and $q_{j2}$ are left linearly dependent, a contradiction to $Q_{j}$ being invertible. Therefore, we must have $\mathcal{M}=\mathcal{M}_{1}^{\prime}$ and hence $\theta_{ij}^{(1)}({\mathbb{D}}^{2\times 2})\subseteq\mathcal{M}_{1}^{\prime}$. Similarly, $\theta_{ij}^{(2)}({\mathbb{D}}^{2\times 2})\subseteq\mathcal{M}_{2}^{\prime}$. It follows that $\phi\left(x_{ik}E_{ik}+x_{ij}E_{ij}\right)\in\mathcal{M}_{1}^{\prime}\cap\mathcal{M}_{2}^{\prime}=\{0\}$ for all $x_{ik},x_{ij}\in\mathbb{D}$, a contradiction. Thus, $\theta_{ij}^{(r)}$ is non-degenerate for some $r\in\{1,2\}$.

By Theorem 3.4 and (12), we have

[TABLE]

for all $x_{rk},x_{rj},x_{ik},x_{ij}\in\mathbb{D}$, where $S_{i}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$, $T_{j}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ and $\delta:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is a nonzero ring homomorphism. Let $t_{ji}=(t_{ji}^{(1)},\ldots,t_{ji}^{(n^{\prime})})$ be the $i$-th row of $T_{j}$, and let $S_{i}=(s_{i1},\ldots,s_{im^{\prime}})$ where $s_{ij}$ be the $j$-th column of $S_{i}$. Then (19) and (13) imply that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By (20), there are $a,b\in{\mathbb{D}^{\prime}}^{*}$ such that $s_{i1}=\,^{t}e_{r}a$ and $t_{j1}=be_{k}^{\prime}$. Moreover, $x^{\delta}=a^{-1}x^{\tau}b^{-1}$ for all $x\in\mathbb{D}$. Since $1^{\delta}=1^{\tau}=1$, we get that $b^{-1}=a$. Hence

[TABLE]

Therefore, for above two fixed $k,r\in\{1,2\}$, we obtain that

[TABLE]

[TABLE]

[TABLE]

By (21) and (17), we get $q_{j2}^{\prime}=at_{j2}$. Thus (17) can be written as

[TABLE]

By (18) and (22), we have $p_{i2}^{\prime}=s_{i2}a^{-1}$. Hence (18) can be written as

[TABLE]

Let \scriptsize P=\left({}^{t}e_{1},\,^{t}e_{2},s_{32}a^{-1},s_{42}a^{-1},\ldots,s_{m2}a^{-1}\right)=\left(\begin{array}[]{cc}I_{2}&\ast\\ 0&\ast\\ \end{array}\right)\in{\mathbb{D}^{\prime}}^{m^{\prime}\times m}, and let

[TABLE]

Then ${\rm rank}(P)\geq 2$ and ${\rm rank}(Q)\geq 2$. By (13) with (23)-(25), we obtain that

[TABLE]

Since $\phi$ is additive, we get (10). $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

By Theorem 3.6 and Theorem 3.2, it is clear that Theorem 3.1 holds.

4 Graph homomorphisms on $n\times n$ matrices

In this section, we will expand [24, Theorem 4.1] (which is due to Šemrl) to the cases of two division rings and $n=2$.

Theorem 4.1

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$. Then either there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, a nonzero ring homomorphism $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, and a matrix $L\in{\mathbb{D}^{\prime}}^{n\times n}$ with the property that $I_{n}+X^{\tau}L\in GL_{n}(\mathbb{D}^{\prime})$ for every $X\in{\mathbb{D}}^{n\times n}$, such that

[TABLE]

or there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, a nonzero ring anti-homomorphism $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, and a matrix $L\in{\mathbb{D}^{\prime}}^{n\times n}$ with the property that $I_{n}+L\,^{t}X^{\sigma}\in GL_{n}(\mathbb{D}^{\prime})$ for every $X\in{\mathbb{D}}^{n\times n}$, such that

[TABLE]

or $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set. In particular, if $\tau$ $[$resp. \sigma$$] is surjective, then $L=0$ and $\tau$ is a ring isomorphism $[$resp. $\sigma$ is a ring anti-isomorphism$]$.

By Theorem 4.1, it is easy to prove the following corollary.

Corollary 4.2

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism, and there exist $A_{0},B_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(B_{0})-\varphi(A_{0}))=n$. Then either there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, a nonzero ring homomorphism $\tau:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, and a matrix $L\in{\mathbb{D}^{\prime}}^{n\times n}$ with the property that $I_{n}+X^{\tau}L\in GL_{n}(\mathbb{D}^{\prime})$ for every $X\in{\mathbb{D}}^{n\times n}$, such that

[TABLE]

or there exist matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$, a nonzero ring anti-homomorphism $\sigma:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$, and a matrix $L\in{\mathbb{D}^{\prime}}^{n\times n}$ with the property that $I_{n}+L\,^{t}X^{\sigma}\in GL_{n}(\mathbb{D}^{\prime})$ for every $X\in{\mathbb{D}}^{n\times n}$, such that

[TABLE]

or both $\varphi\left(\mathbb{B}_{A_{0}}\right)$ and $\varphi\left(\mathbb{B}_{B_{0}}\right)$ are adjacent sets. In particular, if $\tau$ $[$resp. \sigma$$] is surjective, then $L=0$ and $\tau$ is a ring isomorphism $[$resp. $\sigma$ is a ring anti-isomorphism$]$.

Note that maps (26)-(29) are distance preserving maps (cf. [14, Examples 2.5 and 2.6]. We have:

Corollary 4.3

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism, and there exist $A_{0},B_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(B_{0})-\varphi(A_{0}))=n$. Then either $\varphi$ is a distance preserving map, or both $\varphi\left(\mathbb{B}_{A_{0}}\right)$ and $\varphi\left(\mathbb{B}_{B_{0}}\right)$ are adjacent sets.

In order to prove Theorem 4.1 we need the following knowledge and lemmas.

It is well-known that $\mathbb{D}^{m\times n}$ ($m,n\geq 2$) is a partially ordered set (poset) with partial order defined by $A\leq B$ if ${\rm rank}(B-A)={\rm rank}(B)-{\rm rank}(A)$. This partial order is also called minus order (or minus partial order) of $D^{m\times n}$ (cf. [22]). In particular, we write $A<B$ $\Leftrightarrow$ $A\leq B$ with $A\neq B$.

For any $X\in{\mathbb{D}}^{m\times n}$, a g-inverse of $X$ will be denoted by $X^{-}$ and is understood as a matrix (over $\mathbb{D}$) for which $XX^{-}X=X$. Two matrices $A,B\in{\mathbb{D}}^{m\times n}$ are said to be equivalent, denoted by $A\cong B$, if $B$ may be obtained from $A$ by a finite sequence of elementary row and column operations.

Lemma 4.4

(cf. [22, Section 3.3])*  Let $\mathbb{D}$ be a division ring and let $A,B\in\mathbb{D}^{m\times n}$. Then the following results are equivalent:*

• (a)

$A\leq B$;

• (b)

there are g-inverses $G_{1}$ and $G_{2}$ of $A$ such that $AG_{1}=BG_{1}$ and $G_{2}A=G_{2}B$;

• (c)

there are $P\in GL_{m}(\mathbb{D})$ and $Q\in GL_{n}(\mathbb{D})$ such that $B=P{\rm diag}(I_{r+s},0)Q$ and $A=P{\rm diag}(I_{r},0)Q$;

• (d)

$PAQ\leq PBQ$, where $P\in GL_{m}(\mathbb{D})$ and $Q\in GL_{n}(\mathbb{D})$.

Lemma 4.5

(see [22, Theorem 3.6.5])*  Let $A,B\in\mathbb{D}^{n\times n}$ with $B^{2}=B$. Then $A\leq B$ if and only if $A=A^{2}=AB=BA$.*

Lemma 4.6

Let $A,B,C\in{\mathbb{D}}^{m\times n}$ and $d(A,B)=d(B,C)-d(A,C)$. Suppose that $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism and $d(B,C)=d(\varphi(B),\varphi(C))$. Then $d(\varphi(A),\varphi(B))=d(\varphi(B),\varphi(C))-d(\varphi(A),\varphi(C))$.

*Proof. *We have $d(\varphi(A),\varphi(B))\leq d(A,B)=d(B,C)-d(A,C)=d(\varphi(B),\varphi(C))-d(A,C)\leq d(\varphi(B),\varphi(C))-d(\varphi(A),\varphi(C)).$ Hence

[TABLE]

On the other hand, it is clear that $d(\varphi(A),\varphi(B))\geq d(\varphi(B),\varphi(C))-d(\varphi(A),\varphi(C))$. Thus $d(\varphi(A),\varphi(B))=d(\varphi(B),\varphi(C))-d(\varphi(A),\varphi(C))$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Corollary 4.7

Let $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a graph homomorphism with $\varphi(0)=0$, and let $A,B\in{\mathbb{D}}^{m\times n}$. If $A\leq B$ and ${\rm rank}(\varphi(B))={\rm rank}(B)$, then $\varphi(A)\leq\varphi(B)$ and ${\rm rank}(\varphi(A))={\rm rank}(A)$.

*Proof. *Assume that $A\leq B$ and ${\rm rank}(\varphi(B))={\rm rank}(B)$. Then $d(A,B)=d(B,0)-d(A,0)$ and $d(B,0)=d(\varphi(B),\varphi(0))$. By Lemma 4.6, $d(\varphi(A),\varphi(B))=d(\varphi(B),\varphi(0))-d(\varphi(A),\varphi(0))$, hence $\varphi(A)\leq\varphi(B)$. It follows that ${\rm rank}(\varphi(A))=d(\varphi(A),\varphi(0))=d(\varphi(B),\varphi(0))-d(\varphi(A),\varphi(B))\geq{\rm rank}(B)-d(A,B)={\rm rank}(A)$. Since ${\rm rank}(\varphi(A))\leq{\rm rank}(A)$, we obtain ${\rm rank}(\varphi(A))={\rm rank}(A)$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 4.8

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings, and let $n,m^{\prime},n^{\prime}$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that $d(\varphi(A_{0}),0)=n$. Assume that ${\cal M}$ is a maximal set in ${\mathbb{D}}^{n\times n}$ containing [math] and $\varphi(\mathcal{M})\subseteq{\cal M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. Then $\varphi({\cal M})$ is not contained in any line in $AG({\cal M}^{\prime})$, and ${\cal M}^{\prime}$ is the unique maximal set containing $\varphi(\mathcal{M})$ in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$.

*Proof. *Without loss of generality, we assume that ${\cal M}$ is a maximal set of type two, and ${\cal M}^{\prime}$ is a maximal set of type one. By Lemma 1.1, there are invertible matrices $P_{1},P_{2}$ such that $\mathcal{M}={\cal N}_{1}P_{1}$ and $\mathcal{M}^{\prime}=P_{2}{\cal M}^{\prime}_{1}$. Replacing $\varphi$ by the map $X\mapsto P_{2}^{-1}\varphi(XP_{1})$, we have ${\cal M}={\cal N}_{1}$, ${\cal M}^{\prime}={\cal M}_{1}^{\prime}$ and $\varphi({\cal N}_{1})\subseteq{\cal M}^{\prime}_{1}$.

Suppose $\varphi({\cal N}_{1})$ is contained in a line $\ell$ in $AG({\cal M}^{\prime}_{1})$. We show a contradiction as follows. By the parametric equation of a line, there exists an invertible matrix $Q$ such that $\ell=(\mathbb{D}^{\prime}E_{11}+B)Q$, where $E_{11}=E_{11}^{m^{\prime}\times n^{\prime}}$ and $B\in{\cal M}^{\prime}_{1}$. Since $0\in\ell$, we can assume that $B=0$ and $\ell=\mathbb{D}^{\prime}E_{11}Q$. Replacing $\varphi$ by the map $X\mapsto\varphi(X)Q^{-1}$, we have $\varphi(0)=0$ and $\varphi({\cal N}_{1})\subseteq\ell=\mathbb{D}^{\prime}E_{11}$. By the conditions, there is an $A_{0}\in{\mathbb{D}}^{n\times n}$ such that $d(\varphi(A_{0}),0)=n=d(A_{0},0)$. Let $A_{0}=(B_{1},B_{2})$ where $B_{1}\in\mathbb{D}_{2}^{n\times 2}$ and $B_{2}\in\mathbb{D}_{n-2}^{n\times(n-2)}$. When $n=2$, $B_{2}$ is absent. Put $A_{1}=(B_{1},0)\in\mathbb{D}_{2}^{n\times n}$. Then $A_{1}\leq A_{0}$. Using Corollary 4.7, we get ${\rm rank}(\varphi(A_{1}))=2$. Clearly, there are two distinct points $Y_{1},Y_{2}\in{\cal N}_{1}$ such that $A_{1}\sim Y_{i}$, $i=1,2$. Since $\varphi(Y_{i})\subseteq\ell$, $\varphi(A_{1})$ are adjacent to two distinct points in $\mathbb{D}^{\prime}E_{11}$. By Lemma 2.1, $\varphi(A_{1})\in{\cal M}_{1}^{\prime}$ or $\varphi(A_{1})\in{\cal N}_{1}^{\prime}$, and hence ${\rm rank}(\varphi(A_{1}))\leq 1$, a contradiction.

Thus $\varphi({\cal N}_{1})$ is not contained in any line of $AG({\cal M}^{\prime}_{1})$. By Corollary 2.3 and Lemma 2.5, it is easy to see that ${\cal M}^{\prime}$ is the unique maximal set containing $\varphi(\mathcal{M})$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 4.9

(cf. [24])*   Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$ and ${\rm rank}(\varphi(I_{n}))=n$. Then there exist invertible matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ such that*

[TABLE]

*Proof. *Since ${\rm diag}(I_{n-1},0)<I_{n}$ and ${\rm rank}(\varphi(I_{n}))=n$, from Corollary 4.7 we get that

[TABLE]

Using Lemma 4.4(c), there exist $Q_{1}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q_{2}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ such that $\varphi(I_{n})=Q_{1}{\rm diag}(I_{n},0)Q_{2}$ and $\varphi\left({\rm diag}(I_{n-1},0)\right)=Q_{1}{\rm diag}(I_{n-1},0)Q_{2}$. Replacing $\varphi$ by the map $X\mapsto Q_{1}^{-1}\varphi(X)Q_{2}^{-1}$, we have that $\varphi(I_{n})={\rm diag}\left(I_{n},0\right)$ and $\varphi\left({\rm diag}(I_{n-1},0)\right)={\rm diag}\left(I_{n-1},0_{m^{\prime}-n+1,n^{\prime}-n+1}\right)$.

Now, let $n\geq 3$. Since ${\rm diag}(I_{n-2},0)<{\rm diag}(I_{n-1},0)$, Corollary 4.7 implies that ${\rm rank}\left(\varphi\left({\rm diag}(I_{n-2},0\right)\right)=n-2$ and

[TABLE]

Write \scriptsize\varphi\left({\rm diag}(I_{n-2},0)\right)=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $A_{11}\in\mathbb{{D}^{\prime}}^{(n-1)\times(n-1)}$. By (31), Lemma 4.5 and multiplications of matrices, it is easy to see that $A_{12}=0$, $A_{21}=0$ and $A_{11}$ is an idempotent matrix. (Note that when $m^{\prime}\neq n^{\prime}$ and using Lemma 4.5, we can get two suitable square matrices by adding some zero elements on $\varphi\left({\rm diag}(I_{n-2},0)\right)$ and ${\rm diag}(I_{n-1},0_{m^{\prime}-n+1,n^{\prime}-n+1})$.) Since $\varphi\left({\rm diag}(I_{n-2},0)\right)\sim{\rm diag}(I_{n-1},0_{m^{\prime}-n+1,n^{\prime}-n+1})$, we have $A_{22}=0$. Hence $\varphi\left({\rm diag}(I_{n-2},0)\right)={\rm diag}\left(A_{11},0_{m^{\prime}-n+1,n^{\prime}-n+1}\right)$, where $A_{11}$ is an idempotent matrix of rank $n-2$. There is $T\in GL_{n-1}(\mathbb{D}^{\prime})$ such that $A_{11}=T^{-1}{\rm diag}(I_{n-2},0)T$. Let $Q_{3}={\rm diag}(T^{-1},I_{m^{\prime}-n+1})$ and $Q_{4}={\rm diag}(T,I_{n^{\prime}-n+1})$. Replacing $\varphi$ by the map $X\mapsto Q_{3}^{-1}\varphi(X)Q_{4}^{-1}$, we get that $\varphi(I_{n})={\rm diag}(I_{n},0_{m^{\prime}-n,n^{\prime}-n})$, $\varphi\left({\rm diag}(I_{n-1},0)\right)={\rm diag}\left(I_{n-1},0_{m^{\prime}-n+1,n^{\prime}-n+1}\right)$ and $\varphi\left({\rm diag}(I_{n-2},0)\right)={\rm diag}\left(I_{n-2},0_{m^{\prime}-n+2,n^{\prime}-n+2}\right)$.

Similarly, after by some transformations of the form $X\mapsto P^{\prime}\varphi(X)Q^{\prime}$, we can get $\varphi\left({\rm diag}(I_{r},0)\right)={\rm diag}\left(I_{r},0_{m^{\prime}-r,n^{\prime}-r}\right)$, $r=1,\ldots,n$. Thus (30) holds. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

A graph homomorphism $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ with $\varphi(0)=0$ is called to satisfy Condition (I), if for any two maximal sets ${\cal M}$ and ${\cal N}$ of different types in ${\mathbb{D}}^{m\times n}$ with $0\in{\cal M}\cap{\cal N}$, there are two maximal sets ${\cal M}^{\prime}$ and ${\cal N}^{\prime}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi({\cal M})\subseteq{\cal M}^{\prime}$ and $\varphi({\cal N})\subseteq{\cal N}^{\prime}$.

Lemma 4.10

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$. Assume further that $\varphi$ satisfies the Condition (I). Then

[TABLE]

*Proof. *By $\varphi(0)=0$ and (1), $A_{0}$ is invertible. Replacing $\varphi$ by the map $X\mapsto\varphi(A_{0}^{-1}X)$, we can assume that $A_{0}=I_{n}$. By Lemma 4.9, there exist invertible matrices $P\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ such that $\varphi\left({\rm diag}(I_{r},0)\right)=P{\rm diag}(I_{r},0)Q$, $r=1,\ldots,n$. Replacing $\varphi$ by the map $X\mapsto P^{-1}\varphi(X)Q^{-1}$, we get that

[TABLE]

Write $E_{ij}=E_{ij}^{n\times n}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. Note that $\varphi(E_{11})=E_{11}^{\prime}$. Since $\varphi$ satisfies the Condition (I), from Corollary 2.2 we have either $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$, or $\varphi(\mathcal{M}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. We prove (32) only for the first case; the second case is similar. From now on we assume that

[TABLE]

For any $x\in\mathbb{D}^{*}$, by $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$, we can assume that \scriptsize\varphi(E_{11}+xE_{12})=\left(\begin{array}[]{ccc}1^{*}&x^{\sigma_{1}}&0^{*}\\ 0&0&0\\ \end{array}\right). Since $E_{11}+xE_{12}\sim E_{11}+E_{22}$, it follows from (33) that \scriptsize{\rm rank}\left(\begin{array}[]{ccc}1^{*}-1&x^{\sigma_{1}}&0^{*}\\ 0&-1&0\\ \end{array}\right)=1. Thus $1^{*}=1$ and $0^{*}=0$. Therefore,

[TABLE]

where $\sigma_{1}:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is an injective map with $0^{\sigma_{1}}=0$. Similarly, we have

[TABLE]

where $\mu_{1}:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is an injective map with $0^{\mu_{1}}=0$.

For $2\leq k\leq n$ and $x_{i},y_{i}\in\mathbb{D}$ ($i=2,\ldots,k$), we have $E_{11}+x_{2}E_{12}+\cdots+x_{k}E_{1k}<{\rm diag}\left(I_{k},0\right)$ and $E_{11}+y_{2}E_{21}+\cdots+y_{k}E_{k1}<{\rm diag}\left(I_{k},0\right)$. Thus Corollary 4.7 and (33) imply that $\varphi(E_{11}+x_{2}E_{12}+\cdots+x_{k}E_{1k})<{\rm diag}\left(I_{k},0_{m^{\prime}-k,n^{\prime}-k}\right)$ and $\varphi(E_{11}+y_{2}E_{21}+\cdots+y_{k}E_{k1})<{\rm diag}\left(I_{k},0_{m^{\prime}-k,n^{\prime}-k}\right)$. Using (34), it is easy to see that

[TABLE]

[TABLE]

where $x_{j}^{*}$, $y_{j}^{*}\in\mathbb{D}^{\prime}$, $j=2,\ldots,k$. Moreover, by $\varphi(E_{11})=E_{11}^{\prime}$ and the adjacency, $(x_{2},\ldots,x_{k})\neq 0$ [resp. ${}^{t}(y_{2},\ldots,y_{k})\neq 0$] if and only if $(x_{2}^{*},\ldots,x_{k}^{*})\neq 0$ [resp. ${}^{t}(y_{2}^{*},\ldots,y_{k}^{*})\neq 0$].

For any $1\leq r\leq n-2$ and $2\leq k\leq n-r$, we let \scriptsize A_{k}=E_{11}^{k\times k}+x_{2}E_{12}^{k\times k}+\cdots+x_{k}E_{1k}^{k\times k}=\left(\begin{array}[]{cc}1&\alpha_{k}\\ 0&0\\ \end{array}\right), where $x_{i}\in\mathbb{D}$ and $\alpha_{k}:=(x_{2},\ldots,x_{k})\neq 0$. Then $A_{k}$ is a $k\times k$ idempotent matrices of rank one. By Corollary 4.7 and (33), we have that

[TABLE]

and ${\rm rank}\left(\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)\right)=r+1$. Write \scriptsize\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $A_{11}\in\mathbb{{D}^{\prime}}^{(r+k)\times(r+k)}$. By (33), $\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)$ is adjacent with both ${\rm diag}\left(I_{r+1},0\right)$ and ${\rm diag}\left(I_{r},0\right)$. It follows from Lemma 2.1 that $A_{22}=0$. By $\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)<{\rm diag}\left(I_{r+k},0\right)$ and Lemma 4.5, we get that $A_{12}=0$, $A_{21}=0$ and $A_{11}$ is an idempotent matrix of rank $r+1$. (Note that when $m^{\prime}\neq n^{\prime}$ and using Lemma 4.5, we can get two suitable square matrices by adding some zero elements on $\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)$ and ${\rm diag}(I_{r+k},0)$.) Thus $\varphi\left({\rm diag}\left(I_{r},A_{k},0\right)\right)={\rm diag}\left(A_{11},0\right)$.

Write \scriptsize A_{11}=\left(\begin{array}[]{cc}B_{11}&B_{12}\\ B_{21}&B_{22}\\ \end{array}\right) where $B_{11}\in\mathbb{{D}^{\prime}}^{r\times r}$ and $B_{22}\in\mathbb{{D}^{\prime}}^{k\times k}$. Using ${\rm diag}\left(I_{r},0_{k}\right)<A_{11}$ and Lemma 4.5, we have similarly that $B_{12}=0$ and $B_{21}=0$. By $B_{22}\neq 0$ and ${\rm diag}\left(I_{r},0_{k}\right)\sim A_{11}$, it is clear that $B_{11}=I_{r}$. Therefore, we obtain

[TABLE]

where $B_{22}\in\mathbb{{D}^{\prime}}^{k\times k}$ is an idempotent matrix of rank one.

Since ${\rm diag}\left(I_{r},A_{k},0\right)\sim{\rm diag}\left(I_{r+1},0\right)$, ${\rm diag}\left(I_{r},B_{22},0\right)\sim{\rm diag}\left(I_{r+1},0\right)$, and hence $B_{22}\sim E_{11}^{k\times k}$. By Lemma 2.1, we obtain either \scriptsize B_{22}=\left(\begin{array}[]{cc}1&\alpha_{k}^{*}\\ 0&0\\ \end{array}\right) or \scriptsize B_{22}=\left(\begin{array}[]{cc}1&0\\ {}^{t}\alpha_{k}^{*}&0\\ \end{array}\right), where $0\neq\alpha_{k}^{*}\in{\mathbb{D}^{\prime}}^{k-1}$. Since $E_{11}+E_{r+1,1}\leq{\rm diag}\left(I_{r},A_{k},0\right)$, Corollary 4.7 implies that $\small\varphi(E_{11}+E_{r+1,1})\leq\varphi\left({\rm diag}(I_{r},A_{k},0)\right)$. From (38) we must have \scriptsize B_{22}=\left(\begin{array}[]{cc}1&\alpha_{k}^{*}\\ 0&0\\ \end{array}\right). Therefore, for any $0\neq\alpha_{k}\in\mathbb{D}^{k-1}$, there is $0\neq\alpha_{k}^{*}\in{\mathbb{D}^{\prime}}^{k-1}$ such that

[TABLE]

Similarly, for any $0\neq\beta_{k}\in\,^{k-1}\mathbb{D}$, there is $0\neq\beta_{k}^{*}\in\,^{k-1}{\mathbb{D}^{\prime}}$ such that

[TABLE]

For any $2\leq k\leq n$, by (37) we write

[TABLE]

where $a_{1j}^{(k)}\in\mathbb{D}^{\prime}$ and $(a_{12}^{(k)},\ldots,a_{1k}^{(k)})\neq(0,\ldots,0)$. From (35) we have $a_{12}^{(2)}\neq 0$.

We prove $a_{1k}^{(k)}\neq 0$ ($3\leq k\leq n$) as follows. For any $3\leq k\leq n$ and $2\leq r<k$, by (39), we let

[TABLE]

where $a_{rj}^{(k)}\in\mathbb{D}^{\prime}$ and $(a_{r,r+1}^{(k)},\ldots,a_{rk}^{(k)})\neq(0,\ldots,0)$.

By (33) and the adjacency, when $r=k-1$ we have $a_{k-1,k}^{(k)}\neq 0$. We show $a_{rk}^{(k)}\neq 0$, $r=2,\ldots,k-2$.

Since $\varphi\left(E_{11}+\cdots+E_{k-1,k-1}+E_{k-1,k}\right)\sim\varphi\left(E_{11}+\cdots+E_{k-2,k-2}+E_{k-2,k-1}+E_{k-2,k}\right),$ we get

[TABLE]

which implies that \small{\rm rank}\left(\begin{array}[]{cc}a_{k-2,k-1}^{(k)}&a_{k-2,k}^{(k)}\\ 1&a_{k-1,k}^{(k)}\\ \end{array}\right)=1. By $(a_{k-2,k-1}^{(k)},a_{k-2,k}^{(k)})\neq(0,0)$ and $a_{k-1,k}^{(k)}\neq 0$, we have $a_{k-2,k}^{(k)}\neq 0.$

Since $\varphi\left(E_{11}+\cdots+E_{k-2,k-2}+E_{k-2,k-1}+E_{k-2,k}\right)\sim\varphi\left(E_{11}+\cdots+E_{k-3,k-3}+\sum_{j=k-2}^{k}E_{k-3,j}\right),$ one has

[TABLE]

Similarly, we have $a_{k-3,k}^{(k)}\neq 0.$ In the similar way, we can prove that $a_{k-4,k}^{(k)}\neq 0$, $\ldots$, $a_{2k}^{(k)}\neq 0$.

Since $\varphi\left(E_{11}+\sum_{j=2}^{k}E_{1j}\right)\sim\varphi\left(E_{11}+E_{22}+\sum_{j=3}^{k}E_{2j}\right),$ $E_{11}^{\prime}+\sum_{j=2}^{k}a_{1j}^{(k)}E_{1j}^{\prime}\sim E_{11}^{\prime}+E_{22}^{\prime}+\sum_{j=3}^{k}a_{2j}^{(k)}E_{2j}^{\prime}.$ It follows that

[TABLE]

By $(a_{12}^{(k)},\ldots,a_{1k}^{(k)})\neq(0,\ldots,0)$ and $a_{2k}^{(k)}\neq 0$, we must have $a_{1k}^{(k)}\neq 0$. Therefore, we have proved that

[TABLE]

Let $\gamma_{1}=E_{11}$, $\gamma_{k}=E_{11}+\sum_{j=2}^{k}E_{1j}$, $\gamma_{1}^{*}=E_{11}^{\prime}$ and $\gamma_{k}^{*}=E_{11}^{\prime}+\sum_{j=2}^{k}a_{1j}^{(k)}E_{1j}^{\prime}$, $k=2,\ldots,n$. Then $\gamma_{i}\in\mathcal{M}_{1}$, $\gamma_{i}^{*}\in\mathcal{M}_{1}^{\prime}$, $\gamma_{1},\ldots,\gamma_{n}$ are left linearly independent over $\mathbb{D}$, and $\gamma_{1}^{*},\ldots,\gamma_{n}^{*}$ are left linearly independent over $\mathbb{D}^{\prime}$. By (41) we have $\varphi(\gamma_{k})=\gamma_{k}^{*}$, $k=1,\ldots,n$, and hence ${\rm dim}(\varphi(\mathcal{M}_{1}))\geq n$. By Lemma 2.7, we get ${\rm dim}(\varphi(\mathcal{M}_{1}))\leq n$. Hence ${\rm dim}(\varphi(\mathcal{M}_{1}))=n$.

Similarly, we can prove ${\rm dim}(\varphi(\mathcal{N}_{1}))=n$. Hence ${\rm dim}(\varphi(\mathcal{M}_{1}))={\rm dim}(\varphi(\mathcal{N}_{1}))=n.$ $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 4.11

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$. Assume further that $\varphi$ does not satisfies the Condition (I). Then there exist two maximal sets ${\cal M}^{\prime}$ and ${\cal N}^{\prime}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ such that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}^{\prime}\cup{\cal N}^{\prime}$. Moreover, if $|\mathbb{D}|\geq 4$, then $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set.

*Proof. *Step 1.  Since $\varphi$ does not satisfies the Condition (I), there are two maximal sets ${\cal M}$ and ${\cal R}$ of different types in ${\mathbb{D}}^{m\times n}$ with $0\in{\cal M}\cap{\cal R}$, such that $\varphi({\cal M})\subseteq{\cal M}^{\prime}$ and $\varphi({\cal R})\subseteq{\cal R}^{\prime}$, where ${\cal M}^{\prime}$ and ${\cal R}^{\prime}$ are two maximal sets of the same type containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. Since $|{\cal M}\cap{\cal R}|\geq 2$, $|{\cal M}^{\prime}\cap{\cal R}^{\prime}|\geq 2$. It follows from Lemma 2.3 that ${\cal M}^{\prime}={\cal R}^{\prime}$. Thus

[TABLE]

Without loss of generality, we assume that both $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are of type one. Hence $\mathcal{R}$ is of type two. By Lemma 2.6(b), every non-degenerate graph homomorphism satisfies the Condition (I). Thus $\varphi$ is degenerate.

We prove that there exist a type two maximal set ${\cal N}^{\prime}$ such that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}^{\prime}\cup{\cal N}^{\prime}$ as follows. Suppose $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set. By (42) and Lemma 4.8, it is easy to see that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}^{\prime}$. Now, suppose $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is not an adjacent set. Then there exists a type one maximal set ${\cal N}$ containing [math] in ${\mathbb{D}}^{n\times n}$, such that ${\cal M}\neq{\cal N}$, $\varphi({\cal N})\subseteq{\cal N}^{\prime}$ and ${\cal N}^{\prime}\neq{\cal M}^{\prime}$, where ${\cal N}^{\prime}$ is a maximal set containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. Otherwise, for any type one maximal set ${\cal N}$ containing [math] in ${\mathbb{D}}^{n\times n}$, we have $\varphi({\cal N})\subseteq{\cal M}^{\prime}$. Since every matrix in $\mathbb{D}_{\leq 1}^{n\times n}$ is contained in some type one maximal set containing [math], we get $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}^{\prime}$, a contradiction to assumption. By Corollary 2.3, we have $|{\cal N}\cap{\cal R}|\geq 2$, hence $|{\cal N}^{\prime}\cap{\cal M}^{\prime}|\geq 2$. It follows that ${\cal N}^{\prime}$ is of type two. Let ${\cal S}$ be any type two maximal set containing [math] in ${\mathbb{D}}^{n\times n}$, and let $\varphi({\cal S})\subseteq{\cal S}^{\prime}$ where ${\cal S}^{\prime}$ is a maximal set containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. By Corollary 2.3, we have $|{\cal S}\cap{\cal M}|\geq 2$ and $|{\cal S}\cap{\cal N}|\geq 2$, hence $|{\cal S}^{\prime}\cap{\cal M}^{\prime}|\geq 2$ and $|{\cal S}^{\prime}\cap{\cal N}^{\prime}|\geq 2$. Applying Corollary 2.3 again, we get either ${\cal S}^{\prime}={\cal M}^{\prime}$ or ${\cal S}^{\prime}={\cal N}^{\prime}$, and hence ${\cal S}^{\prime}\subseteq{\cal M}^{\prime}\cup{\cal N}^{\prime}$. Consequently, $\varphi({\cal S})\subseteq{\cal M}^{\prime}\cup{\cal N}^{\prime}$ for any type two maximal set ${\cal S}$ containing [math] in ${\mathbb{D}}^{n\times n}$. Since every matrix in $\mathbb{D}_{\leq 1}^{n\times n}$ is contained in some type two maximal set containing [math], we get $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}^{\prime}\cup{\cal N}^{\prime}$. Therefore, we always have

[TABLE]

Step 2.  From now on we assume $|\mathbb{D}|\geq 4$. In this step, we will prove that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set.

By Lemma 2.4(i), there are two invertible matrices $P_{1}$ and $Q_{1}$ over $\mathbb{D}$, such that $\mathcal{M}=P_{1}\mathcal{M}_{1}Q_{1}$ and $\mathcal{R}=P_{1}\mathcal{N}_{1}Q_{1}$. Also, there is an invertible matrix $P_{2}$ over $\mathbb{D}^{\prime}$, such that $\mathcal{M}^{\prime}=P_{2}\mathcal{M}_{1}^{\prime}$. Replacing $\varphi$ by the map $X\mapsto P_{2}^{-1}\varphi(P_{1}XQ_{1})$, (42) becomes

[TABLE]

Suppose that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is not an adjacent set. We show a contradiction as follows.

There exists a type one maximal set ${\cal R}_{1}$ containing [math] in ${\mathbb{D}}^{n\times n}$ with ${\cal R}_{1}\neq{\cal M}_{1}$, such that $\varphi({\cal R}_{1})\nsubseteq{\cal M}_{1}^{\prime}$. Otherwise, for any type one maximal set ${\cal R}_{1}$ containing [math] in ${\mathbb{D}}^{n\times n}$, we have $\varphi({\cal R}_{1})\subseteq{\cal M}_{1}^{\prime}$. Since every matrix in $\mathbb{D}_{\leq 1}^{n\times n}$ is contained in some type one maximal set containing [math], we get $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)\subseteq{\cal M}_{1}^{\prime}$, a contradiction. By (43), it is easy to see that

[TABLE]

Using Lemma 2.4(ii), there is $P_{3}\in GL_{m}(\mathbb{D})$ such that ${\cal R}_{1}=P_{3}{\cal M}_{2}$ and ${\cal M}_{1}=P_{3}{\cal M}_{1}$. On the other hand, there is $Q_{2}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$ such that ${\cal N}^{\prime}={\cal N}^{\prime}_{1}Q_{2}$. Modifying the map $\varphi$ by the map $X\mapsto\varphi(P_{3}X)Q_{2}^{-1}$. We get $\varphi({\cal M}_{2})\subseteq{\cal N}_{1}^{\prime}$ and (44) holds. Therefore, we may assume that

[TABLE]

By the conditions of this lemma, there exists $A_{0}\in GL_{n}(\mathbb{D})$ such that ${\rm rank}(\varphi(A_{0}))=n$. Write \scriptsize A_{0}=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{n}\\ \end{array}\right)=\left(\beta_{1},\ldots,\beta_{n}\right) where $\alpha_{i}\in{\mathbb{D}}^{n}$ and $\beta_{i}\in\,^{n}{\mathbb{D}}$; \scriptsize\varphi(A_{0})=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $A_{11}\in{\mathbb{D}^{\prime}}^{1\times 1}$ and $A_{22}\in{\mathbb{D}^{\prime}}^{(m^{\prime}-1)\times(n^{\prime}-1)}$. Let \scriptsize B=\left(\begin{array}[]{c}\alpha_{1}\\ 0_{n-1,n}\\ \end{array}\right), $\small C=\left(\beta_{1},0,\ldots,0\right)$ and \scriptsize J_{n}=\left(\begin{array}[]{ccc}0&0&1\\ 0&I_{n-2}&0\\ 1&0&0\\ \end{array}\right). Then $B<A_{0}$ and $C<A_{0}$. By Corollary 4.7, we have $\varphi(B)<\varphi(A_{0})$ and $\varphi(C)<\varphi(A_{0})$. Since $\varphi(B)\in\mathcal{M}_{1}^{\prime}$ and $\varphi(C)\in\mathcal{N}_{1}^{\prime}$, it is clear that

[TABLE]

By (46) and ${\rm rank}(\varphi(A_{0}))=n$, we have either $A_{22}\cong{\rm diag}(I_{n-1},0)$ or $A_{22}\cong{\rm diag}(I_{n-2},0)$. By appropriate elementary row and column operations on matrices, we can obtain either ${\rm diag}(I_{n},0)$ (if $A_{22}\cong{\rm diag}(I_{n-1},0)$) or ${\rm diag}(J_{n},0)$ (if $A_{22}\cong{\rm diag}(I_{n-2},0)$) from $\varphi(A_{0})$. Since our matrix elementary operations do not change ${\cal N}_{1}^{\prime}$ and ${\cal M}_{1}^{\prime}$, (45) still holds. Thus, without loss of generality we may assume either $\varphi(A_{0})={\rm diag}(I_{n},0)$ or $\varphi(A_{0})={\rm diag}(J_{n},0)$.

Case 1.   $\varphi(A_{0})={\rm diag}(I_{n},0)$. Since \scriptsize\left(\begin{array}[]{c}0\\ \alpha_{2}+\lambda\alpha_{1}\\ 0_{n-2,n}\\ \end{array}\right)<A_{0} for all $\lambda\in\mathbb{D}^{*}$, Corollary 4.7 implies that \scriptsize\varphi\left(\begin{array}[]{c}0\\ \alpha_{2}+\lambda\alpha_{1}\\ 0_{n-2,\,n}\\ \end{array}\right)<\varphi(A_{0}), $\lambda\in\mathbb{D}^{*}$. Thus $\varphi({\cal M}_{2})\subseteq{\cal N}_{1}^{\prime}$ implies that \scriptsize\varphi\left(\begin{array}[]{c}0\\ \alpha_{2}+\lambda\alpha_{1}\\ 0_{n-2,\,n}\\ \end{array}\right)=\left(\begin{array}[]{cc}1&0\\ Z_{\lambda}&0_{m^{\prime}-1,n^{\prime}-1}\\ \end{array}\right), where $Z_{\lambda}\in\,^{m^{\prime}-1}{\mathbb{D}^{\prime}}$, $\lambda\in\mathbb{D}^{*}$. Since \scriptsize\varphi\left(\begin{array}[]{c}\lambda^{-1}\alpha_{2}+\alpha_{1}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)<\varphi(A_{0}) for all $\lambda\in\mathbb{D}^{*}$, it follows from $\varphi({\cal M}_{1})\subseteq{\cal M}_{1}^{\prime}$ that \scriptsize\varphi\left(\begin{array}[]{c}\lambda^{-1}\alpha_{2}+\alpha_{1}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)=\left(\begin{array}[]{cc}1&T_{\lambda}\\ 0&0\\ \end{array}\right), where $T_{\lambda}\in{\mathbb{D}^{\prime}}^{n^{\prime}-1}$, $\lambda\in\mathbb{D}^{*}$. By $|\mathbb{D}^{*}|\geq 3$ and the adjacency, there is $\lambda_{0}\in\mathbb{D}^{*}$ such that $Z_{\lambda_{0}}\neq 0$ and $T_{\lambda_{0}}\neq 0$. By \scriptsize\varphi\left(\begin{array}[]{c}\lambda^{-1}_{0}\alpha_{2}+\alpha_{1}\\ 0\\ 0\\ \end{array}\right)\sim\varphi\left(\begin{array}[]{c}0\\ \alpha_{2}+\lambda_{0}\alpha_{1}\\ 0_{n-2,n}\\ \end{array}\right), we obtain \scriptsize\left(\begin{array}[]{cc}1&T_{\lambda_{0}}\\ 0&0\\ \end{array}\right)\sim\left(\begin{array}[]{cc}1&0\\ Z_{\lambda_{0}}&0_{m^{\prime}-1,n^{\prime}-2}\\ \end{array}\right), a contradiction.

Case 2.   $\varphi(A_{0})={\rm diag}(J_{n},0)$. By \scriptsize\left(\begin{array}[]{c}\alpha_{1}+\alpha_{2}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)<A_{0} and Corollary 4.7, \scriptsize\varphi\left(\begin{array}[]{c}\alpha_{1}+\alpha_{2}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)<\varphi(A_{0}). It follows from $\varphi({\cal M}_{1})\subseteq{\cal M}_{1}^{\prime}$ that \scriptsize\varphi\left(\begin{array}[]{c}\alpha_{1}+\alpha_{2}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)=\left(\begin{array}[]{ccc}\alpha&1&0\\ 0&0&0\\ 0&0&0_{m^{\prime}-n,n^{\prime}-n}\\ \end{array}\right) where $\alpha\in{\mathbb{D}^{\prime}}^{n-1}$. Since \scriptsize\left(\begin{array}[]{c}0\\ \alpha_{1}+\alpha_{2}\\ 0_{n-2,\,n}\\ \end{array}\right)<A_{0} and Corollary 4.7, \scriptsize\varphi\left(\begin{array}[]{c}0\\ \alpha_{1}+\alpha_{2}\\ 0_{n-2,n}\\ \end{array}\right)<\varphi(A_{0}). By $\varphi({\cal M}_{2})\subseteq{\cal N}_{1}^{\prime}$, we get \scriptsize\varphi\left(\begin{array}[]{c}0\\ \alpha_{1}+\alpha_{2}\\ 0_{n-2,n}\\ \end{array}\right)=\left(\begin{array}[]{ccc}\beta&0&0\\ 1&0&0\\ 0&0&0_{m^{\prime}-n,n^{\prime}-n}\\ \end{array}\right) where $\beta\in\,^{n-1}\mathbb{D}^{\prime}$. Since \small\varphi\left(\begin{array}[]{c}\alpha_{1}+\alpha_{2}\\ 0\\ 0_{n-2,n}\\ \end{array}\right)\sim\varphi\left(\begin{array}[]{c}0\\ \alpha_{1}+\alpha_{2}\\ 0_{n-2,n}\\ \end{array}\right), \small\left(\begin{array}[]{ccc}\alpha&1&0\\ 0&0&0\\ 0&0&0_{m^{\prime}-n,n^{\prime}-n}\\ \end{array}\right)\sim\left(\begin{array}[]{ccc}\beta&0&0\\ 1&0&0\\ 0&0&0_{m^{\prime}-n,n^{\prime}-n}\\ \end{array}\right), a contradiction.

Thus both Case 1 and Case 2 cannot occur. It follows that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 4.12

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m^{\prime},n^{\prime},n$ be integers with $m^{\prime},n^{\prime}\geq n\geq 2$. Suppose that $\varphi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$. Assume further that $\varphi$ satisfies the Condition (I). Then $\varphi$ is non-degenerate.

*Proof. * We prove that $\varphi$ is non-degenerate by contradiction. Suppose that $\varphi$ is degenerate. Then there exists a matrix $A\in\mathbb{D}^{n\times n}_{\leq 1}$ and there are two maximal sets $\mathcal{R}$ and $\mathcal{S}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ such that

[TABLE]

Without loss of generality, we assume that $\mathcal{R}$ is of type one and $\mathcal{S}$ is of type two. By ${\rm rank}(\varphi(A_{0}))=n$ and (1), we have ${\rm rank}(A_{0})=n$. Since $d(\varphi(A),0)\leq 1$, we have either $d(\varphi(A_{0}),\varphi(A))=n$ or $d(\varphi(A_{0}),\varphi(A))=n-1$. Note either $d(A_{0},A)=n$ or $d(A_{0},A)=n-1$. Let $E_{ij}=E_{ij}^{n\times n}$ and $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$. We distinguish the following cases to show a contradiction.

Case 1.  $d(A_{0},A)=n$ and $d(\varphi(A_{0}),\varphi(A))=n-1$. Then ${\rm rank}(\varphi(A))={\rm rank}(A)=1$ and $\varphi(A)<\varphi(A_{0})$. By Lemma 4.4(c), there are invertible matrices $P_{1},Q_{1}$ over $\mathbb{D}^{\prime}$ such that $\varphi(A)=P_{1}E_{11}^{\prime}Q_{1}$ and $\varphi(A_{0})=P_{1}{\rm diag}(I_{n},0)Q_{1}$. Also, there are invertible matrices $P_{2},Q_{2}$ over $\mathbb{D}$ such that $P_{2}AQ_{2}=E_{11}$. Let \scriptsize P_{2}A_{0}Q_{2}=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $A_{11}\in{\mathbb{D}}^{1\times 1}$ and $A_{22}\in{\mathbb{D}}^{(n-1)\times(n-1)}$, Put \scriptsize J_{n}=\left(\begin{array}[]{ccc}0&0&1\\ 0&I_{n-2}&0\\ 1&0&0\\ \end{array}\right).

By ${\rm rank}(A_{0})={\rm rank}(A_{0}-A)=n$, it is clear that $A_{22}\cong I_{n-1}$ or $A_{22}\cong{\rm diag}(I_{n-2},0)$. Using appropriate elementary row and column operations on matrices, we can obtain either ${\rm diag}(a,I_{n-1})$ where $a\notin\{0,1\}$ (if $A_{22}\cong I_{n-1}$) or $J_{n}$ (if $A_{22}\cong{\rm diag}(I_{n-2},0)$) from $P_{2}A_{0}Q_{2}$. Moreover, $P_{2}AQ_{2}=E_{11}$ is unchanged. Thus, there are invertible matrices $P_{3},Q_{3}$ over $\mathbb{D}$, such that $P_{3}AQ_{3}=E_{11}$ and either $P_{3}A_{0}Q_{3}={\rm diag}(a,I_{n-1})$ or $P_{3}A_{0}Q_{3}=J_{n}$. Replacing $\varphi$ by the map $X\mapsto P_{1}^{-1}\varphi(P_{3}^{-1}XQ_{3}^{-1})Q_{1}^{-1}$, we have that $A=E_{11}$, and either $A_{0}={\rm diag}(a,I_{n-1})$ or $A_{0}=J_{n}$. Moreover, $\varphi(E_{11})=E_{11}^{\prime}$ and $\varphi({\rm diag}(a,I_{n-1}))={\rm diag}(I_{n},0)$. On the other hand, (47) becomes $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}+E_{11}\right)\subseteq\mathcal{R}\cup\mathcal{S}$ with $\varphi(E_{11})=E_{11}^{\prime}\in\mathcal{R}\cap\mathcal{S}$.

Recall that $\varphi$ satisfies the Condition (I). By $\varphi(E_{11})=E_{11}^{\prime}$, $\varphi(0)=0$ and Corollary 2.2, we may assume with no loss of generality that

[TABLE]

Since $\mathcal{M}_{1},\mathcal{N}_{1}\subseteq\mathbb{D}_{\leq 1}^{n\times n}+E_{11}$, we get that $\varphi(\mathcal{M}_{1})\subseteq{\cal M}^{\prime}_{1}\cap(\mathcal{R}\cup\mathcal{S})$ and $\varphi(\mathcal{N}_{1})\subseteq{\cal N}^{\prime}_{1}\cap(\mathcal{R}\cup\mathcal{S})$. Suppose that ${\cal M}^{\prime}_{1}\neq\mathcal{R}$. Then ${\cal M}^{\prime}_{1}\cap(\mathcal{R}\cup\mathcal{S})={\cal M}^{\prime}_{1}\cap\mathcal{S}$, thus $\ell:={\cal M}^{\prime}_{1}\cap\mathcal{S}$ is a line in $AG({\cal M}^{\prime}_{1})$ and $\varphi({\cal M}^{\prime}_{1})\subseteq\ell$, a contradiction to Lemma 4.8. Therefore, we must have ${\cal M}^{\prime}_{1}=\mathcal{R}$. Similarly, ${\cal N}^{\prime}_{1}=\mathcal{S}$. It follows that

[TABLE]

Subcase 1.1.  $A_{0}={\rm diag}(a,I_{n-1})$. By (48), we have $\varphi(aE_{11})=a^{*}E_{11}^{\prime}$ where $a^{*}\in{\mathbb{D}^{\prime}}^{*}$. Since $aE_{11}<A_{0}$, Corollary 4.7 implies that $\varphi(aE_{11})<\varphi(A_{0})={\rm diag}(I_{n},0)$. Consequently, $a^{*}=1$ and hence $\varphi(aE_{11})=E_{11}^{\prime}$, a contradiction to $\varphi(aE_{11})\sim\varphi(E_{11})$.

Subcase 1.2.  $A_{0}=J_{n}$. By (49), we get $\varphi\left(E_{nn}+E_{11}\right)\in\mathcal{M}_{1}^{\prime}\cup\mathcal{N}_{1}^{\prime}$. Without loss of generality, we assume that $\varphi\left(E_{nn}+E_{11}\right)\in\mathcal{M}_{1}^{\prime}$. Since $d(E_{nn}+E_{11},J_{n})=n-1$, $d\left(\varphi(E_{nn}+E_{11}),{\rm diag}(I_{n},0)\right)\leq n-1$. It follows that \scriptsize\varphi(E_{nn}+E_{11})=\left(\begin{array}[]{cc}1&\alpha\\ 0&0\\ \end{array}\right), where $0\neq\alpha\in{\mathbb{D}^{\prime}}^{n^{\prime}-1}$ because $\varphi(E_{nn}+E_{11})\sim\varphi(E_{11})=E_{11}^{\prime}$. Since $d(E_{n1}+E_{11},J_{n})=n-1$, $d\left(\varphi(E_{n1}+E_{11}),{\rm diag}(I_{n},0)\right)\leq n-1$. Thus $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ implies that \scriptsize\varphi(E_{n1}+E_{11})=\left(\begin{array}[]{cc}1&0\\ \beta&0\\ \end{array}\right), where $0\neq\beta\in\,^{m^{\prime}-1}{\mathbb{D}^{\prime}}$ because $\varphi(E_{n1}+E_{11})\sim\varphi(E_{11})$. Since $\varphi(E_{nn}+E_{11})\sim\varphi(E_{n1}+E_{11})$, one gets \scriptsize\left(\begin{array}[]{cc}1&\alpha\\ 0&0\\ \end{array}\right)\sim\left(\begin{array}[]{cc}1&0\\ \beta&0\\ \end{array}\right), a contradiction.

By Subcases 1.1-1.2, we always have a contradiction in Case 1.

Case 2.  $d(A_{0},A)=n$ and $d(\varphi(A_{0}),\varphi(A))=n$. Let $\psi(X)=\varphi(X+A)-\varphi(A)$, $X\in{\mathbb{D}}^{n\times n}$. Then $\psi:{\mathbb{D}}^{n\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\psi(0)=0$ and ${\rm rank}(\psi(A_{0}-A))=n$. Moreover, from (47) we have

[TABLE]

where $\mathcal{R}^{\prime}=\mathcal{R}-\varphi(A)$ is a maximal set of type one, and $\mathcal{S}^{\prime}=\mathcal{S}-\varphi(A)$ is a maximal set of type two.

Subcase 2.1.   $\psi$ satisfies the Condition (I). Then, by Lemma 4.10 we have

[TABLE]

By Lemma 1.1, without loss of generality, we may assume either $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$, or $\varphi(\mathcal{M}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. We prove this result only for the first case; the second case is similar. From now on we assume that

[TABLE]

By (50) and (52), one has $\psi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}\cap(\mathcal{R}^{\prime}\cup\mathcal{S}^{\prime})$ and $\psi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}\cap(\mathcal{R}^{\prime}\cup\mathcal{S}^{\prime})$. Clearly, $\mathcal{M}_{1}^{\prime}\cap\mathcal{R}^{\prime}=\{0\}=\mathcal{N}_{1}^{\prime}\cap\mathcal{S}^{\prime}$. We assert $\mathcal{M}_{1}^{\prime}=\mathcal{R}^{\prime}$. Otherwise, we have that $\mathcal{M}_{1}^{\prime}\neq\mathcal{R}^{\prime}$ and $\psi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}\cap(\mathcal{R}^{\prime}\cup\mathcal{S}^{\prime})=\mathcal{M}_{1}^{\prime}\cap\mathcal{S}^{\prime}$. Hence Lemma 2.5 implies that $\ell:=\mathcal{M}_{1}^{\prime}\cap\mathcal{S}^{\prime}$ is a line in $AG({\cal M}^{\prime}_{1})$, a contradiction to Lemma 4.8. Similarly, $\mathcal{N}_{1}^{\prime}=\mathcal{S}^{\prime}$. Therefore,

[TABLE]

By $\psi$ satisfying the Condition (I) and $\psi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$, $\psi$ maps any type two maximal set containing [math] into a type two maximal set. Thus (53) and Lemma 4.8 imply that $\psi$ maps any type two maximal set containing [math] into $\mathcal{N}_{1}^{\prime}$. By (51), let $\{B_{1},\ldots,B_{n}\}$ be a maximal left linear independent subset in $\psi(\mathcal{M}_{1})$, and let $\psi(C_{i})=B_{i}$ where $C_{i}\in\mathcal{M}_{1}$, $i=1,\ldots,n$. Since $C_{i}\sim 0$ and $\psi$ satisfies the Condition (I), there is a type two maximal set $\mathcal{R}_{i}$ containing $C_{i}$ and [math] in $\mathbb{D}^{n\times n}$, such that $\psi(\mathcal{R}_{i})\subseteq\mathcal{N}_{1}^{\prime}$, $i=1,\ldots,n$. Thus, $\{B_{1},\ldots,B_{n}\}\subset(\mathcal{M}_{1}^{\prime}\cap\mathcal{N}_{1}^{\prime})=\mathbb{D}^{\prime}E_{11}^{\prime}$, a contradiction.

Subcase 2.2.   $\psi$ does not satisfies the Condition (I). By Lemma 4.11, $\psi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set. Thus $\varphi\left(\mathbb{B}_{A}\right)$ is an adjacent set. Then, there is a maximal set $\mathcal{M}^{\prime}$ containing $\varphi(A)$ in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that

[TABLE]

Without loss of generality, we assume that $\mathcal{M}^{\prime}$ is of type one.

Since ${\rm rank}(A)\leq 1$, there is a type one maximal set $\mathcal{M}_{A}$ [resp. type two maximal set $\mathcal{S}_{A}$] containing $A$ and [math] in $\mathbb{D}^{n\times n}$. Since $\varphi$ satisfies the Condition (I), there are two maximal sets ${\cal M}_{A}^{\prime}$ and ${\cal S}_{A}^{\prime}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi({\cal M}_{A})\subseteq{\cal M}^{\prime}_{A}$ and $\varphi({\cal S}_{A})\subseteq{\cal S}^{\prime}_{A}$. Without loss of generality, we assume that $\mathcal{M}^{\prime}_{A}$ is of type one and $\mathcal{S}^{\prime}_{A}$ is of type two. Since $\mathcal{S}_{A}\subseteq\mathbb{D}_{\leq 1}^{n\times n}+A$, it follows from (54) that $\varphi(\mathcal{S}_{A})\subseteq({\cal S}^{\prime}_{A}\cap{\cal M}^{\prime})$. By Lemma 2.5, $\ell:={\cal S}^{\prime}_{A}\cap{\cal M}^{\prime}$ is a line in $AG({\cal S}^{\prime}_{A})$, a contradiction to Lemma 4.8.

Therefore, there is always a contradiction in Case 2.

Case 3.  $d(A_{0},A)=n-1$. Then $A<A_{0}$. Since ${\rm rank}(\varphi(A_{0}))={\rm rank}(A_{0})=n$ and Corollary 4.7, we get $\varphi(A)<\varphi(A_{0})$. By Lemma 4.4(c), there are invertible matrices $P_{1},Q_{1}$ over $\mathbb{D}$ such that $A=P_{1}E_{11}Q_{1}$ and $A_{0}=P_{1}I_{n}Q_{1}$, and there are invertible matrices $P_{2},Q_{2}$ over $\mathbb{D}^{\prime}$ such that $\varphi(A)=P_{2}E_{11}^{\prime}Q_{2}$ and $\varphi(A_{0})=P_{2}{\rm diag}(I_{n},0)Q_{2}$. Replacing $\varphi$ by the map $X\mapsto P_{2}^{-1}\varphi(P_{1}XQ_{1})Q_{2}^{-1}$, we have that $A=E_{11}$, $A_{0}=I_{n}$, $\varphi(E_{11})=E_{11}^{\prime}$ and $\varphi(I_{n})={\rm diag}(I_{n},0)$.

Since $\varphi$ satisfies the Condition (I) and $\varphi(E_{11})=E_{11}^{\prime}$, Corollary 2.2 implies either $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$, or $\varphi(\mathcal{M}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. We prove this result only for the first case; the second case is similar. From now on we assume that

[TABLE]

Recalling (47), we have $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}+E_{11}\right)\subseteq\mathcal{R}\cup\mathcal{S}$ with $\varphi(E_{11})=E_{11}^{\prime}\in\mathcal{R}\cap\mathcal{S}$. Since $\mathcal{M}_{1},\mathcal{N}_{1}\subset(\mathbb{D}_{\leq 1}^{n\times n}+E_{11})$, we obtain that $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}\cap(\mathcal{R}\cup\mathcal{S})=(\mathcal{M}_{1}^{\prime}\cap\mathcal{R})\cup(\mathcal{M}_{1}^{\prime}\cap\mathcal{S})$ and $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}\cap(\mathcal{R}\cup\mathcal{S})=(\mathcal{N}_{1}^{\prime}\cap\mathcal{R})\cup(\mathcal{N}_{1}^{\prime}\cap\mathcal{S})$.

We show $\mathcal{M}_{1}^{\prime}=\mathcal{R}$ by contradiction. Suppose that $\mathcal{M}_{1}^{\prime}\neq\mathcal{R}$. Then $\mathcal{M}_{1}^{\prime}\cap\mathcal{R}=\{E_{11}^{\prime}\}$ and hence $\varphi(\mathcal{M}_{1})\subseteq(\mathcal{M}_{1}^{\prime}\cap\mathcal{S})$. By Lemma 2.5, $\ell:=\mathcal{M}_{1}^{\prime}\cap\mathcal{S}$ is a line in $AG({\cal M}^{\prime}_{1})$, a contradiction to Lemma 4.8. Hence $\mathcal{M}_{1}^{\prime}=\mathcal{R}$. Similarly, $\mathcal{N}_{1}^{\prime}=\mathcal{S}$. Thus

[TABLE]

Since $E_{11}+E_{22}\in(\mathbb{D}_{\leq 1}^{n\times n}+E_{11})$, we get $\varphi(E_{11}+E_{22})\in\mathcal{M}_{1}^{\prime}$ or $\varphi(E_{11}+E_{22})\in\mathcal{N}_{1}^{\prime}$. Without loss of generality, we assume that $\varphi(E_{11}+E_{22})\in\mathcal{M}_{1}^{\prime}$. Recall $\varphi(I_{n})={\rm diag}(I_{n},0)$. We have $d(E_{11}+E_{22},I_{n})=n-2$ and $d\left(\varphi(E_{11}+E_{22}),{\rm diag}(I_{n},0)\right)\geq n-1$, a contradiction.

Combining Cases 1-3, we always have a contradiction. Hence $\varphi$ is non-degenerate. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Now, we prove Theorem 4.1 as follows.

Proof of Theorem 4.1.  Suppose the graph homomorphism $\varphi$ satisfies the Condition (I). Then, by Lemmas 4.10 and 4.12, $\varphi$ is non-degenerate and ${\rm dim}(\varphi(\mathcal{M}_{1}))={\rm dim}(\varphi(\mathcal{N}_{1}))=n$. By [14, Theorem 3.1], $\varphi$ is of the form either (26) or (27). Now, we assume that the homomorphism $\varphi$ does not satisfies the Condition (I). Then Lemma 4.11 implies that $\varphi\left(\mathbb{D}_{\leq 1}^{n\times n}\right)$ is an adjacent set. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

5 Degenerate graph homomorphisms

In this section, we discuss the degenerate graph homomorphisms. For a degenerate graph homomorphism $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, there are no explicit algebraic formulas of $\varphi$. Thus, we discuss the ranges and some properties on degenerate graph homomorphisms.

Let $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ be a graph homomorphism. We called that $\varphi$* maps distinct maximal sets of the same type $[$resp. different types$]$ into distinct maximal sets of the same type $[$resp. different types$]$, if for any two distinct maximal sets $\mathcal{M}$ and $\mathcal{N}$ of the same type [resp. different types] in $\mathbb{D}^{m\times n}$, there are two distinct maximal sets $\mathcal{M}^{\prime}$ and $\mathcal{N}^{\prime}$ of the same type [resp. different types] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ and $\varphi(\mathcal{N})\subseteq\mathcal{N}^{\prime}$. We called that $\varphi$ maps maximal sets of the same type $[$resp. different types$]$ into maximal sets of the same type $[$resp. different types$]$*, if for any two distinct maximal sets $\mathcal{M}$ and $\mathcal{N}$ of the same type $[$resp. different types$]$ in $\mathbb{D}^{m\times n}$, there are two maximal sets $\mathcal{M}^{\prime}$ and $\mathcal{N}^{\prime}$ of the same type [resp. different types] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ and $\varphi(\mathcal{N})\subseteq\mathcal{N}^{\prime}$ (where $\mathcal{N}^{\prime}$ and $\mathcal{N}^{\prime}$ may be equal if $\mathcal{M}$ and $\mathcal{N}$ are of the same type).

Our main results in this section are the following two theorems.

Theorem 5.1

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with $m^{\prime},n^{\prime}\geq{\rm min}\{m,n\}$. Suppose $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a degenerate graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{m\times n}$ such that ${\rm rank}(\varphi(A_{0}))={\rm min}\{m,n\}$. Then $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ and $\varphi(\mathbb{B}_{A_{0}})$ are two adjacent sets.

Theorem 5.2

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with ${\rm min}\{m,n\}=2$. Assume that $\varphi:\mathbb{D}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a degenerate graph homomorphism. Then there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that

[TABLE]

where $R\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is fixed.

To prove Theorems 5.1 and 5.2, we need the following lemmas.

Lemma 5.3

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with $n,m^{\prime},n^{\prime}\geq m$. Suppose that $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{m\times n}$ such that ${\rm rank}(\varphi(A_{0}))=m$. Assume further that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set. Then:

• (i)

$\varphi$* maps distinct maximal sets of type one containing [math] into distinct maximal sets of the same type, and $\varphi$ maps maximal sets of different types [resp. the same type] containing [math] into maximal sets of different types [resp. the same type]. Moreover, if ${\cal M}$ is a maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then $\varphi({\cal M})$ is not contained in any line in $AG({\cal M}^{\prime})$;*

• (ii)

if $A\in\mathbb{D}_{1}^{m\times n}$ and $d(A_{0},A)=m-1$, then $\varphi\left(\mathbb{B}_{A}\right)$ is not contained in a union of two maximal sets of different types containing $\varphi(A)$.

*Proof. *By Theorem 4.1 or Corollary 4.3, without loss of generality, we assume that $n>m$. There is $Q_{1}\in GL_{n}(\mathbb{D})$ such that $A_{0}=(A_{1},0)Q_{1}$ where $A_{1}\in GL_{m}(\mathbb{D})$. Let $\psi(X)=\varphi(XQ_{1})$ for all $X\in{\mathbb{D}}^{m\times n}$. Then $\psi$ is a graph homomorphism with $\psi(0)=0$, and ${\rm rank}(\psi(A_{1},0))=m={\rm rank}(A_{1},0)$. Since $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set, $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set. Definite the map $g:{\mathbb{D}}^{m\times m}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ by

[TABLE]

Then $g$ is a graph homomorphism with $g(0)=0$, ${\rm rank}(g(A_{1}))=m={\rm rank}(A_{1})$. By Theorem 4.1 or Corollary 4.3, either $g\left(\mathbb{D}_{\leq 1}^{m\times m}\right)$ is an adjacent set, or $g$ is a distance preserving map.

We assert that $g\left(\mathbb{D}_{\leq 1}^{m\times m}\right)$ is not an adjacent set. Otherwise, $g(\mathbb{D}_{\leq 1}^{m\times m})\subseteq\mathcal{R}^{\prime}$ where $\mathcal{R}^{\prime}$ is a maximal set containing [math] in $\mathbb{D^{\prime}}^{m^{\prime}\times n^{\prime}}$. We show a contradiction as follows.

By Lemma 1.1, $\mathcal{R}^{\prime}=P_{2}\mathcal{M}^{\prime}_{1}Q_{2}$ or $\mathcal{R}^{\prime}=P_{2}\mathcal{N}^{\prime}_{1}Q_{2}$ where $P_{2}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q_{2}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$. Replacing the map $g$ by the map $X\mapsto P_{2}^{-1}g(X)Q_{2}^{-1}$, we have either $\mathcal{R}^{\prime}=\mathcal{M}^{\prime}_{1}$ or $\mathcal{R}^{\prime}=\mathcal{N}^{\prime}_{1}$. Without loss of generality, we may assume that $\mathcal{R}^{\prime}=\mathcal{M}_{1}^{\prime}$. Then

[TABLE]

Thus, we obtain

[TABLE]

Set $\left(\mathbb{D}_{\leq 1}^{m\times m},0\right)=\left\{(X,0):X\in\mathbb{D}_{\leq 1}^{m\times m}\right\}\subset\mathbb{D}_{\leq 1}^{m\times n}$. By (56), we have $\psi\left(\mathbb{D}_{\leq 1}^{m\times m},0\right)\subseteq\mathcal{M}_{1}^{\prime}$. Let $\mathcal{S}$ be any maximal set of type one containing [math] in $\mathbb{D}^{m\times n}$, and let $\psi(\mathcal{S})\subseteq{\cal S}^{\prime}$ where ${\cal S}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. Then $|\mathcal{S}\cap\mathcal{N}_{1}|\geq 2$. Thus (57) implies that $|\mathcal{S}^{\prime}\cap\mathcal{M}_{1}^{\prime}|\geq 2$. Using Corollary 2.3, either ${\cal S}^{\prime}={\cal M}^{\prime}_{1}$ or ${\cal S}^{\prime}$ is of type two. Put $(\mathcal{S}^{(m)},0)=\mathcal{S}\cap\left(\mathbb{D}_{\leq 1}^{m\times m},0\right)$. Then $\mathcal{S}^{(m)}$ is a maximal set of type one in $\mathbb{D}^{m\times m}$. From (56) we get $g(\mathcal{S}^{(m)})\subseteq\mathcal{M}_{1}^{\prime}$. Since $g(\mathcal{S}^{(m)})\subseteq\mathcal{S}^{\prime}$, $g(\mathcal{S}^{(m)})\subseteq\mathcal{M}_{1}^{\prime}\cap{\cal S}^{\prime}$. If ${\cal S}^{\prime}$ is of type two, then $\ell:=\mathcal{M}_{1}^{\prime}\cap{\cal S}^{\prime}$ is a line in $AG(\mathcal{M}_{1}^{\prime})$, which is a contradiction because Lemma 4.8. Thus, ${\cal S}^{\prime}={\cal M}^{\prime}_{1}$. Then, $\psi$ maps every maximal set of type one containing [math] in $\mathbb{D}^{m\times n}$ into $\mathcal{M}_{1}^{\prime}$. Since every matrix of rank one is contained in a maximal set of type one containing [math], we obtain $\psi(\mathbb{D}_{\leq 1}^{m\times n})\subseteq\mathcal{M}_{1}^{\prime}$. Since $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set, we get a contradiction.

Therefore, $g$ is a distance preserving map, and hence $g$ is non-degenerate.

(i).  By Lemma 2.6(b), $g$ maps two distinct maximal sets of different types [resp. the same type] containing [math] into two distinct maximal sets of different types [resp. the same type]. Since $g(Y)=\varphi((Y,0)Q_{1})$ ($Y\in{\mathbb{D}}^{m\times m}$), it is easy to see that $\varphi$ maps two distinct maximal sets of type one containing [math] into two distinct maximal sets of the same type.

Let ${\cal M}$ be any type one maximal set containing [math] in ${\mathbb{D}}^{m\times n}$, and let $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. Set $(\mathcal{M}^{(m)},0)=\mathcal{M}\cap\left(\mathbb{D}_{\leq 1}^{m\times m},0\right)$. Then $\mathcal{M}^{(m)}$ is a maximal set of type one containing [math] in $\mathbb{D}^{m\times m}$. Recall that $g(Y)=\varphi((Y,0)Q_{1})$ ($Y\in{\mathbb{D}}^{m\times m}$). We get $(\mathcal{M}^{(m)},0)Q_{1}\subseteq\mathcal{M}Q_{1}=\mathcal{M}$. Thus,

[TABLE]

By Lemma 2.6(c), $g(\mathcal{M}^{(m)})$ is not contained in any line in $AG({\cal M}^{\prime})$. It follows that $\varphi(\mathcal{M})$ is not contained in any line in $AG({\cal M}^{\prime})$.

Thus, if ${\cal M}$ is a type one maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then $\varphi({\cal M})$ is not contained in any line in $AG({\cal M}^{\prime})$.

Next, we discuss maximal sets of type two. Let ${\cal S}_{1}$, ${\cal S}_{2}$ be any two distinct maximal sets of type two containing [math] in ${\mathbb{D}}^{m\times n}$, and let $\varphi(\mathcal{S}_{i})\subseteq\mathcal{S}_{i}^{\prime}$ where ${\cal S}_{i}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, $i=1,2$. Assume that $\mathcal{R}_{1},\mathcal{R}_{2}$ are any two distinct type one maximal sets containing [math] in ${\mathbb{D}}^{m\times n}$. By the above results, there are two distinct maximal sets $\mathcal{R}_{1}^{\prime},\mathcal{R}_{2}^{\prime}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\mathcal{R}_{1}^{\prime}$ and $\mathcal{R}_{2}^{\prime}$ are of the same type and $\varphi(\mathcal{R}_{i})\subseteq\mathcal{R}_{i}^{\prime}$, $i=1,2$. By Corollary 2.3, we have $|\mathcal{S}_{i}^{\prime}\cap\mathcal{R}_{j}^{\prime}|\geq 2$, $i,j=1,2$. Thus $\mathcal{R}_{1}^{\prime}\neq\mathcal{R}_{2}^{\prime}$ implies that $\mathcal{S}_{i}^{\prime}$ and $\mathcal{R}_{1}^{\prime}$ are of different types $i=1,2$. Consequently ${\cal S}_{1}^{\prime}$ and ${\cal S}_{2}^{\prime}$ are of the same type.

Therefore, we have proved that $\varphi$ maps two distinct maximal sets of different types [resp. the same type] containing [math] into two maximal sets of different types [resp. the same type].

On the other hand, by Lemma 2.5, $\ell_{1}:=\mathcal{S}_{1}\cap\mathcal{R}_{1}$ and $\ell_{2}:=\mathcal{S}_{1}\cap\mathcal{R}_{2}$ are two distinct lines in $AG(\mathcal{S}_{1})$. Also, $\ell_{1}^{\prime}:=\mathcal{S}_{1}^{\prime}\cap\mathcal{R}_{1}^{\prime}$ and $\ell_{2}^{\prime}:=\mathcal{S}_{2}^{\prime}\cap\mathcal{R}_{2}^{\prime}$ are two distinct lines in $AG(\mathcal{S}_{1}^{\prime})$. Since $\varphi(\ell_{i})\subseteq\ell_{i}^{\prime}$, $i=1,2$, it follows that $\varphi({\cal S}_{1})$ is not contained in any line in $AG({\cal S}_{1}^{\prime})$ because two different lines have at most a common point. Thus, if ${\cal S}_{1}$ is a type two maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{S}_{1})\subseteq\mathcal{S}_{1}^{\prime}$ where ${\cal S}_{1}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then $\varphi({\cal S}_{1})$ is not contained in any line in $AG({\cal S}_{1}^{\prime})$.

Therefore, if ${\cal M}$ is a maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, we have prove that $\varphi({\cal M})$ is not contained in any line in $AG({\cal M}^{\prime})$. Thus, the (i) of this lemma is proved.

(ii).   Suppose $A\in\mathbb{D}_{1}^{m\times n}$ and $d(A_{0},A)=m-1$. Then $A<A_{0}$. By Lemma 4.4(c), there are $P_{3}\in GL_{m}(\mathbb{D})$ and $Q_{3}\in GL_{n}(\mathbb{D})$ such that $A=P_{3}E_{11}Q_{3}$ and $A_{0}=P_{3}(I_{m},0)Q_{3}$. Replacing $\varphi$ by the map $X\mapsto\varphi(P_{3}XQ_{3})$, we can assume that $A_{0}=(I_{m},0)$ and $A=E_{11}$.

Definite the map $f:{\mathbb{D}}^{m\times m}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ by $f(Y)=\varphi(Y,0)$, $Y\in{\mathbb{D}}^{m\times m}$. Then $f$ is a graph homomorphism such that $f(0)=0$ and ${\rm rank}(f(I_{m}))=m$. By Theorem 4.1 or Corollary 4.3, either $f\left(\mathbb{D}_{\leq 1}^{m\times m}\right)$ is an adjacent set or $f$ is a distance preserving map. Similar to the proof on the homomorphism $g$, we can prove that $f$ is a distance preserving map. Since every distance preserving map carries distinct maximal sets into distinct maximal sets, $f\left(\mathbb{D}_{\leq 1}^{m\times m}+E_{11}^{m\times m}\right)$ is not contained in a union of two maximal sets of different types containing $f(E_{11}^{m\times m})$. Since $f\left(\mathbb{D}_{\leq 1}^{m\times m}+E_{11}^{m\times m}\right)=\varphi\left(\mathbb{D}_{\leq 1}^{m\times m}+E_{11}^{m\times m},\,0\right)\subseteq\varphi\left(\mathbb{B}_{E_{11}}\right),$ $\varphi\left(\mathbb{B}_{E_{11}}\right)$ is not contained in a union of two maximal sets of different types containing $\varphi(E_{11})$. Thus the (ii) of this lemma is proved. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

By the symmetry of rows and columns of a matrix, we have similarly the following lemma.

Lemma 5.4

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with $m,m^{\prime},n^{\prime}\geq n$. Suppose that $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{m\times n}$ such that ${\rm rank}(\varphi(A_{0}))=n$. Assume further that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set. Then:

• (i)

$\varphi$* maps distinct type two maximal sets containing [math] into distinct maximal sets of the same type, and $\varphi$ maps maximal sets of different types [resp. the same type] containing [math] into maximal sets of different types [resp. the same type]. Moreover, if ${\cal M}$ is a maximal set containing [math] in ${\mathbb{D}}^{m\times n}$ and $\varphi(\mathcal{M})\subseteq\mathcal{M}^{\prime}$ where ${\cal M}^{\prime}$ is a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, then $\varphi({\cal M})$ is not contained in any line in $AG({\cal M}^{\prime})$;*

• (ii)

if $A\in\mathbb{D}_{1}^{m\times n}$ and $d(A_{0},A)=n-1$, then $\varphi\left(\mathbb{B}_{A}\right)$ is not contained in a union of two maximal sets of different types containing $\varphi(A)$.

Now, we prove Theorem 5.1 as follows.

Proof of Theorem 5.1.  We prove this theorem only for the case of $m={\rm min}\{m,n\}$; the case of $n={\rm min}\{m,n\}$ is similar by using Lemma 5.4. Since $\varphi$ is degenerate, there exists a matrix $A\in\mathbb{D}^{m\times n}_{\leq 1}$ and there are two maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that

[TABLE]

Without loss of generality, we assume that $\mathcal{M}$ is of type one and $\mathcal{N}$ is of type two. Since ${\rm rank}(\varphi(A_{0}))=m$ and ${\rm rank}(\varphi(A))\leq 1$, we have either $d(\varphi(A_{0}),\varphi(A))=m$ or $m-1$.

Step 1.  In this step, we will prove that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set by contradiction. By Corollary 4.3, without loss of generality, we assume that $n>m$. Suppose that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set. We distinguish the following two cases to show a contradiction.

Case 1.1.  $d(\varphi(A_{0}),\varphi(A))=m$. Let $\psi(X)=\varphi(X+A)-\varphi(A)$, $X\in{\mathbb{D}}^{m\times n}$. Then $\psi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\psi(0)=0$ and ${\rm rank}(\psi(A_{0}-A))=m$. We show that $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set. Otherwise, if $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set, then by Lemma 5.3(i), $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not contained in a union of two maximal sets of different types containing [math], which implies that $\varphi\left(\mathbb{B}_{A}\right)$ is not contained in a union of two maximal sets of different types containing $\varphi(A)$, a contradiction to (58). Therefore, $\psi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set. It follows that $\varphi\left(\mathbb{B}_{A}\right)$ is an adjacent set. Thus, there exists a maximal set $\mathcal{S}$ such that

[TABLE]

Since $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is not an adjacent set, $A$ is of rank one. There is a type one maximal set $\mathcal{M}_{A}$ [resp. type two maximal set $\mathcal{S}_{A}$] containing $A$ and [math] in $\mathbb{D}^{m\times n}$. By Lemma 5.3(i), we have $\varphi({\cal M}_{A})\subseteq{\cal M}^{\prime}_{A}$ and $\varphi({\cal S}_{A})\subseteq{\cal S}^{\prime}_{A}$, where ${\cal M}_{A}^{\prime}$ and ${\cal S}_{A}^{\prime}$ are two maximal sets of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ containing $\varphi(A)$ and [math]. Since $\mathcal{M}_{A},\mathcal{S}_{A}\subseteq\mathbb{B}_{A}$, we get $\varphi(\mathcal{M}_{A})\subseteq({\cal M}^{\prime}_{A}\cap{\cal S})$ and $\varphi(\mathcal{S}_{A})\subseteq({\cal S}^{\prime}_{A}\cap{\cal S})$. Without loss of generality, we assume that $\mathcal{M}^{\prime}_{A}$ and ${\cal S}$ are of the same type. Then Corollary 2.3 implies that ${\cal M}^{\prime}_{A}\cap{\cal S}=\{\varphi(A)\}$ and $\varphi(\mathcal{M}_{A})\subseteq\{\varphi(A)\}$, a contradiction.

Case 1.2.  $d(\varphi(A_{0}),\varphi(A))=m-1$. Then, we have ${\rm rank}(\varphi(A))={\rm rank}(A)=1$ and $\varphi(A)<\varphi(A_{0})$. Write $E_{ij}^{\prime}=E_{ij}^{m^{\prime}\times n^{\prime}}$ and $E_{ij}=E_{ij}^{m\times n}$. By Lemma 4.4(c), there are invertible matrices $P_{1},Q_{1}$ over $\mathbb{D}^{\prime}$ such that

[TABLE]

Also, there are invertible matrices $P_{2},Q_{2}$ over $\mathbb{D}$ such that $P_{2}AQ_{2}=E_{11}$. Let \scriptsize P_{2}A_{0}Q_{2}=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $A_{11}\in{\mathbb{D}}^{1\times 1}$ and $A_{22}\in{\mathbb{D}}^{(m-1)\times(n-1)}$. Clearly, $d(A_{0},A)=m$ or $m-1$. By Lemma 5.3(ii) and (58), we have $d(A_{0},A)=m$. Thus, either $A_{22}\cong{\rm diag}(I_{m-1},0)$ or $A_{22}\cong{\rm diag}(I_{m-2},0)$. Put \scriptsize J_{m}=\left(\begin{array}[]{ccc}0&0&1\\ 0&I_{m-2}&0\\ 1&0&0\\ \end{array}\right).

When $A_{22}\cong{\rm diag}(I_{m-1},0)$, by appropriate elementary row and column operations on matrices, we can obtain either $\left({\rm diag}(a,I_{m-1}),0\right)$ (where $a\in\mathbb{D}^{*}$ with $a\neq 1$) or \scriptsize\left(\begin{array}[]{cccc}0_{1}&0&1&0\\ 0&I_{m-1}&0&0\\ \end{array}\right) from $P_{2}A_{0}Q_{2}$. Moreover, $P_{2}AQ_{2}=E_{11}$ is unchanged.

When $A_{22}\cong{\rm diag}(I_{m-2},0)$, by appropriate elementary row and column operations on matrices, we can get $(J_{m},0)$ from $P_{2}A_{0}Q_{2}$. Moreover, $P_{2}AQ_{2}=E_{11}$ is unchanged.

Thus, there are invertible matrices $P_{3},Q_{3}$ over $\mathbb{D}$, such that $P_{3}AQ_{3}=E_{11}$ and either

[TABLE]

Replacing $\varphi$ by the map $X\mapsto P_{1}^{-1}\varphi(P_{3}^{-1}XQ_{3}^{-1})Q_{1}^{-1}$, we have $A=E_{11}$, and either $A_{0}=\left({\rm diag}(a,I_{m-1}),0\right)$, or $(J_{m},0)$, or \scriptsize\left(\begin{array}[]{cccc}0_{1}&0&1&0\\ 0&I_{m-1}&0&0\\ \end{array}\right). Moreover, we have

[TABLE]

On the other hand, (58) becomes

[TABLE]

In $\mathbb{D}^{m\times n}$ [resp. ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$], by Corollary 2.2, there are only two maximal sets of different types containing $E_{11}$ and [math] [resp. $E_{11}^{\prime}$ and [math]], they are $\mathcal{M}_{1}$ and $\mathcal{N}_{1}$ [resp. $\mathcal{M}_{1}^{\prime}$ and $\mathcal{N}_{1}^{\prime}$]. Thus, by Lemma 5.3(i) and $\varphi(E_{11})=E_{11}^{\prime}$, we have either $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$, or $\varphi(\mathcal{M}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ with $\varphi(\mathcal{N}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. Without loss of generality, we assume that

[TABLE]

Since $\mathcal{M}_{1},\mathcal{N}_{1}\subseteq(\mathbb{D}_{\leq 1}^{m\times n}+E_{11})$, we get that $\varphi(\mathcal{M}_{1})\subseteq{\cal M}^{\prime}_{1}\cap(\mathcal{M}\cup\mathcal{N})$ and $\varphi(\mathcal{N}_{1})\subseteq{\cal N}^{\prime}_{1}\cap(\mathcal{M}\cup\mathcal{N})$. Suppose that ${\cal M}^{\prime}_{1}\neq\mathcal{M}$. Then ${\cal M}^{\prime}_{1}\cap\mathcal{M}=\{E_{11}^{\prime}\}$ and ${\cal M}^{\prime}_{1}\cap(\mathcal{M}\cup\mathcal{N})={\cal M}^{\prime}_{1}\cap\mathcal{N}$, thus $\ell:={\cal M}^{\prime}_{1}\cap\mathcal{N}$ is a line in $AG({\cal M}^{\prime}_{1})$ and $\varphi({\cal M}^{\prime}_{1})\subseteq\ell$. By Lemma 5.3(i), this is a contradiction. Therefore, we must have ${\cal M}^{\prime}_{1}=\mathcal{M}$. Similarly, ${\cal N}^{\prime}_{1}=\mathcal{N}$. It follows that

[TABLE]

In order to give a contradiction, we distinguish the following subcases.

Subcase 1.2.1.  $A_{0}=\left({\rm diag}(a,I_{m-1}),0\right)$ where $a\in\mathbb{D}^{*}$ with $a\neq 1$. By (60), we have $\varphi(aE_{11})=a^{*}E_{11}^{\prime}$ where $a^{*}\in{\mathbb{D}^{\prime}}^{*}$. Since $aE_{11}<A_{0}$, Corollary 4.7 and (59) imply that $\varphi(aE_{11})<\varphi(A_{0})={\rm diag}(I_{m},0)$. Thus, $a^{*}=1$ and hence $\varphi(aE_{11})=E_{11}^{\prime}$. Since $\varphi(aE_{11})\sim\varphi(E_{11})=E_{11}^{\prime}$, we have a contradiction.

Subcase 1.2.2.  $A_{0}=(J_{m},0)$. By (61), we get $\varphi\left(E_{mm}+E_{11}\right)\in\mathcal{M}_{1}^{\prime}\cup\mathcal{N}_{1}^{\prime}$. Without loss of generality, we assume that $\varphi\left(E_{mm}+E_{11}\right)\in\mathcal{M}_{1}^{\prime}$. Since $d\left(E_{mm}+E_{11},(J_{m},0)\right)=m-1$, it follows from (59) that $d\left(\varphi(E_{mm}+E_{11}),{\rm diag}(I_{m},0)\right)\leq m-1$. Thus \scriptsize\varphi(E_{mm}+E_{11})=\left(\begin{array}[]{cc}1&\alpha\\ 0&0\\ \end{array}\right) where $0\neq\alpha\in{\mathbb{D}^{\prime}}^{n^{\prime}-1}$ because $\varphi(E_{mm}+E_{11})\sim\varphi(E_{11})=E_{11}^{\prime}$. Since $d\left(E_{m1}+E_{11},(J_{m},0)\right)=m-1$, from (59) we have $d\left(\varphi(E_{m1}+E_{11}),{\rm diag}(I_{m},0)\right)\leq m-1$. Thus $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ implies that \scriptsize\varphi(E_{m1}+E_{11})=\left(\begin{array}[]{cc}1&0\\ \beta&0\\ \end{array}\right) where $0\neq\beta\in\,^{m^{\prime}-1}{\mathbb{D}^{\prime}}$ because $\varphi(E_{m1}+E_{11})\sim E_{11}^{\prime}$. By $\varphi(E_{mm}+E_{11})\sim\varphi(E_{m1}+E_{11})$, one gets \scriptsize\left(\begin{array}[]{cc}1&\alpha\\ 0&0\\ \end{array}\right)\sim\left(\begin{array}[]{cc}1&0\\ \beta&0\\ \end{array}\right), a contradiction.

Subcase 1.2.3.   \scriptsize A_{0}=\left(\begin{array}[]{cccc}0_{1}&0&1&0\\ 0&I_{m-1}&0&0\\ \end{array}\right). Let $B=E_{11}+E_{mm}$. Since $B\in\mathbb{B}_{E_{11}}$, from (61) we have $\varphi(B)\in\mathcal{M}_{1}^{\prime}$ or $\varphi(B)\in\mathcal{N}_{1}^{\prime}$.

Assume that $\varphi(B)\in\mathcal{M}_{1}^{\prime}$. Since $d(B,A_{0})=m-1$, by (59) we get $d(\varphi(B),{\rm diag}(I_{m},0))\leq m-1$. On the other hand, we have $\varphi(B)\sim\varphi(E_{11})=E_{11}^{\prime}$, which implies that \scriptsize\varphi(B)=\left(\begin{array}[]{cc}1&\alpha\\ 0&0\\ \end{array}\right) where $0\neq\alpha\in\,{\mathbb{D}^{\prime}}^{n^{\prime}-1}$. Let $C_{1}=E_{11}+E_{m1}$. Since $C_{1}\in\mathcal{N}_{1}$ and $\varphi(C_{1})\sim\varphi(B)$, it follows from $\varphi(\mathcal{N}_{1})\subseteq\mathcal{N}_{1}^{\prime}$ that $\varphi(C_{1})=c_{1}E_{11}^{\prime}$, where $c_{1}\in{\mathbb{D}^{\prime}}^{*}$. Let $\mathcal{M}_{C}$ be the type one maximal set containing $C_{1}$ and [math] in $\mathbb{D}^{m\times n}$. Then $\mathcal{M}_{C}\neq\mathcal{M}_{1}$. Suppose $\varphi(\mathcal{M}_{C})\subseteq\mathcal{M}_{C}^{\prime}$ where $\mathcal{M}_{C}^{\prime}$ is a maximal set containing $\varphi(C_{1})$ and [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$. By Lemma 5.3(i) and (60), $\mathcal{M}_{C}^{\prime}$ is of type one and $\mathcal{M}_{C}^{\prime}\neq\mathcal{M}_{1}^{\prime}$. Thus, $\{0,\varphi(C_{1})\}\subseteq\mathcal{M}_{C}^{\prime}\cap\mathcal{M}_{1}^{\prime}$, which is a contradiction to Corollary 2.3.

Assume that $\varphi(B)\in\mathcal{N}_{1}^{\prime}$. Similarly, we have \scriptsize\varphi(B)=\left(\begin{array}[]{cc}1&0\\ \beta&0\\ \end{array}\right) where $0\neq\beta\in\,^{m^{\prime}-1}{\mathbb{D}^{\prime}}$. Let $B_{1}=E_{11}+E_{m1}+E_{mm}-E_{m,m+1}$. Since $B_{1}\in\mathbb{B}_{E_{11}}$, by (61) we get $\varphi(B_{1})\in\mathcal{M}_{1}^{\prime}$ or $\varphi(B_{1})\in\mathcal{N}_{1}^{\prime}$. By $d(B_{1},A_{0})=m-1$ and (59), we have $d(\varphi(B_{1}),{\rm diag}(I_{m},0))\leq m-1$. Thus \scriptsize\varphi(B_{1})=\left(\begin{array}[]{cc}1&\alpha_{1}\\ 0&0\\ \end{array}\right) or \scriptsize\varphi(B_{1})=\left(\begin{array}[]{cc}1&0\\ \beta_{1}&0\\ \end{array}\right), where $0\neq\alpha_{1}\in\,{\mathbb{D}^{\prime}}^{n^{\prime}-1}$ and $0\neq\beta_{1}\in\,^{m^{\prime}-1}{\mathbb{D}^{\prime}}$. Since $\varphi(B_{1})\sim\varphi(B)$, we must have \scriptsize\varphi(B_{1})=\left(\begin{array}[]{cc}1&0\\ \beta_{1}&0\\ \end{array}\right). Let $B_{2}=-E_{1m}+E_{1,m+1}\in\mathcal{M}_{1}$. Then $B_{2}\sim B_{1}$ and $B_{2}\sim E_{11}$, hence (59) and (60) imply that $\varphi(B_{2})=b_{2}E_{11}^{\prime}$, where $b_{2}\in{\mathbb{D}^{\prime}}^{*}$ with $b_{2}\neq 1$. However, we have $d(B_{2},A_{0})=m-1$ and $d(\varphi(B_{2}),\varphi(A_{0}))=d(b_{2}E_{11}^{\prime},{\rm diag}(I_{m},0)))=m$, a contradiction.

Combining Case 1.1 with Case 1.2, there is always a contradiction. Therefore, $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ must be an adjacent set.

Step 2.  In this step, we will prove that $\varphi(\mathbb{B}_{A_{0}})$ is an adjacent set. By Step 1, $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ is an adjacent set. Thus, there is a maximal set $\mathcal{M}^{\prime}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}^{\prime}$. There is an invertible matrix $Q$ over $\mathbb{D}$ such that $A_{0}=(A_{1},0)Q$ where $A_{1}\in GL_{m}(\mathbb{D})$.

Let $\varphi^{\prime}(X)=\varphi(XQ)$ for all $X\in\mathbb{D}^{m\times n}$. Then $\varphi^{\prime}$ is a graph homomorphism, ${\rm rank}(\varphi^{\prime}(A_{1},0))=m$ and

[TABLE]

Let $f(Y)=\varphi^{\prime}(Y+A_{1},0)-\varphi^{\prime}(A_{1},0)$, $Y\in\mathbb{D}^{m\times m}$. Then $f$ is a graph homomorphism such that $f(0)=0$ and ${\rm rank}(f(-A_{1}))=m$. By (62), it is clear that $f$ does not preserve distance $2$. It follows from Corollary 4.3 that $f\left(\mathbb{D}_{\leq 1}^{m\times m}\right)$ is an adjacent set. Hence there is a maximal set $\mathcal{N}^{\prime}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that

[TABLE]

Let $h(X)=\varphi^{\prime}(X+(A_{1},0))-\varphi^{\prime}(A_{1},0)$, $X\in\mathbb{D}^{m\times n}$. Then $h$ is a graph homomorphism with $h(0)=0$. Moreover, $h(Y,0)=f(Y)$ for all $Y\in\mathbb{D}^{m\times m}$. We assert $h\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{N}^{\prime}$. Otherwise, there exists a type one maximal set $\mathcal{R}$ containing [math] in ${\mathbb{D}}^{m\times n}$, such that $h(\mathcal{R})\subseteq\mathcal{S}$ where $\mathcal{S}$ is a maximal set containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ and $\mathcal{S}\neq\mathcal{N}^{\prime}$. Let $\left(\mathcal{R}^{(m)},0\right)=\mathcal{R}\cap\left(\mathbb{D}_{\leq 1}^{m\times m},0\right)$. Then $\mathcal{R}^{(m)}$ is a type one maximal set containing [math] in $\mathbb{D}^{m\times m}$, and hence $f(\mathcal{R}^{(m)})\subseteq\mathcal{N}^{\prime}\cap\mathcal{S}$. By Corollary 2.3, $\mathcal{S}$ and $\mathcal{N}^{\prime}$ must be of different type. Thus $\ell:=\mathcal{N}^{\prime}\cap{\cal S}$ is a line in $AG(\mathcal{N}^{\prime})$ by Lemma 2.5, a contradiction to Lemma 4.8. Therefore, we obtain $h\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{N}^{\prime}$, and hence $\varphi^{\prime}\left(\mathbb{B}_{(A_{1},0)}\right)\subseteq\mathcal{N}^{\prime}+\varphi^{\prime}(A_{1},0)$. Consequently, $\varphi(\mathbb{B}_{A_{0}})\subseteq\mathcal{N}^{\prime}+\varphi(A_{0})$ and hence $\varphi(\mathbb{B}_{A_{0}})$ is an adjacent set. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Next, we will prove Theorem 5.2. We need the following lemma.

Lemma 5.5

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with ${\rm min}\{m,n\}=2$. Suppose $\varphi:\mathbb{D}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a degenerate graph homomorphism with $\varphi(0)=0$, and there exists $A\in{\mathbb{D}}_{2}^{m\times n}$ such that ${\rm rank}(\varphi(A))=2$. Then there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $0\in\mathcal{M}\cap\mathcal{N}$ and

[TABLE]

where $R\in\mathcal{M}$ is fixed and ${\rm rank}(R)=1$. Moreover, there is $\mathcal{M}^{\prime}\in\{\mathcal{M},\mathcal{N}\}$ such that $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}^{\prime}$ and $\varphi(\mathbb{N}_{A})\subseteq\mathcal{M}^{\prime}\cap(\mathcal{N}^{\prime}+R)$ where $\{\mathcal{M}^{\prime},\mathcal{N}^{\prime}\}=\{\mathcal{M},\mathcal{N}\}$.

*Proof. * We prove this lemma only for the case $m={\rm min}\{m,n\}=2$; the case $n={\rm min}\{m,n\}=2$ is similar. From now on we assume that $m={\rm min}\{m,n\}=2$.

Step 1.  By Theorem 5.1, $\varphi\left(\mathbb{D}_{\leq 1}^{2\times n}\right)$ and $\varphi(\mathbb{B}_{A})$ are two adjacent sets. Then, there are maximal sets $\mathcal{M}$ and $\mathcal{N}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $\varphi\left(\mathbb{D}_{\leq 1}^{2\times n}\right)\subseteq\mathcal{M}$ and $\varphi(\mathbb{B}_{A})\subseteq\mathcal{N}+\varphi(A)$. By Lemma 1.1, $\mathcal{M}=P_{1}\mathcal{M}^{\prime}_{1}Q_{1}$ or $\mathcal{M}=P_{1}\mathcal{N}^{\prime}_{1}Q_{1}$ where $P_{1}\in GL_{m^{\prime}}(\mathbb{D}^{\prime})$ and $Q_{1}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$. Replacing $\varphi$ by the map $X\mapsto P_{1}^{-1}\varphi(X)Q_{1}^{-1}$, we have either $\mathcal{M}=\mathcal{M}^{\prime}_{1}$ or $\mathcal{M}=\mathcal{N}^{\prime}_{1}$. We prove this lemma only for the case of $\mathcal{M}=\mathcal{M}^{\prime}_{1}$; the case of $\mathcal{M}=\mathcal{N}^{\prime}_{1}$ is similar. Now, we assume that $\mathcal{M}=\mathcal{M}^{\prime}_{1}$.

Write \scriptsize A=\left(\begin{array}[]{c}\alpha_{1}\\ \alpha_{2}\\ \end{array}\right) where $\alpha_{1},\alpha_{2}\in\mathbb{D}^{n}$ are left linearly independent. Set \scriptsize C_{\lambda}=\left(\begin{array}[]{c}-\alpha_{1}+\lambda\alpha_{2}\\ 0\\ \end{array}\right) where $\lambda\in\mathbb{D}$. Since $C_{\lambda}+A\in\mathbb{B}_{A}\cap\mathbb{D}_{\leq 1}^{2\times n}$ for all $\lambda\in\mathbb{D}$, $|(\mathcal{N}+\varphi(A))\cap\mathcal{M}_{1}^{\prime}|\geq 2$. Since ${\rm rank}(\varphi(A))=2$, $\mathcal{N}+\varphi(A)\neq\mathcal{M}_{1}^{\prime}$. Thus Corollary 2.3 implies that $\mathcal{N}$ is of type two. By Lemma 1.1, ${\cal N}={\cal N}_{1}^{\prime}Q_{2}$ where $Q_{2}\in GL_{n^{\prime}}(\mathbb{D}^{\prime})$. Replacing $\varphi$ by the map $X\mapsto\varphi(X)Q_{2}^{-1}$ ($X\in\mathbb{D}^{2\times n}$), we may assume with no loss of generality that $\mathcal{N}=\mathcal{N}_{1}^{\prime}$. Thus

[TABLE]

We prove $\varphi(\mathbb{N}_{A})\subseteq\mathcal{M}_{1}^{\prime}$ as follows.

Write \scriptsize\varphi(A)=\left(\begin{array}[]{cc}a_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{array}\right) where $a_{11}\in\mathbb{D}^{\prime}$. Since $C_{\lambda}+A\in\mathbb{N}_{A}\cap\mathbb{D}_{\leq 1}^{2\times n}$ for all $\lambda\in\mathbb{D}$, $\varphi(C_{\lambda}+A)\in(\mathcal{N}_{1}^{\prime}+\varphi(A))\cap\mathcal{M}_{1}^{\prime}$ for all $\lambda\in\mathbb{D}$. Hence $\varphi(A)$ is of the form

[TABLE]

Otherwise, $A_{22}\neq 0$ implies that $(\mathcal{N}_{1}^{\prime}+\varphi(A))\cap\mathcal{M}_{1}^{\prime}=\emptyset$, a contradiction.

Now, we affirm that ${\rm rank}(\varphi(Z))=1$ for every $Z\in\mathbb{N}_{A}$. Suppose that $Z\in\mathbb{N}_{A}$ and ${\rm rank}(\varphi(Z))=2$. Then ${\rm rank}(Z)=2$. We show a contradiction as follows. By Theorem 5.1, $\varphi(\mathbb{B}_{Z})$ is an adjacent set. Thus, there exists a maximal set $\mathcal{S}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ such that

[TABLE]

Write \scriptsize Z=\left(\begin{array}[]{c}\delta_{1}\\ \delta_{2}\\ \end{array}\right), where $\delta_{1},\delta_{2}\in\mathbb{D}^{n}$ are left linearly independent. Let \scriptsize T_{\lambda}=\left(\begin{array}[]{c}-\delta_{1}+\lambda\delta_{2}\\ 0\\ \end{array}\right) where $\lambda\in\mathbb{D}$. Since $T_{\lambda}+Z\in\mathbb{B}_{Z}\cap\mathbb{D}_{\leq 1}^{2\times n}$ for all $\lambda\in\mathbb{D}$, $|(\mathcal{S}+\varphi(Z))\cap\mathcal{M}_{1}^{\prime}|\geq 2$. By ${\rm rank}(\varphi(Z))=2$, we have $\mathcal{S}+\varphi(Z)\neq\mathcal{M}_{1}^{\prime}$. Hence Corollary 2.3 implies that $\mathcal{S}+\varphi(Z)$ is a maximal set of type two. Applying Corollary 2.3 again, we get $\left|(\mathcal{S}+\varphi(Z))\cap(\mathcal{N}_{1}^{\prime}+\varphi(A))\right|\leq 1$. Let \scriptsize Z-A=\left(\begin{array}[]{c}\gamma_{1}\\ \gamma_{2}\\ \end{array}\right)\in\mathbb{D}_{1}^{2\times n}, \scriptsize Z_{1}=\left(\begin{array}[]{c}0\\ \gamma_{1}-\gamma_{2}\\ \end{array}\right), \scriptsize Z_{2}=\left(\begin{array}[]{c}\gamma_{2}-\gamma_{1}\\ 0\\ \end{array}\right), \scriptsize Y_{1}=\left(\begin{array}[]{c}\gamma_{1}\\ \gamma_{1}\\ \end{array}\right) and \scriptsize Y_{2}=\left(\begin{array}[]{c}\gamma_{2}\\ \gamma_{2}\\ \end{array}\right). Then $Z_{i}+Z=Y_{i}+A$, $i=1,2$. Moreover, $Z_{1}+Z\sim Z_{2}+Z$. Clearly, $Z_{i}+Z\in\mathbb{B}_{Z}$ and $Y_{i}+A\in\mathbb{B}_{A}$, $i=1,2$. It follows that $|\mathbb{B}_{Z}\cap\mathbb{B}_{A}|\geq 2$ and hence $\left|(\mathcal{S}+\varphi(Z))\cap(\mathcal{N}_{1}^{\prime}+\varphi(A))\right|\geq 2$, a contradiction. Therefore, ${\rm rank}(\varphi(Z))\neq 2$ for all $Z\in\mathbb{N}_{A}$. By ${\rm rank}(\varphi(A))=2$ and $\varphi(A)\sim\varphi(Z)$, one gets ${\rm rank}(\varphi(Z))=1$ for every $Z\in\mathbb{N}_{A}$.

Note that $\mathbb{N}_{A}\subset\mathbb{B}_{A}$. By (64) and (65), we obtain

[TABLE]

Step 2.  Let $B\in{\mathbb{D}}_{2}^{2\times n}$ with $B\neq A$ and ${\rm rank}(\varphi(B))=2$. By Theorem 5.1, $\varphi(\mathbb{B}_{B})$ is an adjacent set. Thus, there is a maximal set $\mathcal{N}^{\prime\prime}$ containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ such that

[TABLE]

Write \scriptsize B=\left(\begin{array}[]{c}\beta_{1}\\ \beta_{2}\\ \end{array}\right) where $\beta_{1},\beta_{2}\in\mathbb{D}^{n}$ are left linearly independent. Set \scriptsize C_{\lambda}=\left(\begin{array}[]{c}-\beta_{1}+\lambda\beta_{2}\\ 0\\ \end{array}\right) where $\lambda\in\mathbb{D}$. Since $C_{\lambda}+B\in\mathbb{B}_{B}\cap\mathbb{D}_{\leq 1}^{2\times n}$ for all $\lambda\in\mathbb{D}$, $|(\mathcal{N}^{\prime\prime}+\varphi(B))\cap\mathcal{M}_{1}^{\prime}|\geq 2$. Since ${\rm rank}(\varphi(B))=2$, $\mathcal{N}^{\prime\prime}+\varphi(B)\neq\mathcal{M}_{1}^{\prime}$. It follows from Corollary 2.3 that $\mathcal{N}^{\prime\prime}$ is of type two. We prove $\mathcal{N}^{\prime\prime}=\mathcal{N}_{1}^{\prime}$ by contradiction as follows.

Suppose that $\mathcal{N}^{\prime\prime}\neq\mathcal{N}_{1}^{\prime}$. Write $\mathcal{N}^{\prime\prime}=\left\{(yq_{1},yq_{2},\ldots,yq_{n}):y\in\,^{m}\mathbb{D}\right\}$ where $q_{j}\in\mathbb{D}^{\prime}$ and $(q_{2},\ldots,q_{n})\neq 0$. Then there is \scriptsize Q_{2}=\left(\begin{array}[]{cc}1&0\\ &*\\ \end{array}\right)\in GL_{n^{\prime}}(\mathbb{D}^{\prime}) such that $\mathcal{N}^{\prime\prime}=\mathcal{N}_{2}^{\prime}Q_{2}$ and $\mathcal{N}^{\prime}_{1}=\mathcal{N}_{1}^{\prime}Q_{2}$. Modifying the map $\varphi$ by the map $X\mapsto\varphi(X)Q_{2}^{-1}$ ($X\in\mathbb{D}^{2\times n}$). We may assume with no loss of generality that $\mathcal{N}^{\prime\prime}=\mathcal{N}_{2}^{\prime}$ and (64)-(66) hold. Thus

[TABLE]

Write \scriptsize\varphi(B)=\left(\begin{array}[]{ccc}b_{11}&b_{12}&B_{13}\\ B_{21}&B_{22}&B_{23}\\ \end{array}\right) where $b_{11},b_{12}\in\mathbb{D}^{\prime}$. Since $C_{\lambda}+B\in\mathbb{B}_{B}\cap\mathbb{D}_{\leq 1}^{2\times n}$ for all $\lambda\in\mathbb{D}$, $\varphi(C_{\lambda}+B)\in(\mathcal{N}_{2}^{\prime}+\varphi(B))\cap\mathcal{M}_{1}^{\prime}$ for all $\lambda\in\mathbb{D}$. Hence $\varphi(B)$ is of the form

[TABLE]

where $B_{22}\neq 0$ and $(b_{11},B_{13})\neq(0,0)$. Otherwise, we have $(B_{21},B_{23})\neq(0,0)$, which implies that $(\mathcal{N}_{2}^{\prime}+\varphi(B))\cap\mathcal{M}_{1}^{\prime}=\emptyset$, a contradiction. By $\varphi(B)\notin\mathcal{M}_{1}^{\prime}$ and (66), we have $B\notin\mathbb{N}_{A}$ and hence $d(A,B)=2$. Write \scriptsize B-A=\left(\begin{array}[]{c}\nu_{1}\\ \nu_{2}\\ \end{array}\right), where $\nu_{1},\nu_{2}\in\mathbb{D}^{n}$ are left linearly independent. Then \scriptsize B\sim\left(\begin{array}[]{c}\nu_{1}+\lambda\nu_{2}\\ 0\\ \end{array}\right)+A\sim A for all $\lambda\in\mathbb{D}$. Let $A_{12}=(a_{12},A_{13})$ where $a_{12}\in\mathbb{D}^{\prime}$. By (66), one has that

[TABLE]

where $\sigma$ is an injection from $\mathbb{D}$ to $\mathbb{D}^{\prime}$. Hence

[TABLE]

which is a contradiction because $B_{22}\neq 0$. Therefore, we must have $\mathcal{N}^{\prime\prime}=\mathcal{N}_{1}^{\prime}$.

Similar to the proof of (69), we have that $d(A,B)=2$ and

[TABLE]

where $\mu$ is an injection from $\mathbb{D}$ to $\mathbb{D}^{\prime}$. Therefore,

[TABLE]

Using Lemma 2.1 and ${\rm rank}(\varphi(B))=2$, it is easy to see that $B_{22}^{\prime}=0$ and $B_{12}^{\prime}=A_{12}$. Consequently, $\mathcal{N}_{1}^{\prime}+\varphi(B)=\mathcal{N}_{1}^{\prime}+\varphi(A)$, and (67) implies that

[TABLE]

Step 3.  Let $T\in{\mathbb{D}}_{2}^{2\times n}$ with ${\rm rank}(\varphi(T))=1$. We show $\varphi(T)\in\mathcal{M}_{1}^{\prime}$ as follows.

Put \scriptsize\varphi(T)=\left(\begin{array}[]{cc}t_{11}&T_{12}\\ T_{21}&T_{22}\\ \end{array}\right) where $t_{11}\in\mathbb{D}^{\prime}$. If $T\sim A$, then $T\in\mathbb{N}_{A}$ and hence $\varphi(T)\in\mathcal{M}_{1}^{\prime}$ by (66). Now we assume $d(T,A)=2$. Write \scriptsize T-A=\left(\begin{array}[]{c}\delta_{1}\\ \delta_{2}\\ \end{array}\right) where $\delta_{1},\delta_{2}\in\mathbb{D}^{n}$ are left linearly independent. Then \scriptsize T\sim\left(\begin{array}[]{c}\delta_{1}+\lambda\delta_{2}\\ 0\\ \end{array}\right)+A\sim A for all $\lambda\in\mathbb{D}$. By (66), we can let \scriptsize\varphi\left(\left(\begin{array}[]{c}\delta_{1}+\lambda\delta_{2}\\ 0\\ \end{array}\right)+A\right)=\left(\begin{array}[]{cc}\lambda^{\tau}&A_{12}\\ 0&0\\ \end{array}\right) for all $\lambda\in\mathbb{D}$, where $\tau$ is an injection from $\mathbb{D}$ to $\mathbb{D}^{\prime}$. Since \scriptsize\varphi(T)\sim\left(\begin{array}[]{cc}\lambda^{\tau}&A_{12}\\ 0&0\\ \end{array}\right) for all $\lambda\in\mathbb{D}$, we get that

[TABLE]

By Lemma 2.1 and $A_{12}\neq 0$, we obtain $(T_{21},T_{22})=0$. Then \scriptsize\varphi(T)=\left(\begin{array}[]{cc}t_{11}&T_{12}\\ 0&0\\ \end{array}\right)\in\mathcal{M}_{1}^{\prime}. Hence

[TABLE]

By (64), (71) and (72), we have that $\varphi(X)\in\mathcal{N}_{1}^{\prime}+\varphi(A)$ whenever ${\rm rank}(\varphi(X))=2$, and $\varphi(X)\in\mathcal{M}_{1}^{\prime}$ whenever ${\rm rank}(\varphi(X))\neq 2$. Thus

[TABLE]

Recall that \scriptsize\varphi(A)=\left(\begin{array}[]{cc}a_{11}&A_{12}\\ A_{21}&0\\ \end{array}\right) where $A_{21}\neq 0$ and $A_{12}\neq 0$. Let \scriptsize R=\left(\begin{array}[]{cc}a_{11}&A_{12}\\ 0&0\\ \end{array}\right)\in\mathcal{M}_{1}^{\prime}. Then ${\rm rank}(R)=1$, $\mathcal{M}_{1}^{\prime}=\mathcal{M}_{1}^{\prime}+R$ and $\mathcal{N}_{1}^{\prime}+\varphi(A)=\mathcal{N}_{1}^{\prime}+R$, and hence $\mathcal{M}_{1}^{\prime}\cup(\mathcal{N}_{1}^{\prime}+\varphi(A))=(\mathcal{M}_{1}^{\prime}+R)\cup(\mathcal{N}_{1}^{\prime}+R)$. Therefore, (63) holds. By (64) and (66), we have $\varphi\left(\mathbb{D}_{\leq 1}^{2\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}$ and $\varphi(\mathbb{N}_{A})\subseteq\mathcal{M}_{1}^{\prime}\cap(\mathcal{N}_{1}^{\prime}+R)$. Thus, there is $\mathcal{M}^{\prime}\in\{\mathcal{M},\mathcal{N}\}$ such that $\varphi\left(\mathbb{D}_{\leq 1}^{2\times n}\right)\subseteq\mathcal{M}^{\prime}$ and $\varphi(\mathbb{N}_{A})\subseteq\mathcal{M}^{\prime}\cap(\mathcal{N}^{\prime}+R)$ where $\{\mathcal{M}^{\prime},\mathcal{N}^{\prime}\}=\{\mathcal{M},\mathcal{N}\}$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Now, we prove Theorem 5.2 as follows.

Proof of Theorem 5.2.   Note that the map $\varphi(X)-\varphi(0)$ is also a degenerate graph homomorphism. By the map $X\mapsto\varphi(X)-\varphi(0)$, we may assume with no loss of generality that $\varphi(0)=0$. We prove this theorem only for the case of $m=2$; the case of $n=2$ is similar. From now on we assume that $m={\rm min}\{m,n\}=2$. Clearly, $1\leq{\rm diam}(\varphi(\mathbb{D}^{2\times n}))\leq 2$.

Case 1.   There exists $A\in{\mathbb{D}}_{2}^{2\times n}$ such that ${\rm rank}(\varphi(A))=2$. By Lemma 5.5, this theorem holds.

Case 2.  There is no $A\in{\mathbb{D}}_{2}^{2\times n}$ such that ${\rm rank}(\varphi(A))=2$. Then

[TABLE]

If ${\rm diam}(\varphi(\mathbb{D}^{2\times n}))=1$, then $\varphi(\mathbb{D}^{2\times n})$ is an adjacent set, and hence this theorem holds. From now on we assume that ${\rm diam}(\varphi(\mathbb{D}^{2\times n}))=2$. Then, there are two matrices $B_{1},B_{2}\in\mathbb{D}^{2\times n}$ such that $d(\varphi(B_{1}),\varphi(B_{2}))=2=d(B_{1},B_{2})$. Let $\psi(X)=\varphi(X+B_{1})-\varphi(B_{1})$, $X\in{\mathbb{D}}^{2\times n}$. Then $\psi:{\mathbb{D}}^{2\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\psi(0)=0$ and ${\rm rank}(\psi(B_{2}-B_{1}))=2$.

Subcase 2.1.  $\psi$ is degenerate. By Lemma 5.5, there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types containing [math], such that $\psi(\mathbb{D}^{2\times n})\subseteq(\mathcal{M}+R_{1})\cup(\mathcal{N}+R_{1})$, where $R_{1}\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is fixed and ${\rm rank}(R_{1})=1$. Hence $\varphi(\mathbb{D}^{2\times n})\subseteq(\mathcal{M}+R_{1}+\varphi(A))\cup(\mathcal{N}+R_{1}+\varphi(B_{1}))$ and this theorem holds.

Subcase 2.2.  $\psi$ is non-degenerate. By Lemma 3.3, there exist invertible matrices $T_{1},T_{2}$ over $\mathbb{D}^{\prime}$ and invertible matrices $T_{3},T_{4}$ over $\mathbb{D}$, such that either

[TABLE]

or

[TABLE]

Without loss of generality, we may assume that (74) holds and all $T_{1},T_{2},T_{3},T_{4}$ are identity matrices. Then $\psi(\mathcal{M}_{i})\subseteq\mathcal{M}_{i}^{\prime}$   with  $\psi(\mathcal{N}_{i})\subseteq\mathcal{N}_{i}^{\prime}$, $i=1,2$. Since $\mathcal{M}_{i}\cap\mathcal{N}_{j}=\mathbb{D}E_{ij}$ and $\mathcal{M}_{i}^{\prime}\cap\mathcal{N}_{j}^{\prime}=\mathbb{D}^{\prime}E_{ij}^{\prime}$, $1\leq i,j\leq 2$, we can let $\psi(xE_{ij})=x^{\sigma_{ij}}E_{ij}^{\prime}$, $x\in\mathbb{D}$, where $\sigma_{ij}:\mathbb{D}\rightarrow\mathbb{D}^{\prime}$ is an injective map with $0^{\sigma_{ij}}=0$. By the definition of $\psi$, we obtain that

[TABLE]

Using (73), we have ${\rm rank}\left(x^{\sigma_{ij}}E_{ij}^{\prime}+\varphi(B_{1})\right)\leq 1$, $x\in\mathbb{D}$, $i,j=1,2$. Applying Lemma 2.1, it is easy to verify that $\varphi(B_{1})=0$. Thus, ${\rm rank}(\varphi(B_{2}))=d(\varphi(B_{2}),\varphi(B_{1}))=2$, a contradiction to (73). Therefore, Subcase 2.2 does not happen. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

By Theorem 5.1, the following corollary is obvious.

Corollary 5.6

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers with $m^{\prime},n^{\prime}\geq{\rm min}\{m,n\}$. Suppose $\varphi:{\mathbb{D}}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\varphi(0)=0$, and there exists $A_{0}\in{\mathbb{D}}^{m\times n}$ such that ${\rm rank}(\varphi(A_{0}))={\rm min}\{m,n\}$. If $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)$ or $\varphi(\mathbb{B}_{A_{0}})$ is not an adjacent set, then $\varphi$ is non-degenerate.

The following result is a generalization of [17, Theorem 1.1] (which is due to Huang and Šemrl).

Corollary 5.7

Let $\mathbb{D},\mathbb{D}^{\prime}$ be division rings with $|\mathbb{D}|\geq 4$, and let $m^{\prime},n^{\prime}\geq 2$. Suppose $\varphi:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism. Then either $\varphi$ is a distance preserving map (which is of the form either (28) or (29)), or there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $0\in\mathcal{M}\cap\mathcal{N}$ and

[TABLE]

where $R\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is fixed.

*Proof. *Assume that ${\rm diam}(\varphi(\mathbb{D}^{2\times 2}))=1$. Then $\varphi(\mathbb{D}^{2\times 2})$ is an adjacent set, and hence (76) holds. From now on we assume that ${\rm diam}(\varphi(\mathbb{D}^{2\times 2}))=2$. Then there are two matrices $A_{0},B_{0}\in\mathbb{D}^{2\times 2}$ such that $d(\varphi(A_{0}),\varphi(B_{0}))={\rm rank}(\varphi(B_{0})-\varphi(A_{0}))=2$. Let $\psi(X)=\varphi(X+A_{0})-\varphi(A_{0})$, $X\in{\mathbb{D}}^{2\times 2}$. Then $\psi:{\mathbb{D}}^{2\times 2}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism with $\psi(0)=0$ and ${\rm rank}(\psi(B_{0}-A_{0}))=2$. By Corollary 4.3, either $\psi$ is a distance preserving map or $\psi\left(\mathbb{D}_{\leq 1}^{2\times 2}\right)$ is an adjacent set.

Assume that $\psi$ is a distance preserving map. Then $\varphi$ is also a distance preserving map. By Corollary 4.2, $\varphi$ is of the form either (28) or (29). Now, we assume that $\psi\left(\mathbb{D}_{\leq 1}^{2\times 2}\right)$ is an adjacent set. By Theorem 5.2 or Lemma 5.5, there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that $0\in\mathcal{M}\cap\mathcal{N}$ and (76) holds. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Finally, we discuss the case of finite fields. For the case of finite fields, we have the following better results.

Theorem 5.8

Let $\mathbb{D},\mathbb{D}^{\prime}$ be two finite fields with $|\mathbb{D}|>|\mathbb{D}^{\prime}|$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers. Then every graph homomorphism from $\mathbb{D}^{m\times n}$ to ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a (vertex) colouring.

*Proof. *Suppose $\varphi:\mathbb{D}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a graph homomorphism. Without loss of generality, we may assume that $\varphi(0)=0$ and $\varphi(\mathcal{M}_{1})\subseteq\mathcal{M}_{1}^{\prime}$. If $\ell$ is a line in $AG(\mathcal{M}_{1})$, then $|\mathbb{D}|>|\mathbb{D}^{\prime}|$ implies that $\varphi(\ell)$ is not contained in any line in $AG(\mathcal{M}_{1}^{\prime})$, and hence $\varphi(\ell)$ contains at least three noncollinear points in $AG(\mathcal{M}_{1}^{\prime})$.

Let \scriptsize A=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{m}\\ \end{array}\right)\in\mathbb{D}_{1}^{m\times n}. Without loss of generality, we assume $\alpha_{m}\neq 0$. Then $\alpha_{i}=k_{i}\alpha_{m}$, $i=1,\ldots,m-1$. We have \scriptsize A\sim\left(\begin{array}[]{c}\lambda\alpha_{m}\\ 0\\ \end{array}\right) for all $\lambda\in\mathbb{D}$. Since \scriptsize\ell_{1}:=\left\{\left(\begin{array}[]{c}\lambda\alpha_{m}\\ 0\\ \end{array}\right):\lambda\in\mathbb{D}\right\} is a line in $AG(\mathcal{M}_{1})$, it follows from above result that $\varphi(A)$ is adjacent with three noncollinear points in $AG(\mathcal{M}_{1}^{\prime})$. By Lemma 2.8, $\varphi(A)\in\mathcal{M}_{1}^{\prime}$ for all $A\in\mathbb{D}_{1}^{m\times n}$. Therefore, we obtain $\varphi\left(\mathbb{D}_{\leq 1}^{m\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}$.

Suppose that $\varphi\left(\mathbb{D}_{\leq\,k-1}^{m\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}$ where $2\leq k\leq m$. Let $B\in\mathbb{D}_{k}^{m\times n}$. Note that $P(\mathbb{D}_{\leq\,k-1}^{m\times n})Q=\mathbb{D}_{\leq\,k-1}^{m\times n}$ for any $P\in GL_{m}(\mathbb{D}$ and $Q\in GL_{n}(\mathbb{D}$. Without loss of generality, we can assume that \scriptsize B=\left(\begin{array}[]{c}B_{1}\\ 0\\ \end{array}\right) where \scriptsize B_{1}=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{k}\\ \end{array}\right)\in\mathbb{D}_{k}^{k\times n}. Put \scriptsize C_{1}=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{k-1}\\ \end{array}\right). Then \scriptsize B\sim\left(\begin{array}[]{c}C_{1}+{\scriptsize\left(\begin{array}[]{c}\lambda\alpha_{k}\\ 0\\ \end{array}\right)}\\ 0\\ \end{array}\right) for all $\lambda\in\mathbb{D}$. Let \scriptsize\ell_{2}=\left\{\left(\begin{array}[]{c}C_{1}+{\scriptsize\left(\begin{array}[]{c}\lambda\alpha_{k}\\ 0\\ \end{array}\right)}\\ 0\\ \end{array}\right):\lambda\in\mathbb{D}\right\}\subset\mathbb{D}_{\leq\,k-1}^{m\times n}. Then $\ell_{2}$ is a line in some affine geometry on a maximal set in $\mathbb{D}^{m\times n}$. By the induction hypothesis, we have $\varphi(\ell_{2})\subseteq\mathcal{M}_{1}^{\prime}$. Since $|\mathbb{D}|>|\mathbb{D}^{\prime}|$, $\varphi(\ell_{2})$ contains at least three noncollinear points in $AG(\mathcal{M}_{1}^{\prime})$. By Lemma 2.8 and $\varphi(B)\sim Y$ for any $Y\in\varphi(\ell_{2})$, we obtain $\varphi(B)\in\mathcal{M}_{1}^{\prime}$ for any $B\in\mathbb{D}_{k}^{m\times n}$. Hence $\varphi\left(\mathbb{D}_{\leq\,k}^{m\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}$. Taking $k=m$, we get $\varphi\left(\mathbb{D}^{m\times n}\right)\subseteq\mathcal{M}_{1}^{\prime}$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $\lceil x\rceil$ denote the smallest integer at least as large as $x$.

Theorem 5.9

Let $\mathbb{D},\mathbb{D}^{\prime}$ be two finite fields with $4\leq|\mathbb{D}|\leq|\mathbb{D}^{\prime}|\leq(|\mathbb{D}|-1)\lceil(|\mathbb{D}|+1)/2\rceil$, and let $m,n,m^{\prime},n^{\prime}\geq 2$ be integers. Suppose $\varphi:\mathbb{D}^{m\times n}\rightarrow{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is a degenerate graph homomorphism. Then there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that

[TABLE]

where $R\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is fixed.

*Proof. *When ${\rm min}\{m,n\}=2$, this theorem holds by Theorem 5.2. From now on we assume that $m,n>2$. By Theorem 5.2, there are two fixed maximal sets $\mathcal{M}$ and $\mathcal{N}$ of different types containing [math] in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, such that \scriptsize\varphi\left(\begin{array}[]{c}\mathbb{D}^{2\times n}\\ 0\\ \end{array}\right)\subseteq(\mathcal{M}+R)\cup(\mathcal{N}+R), where $R\in{\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$ is fixed. Suppose that $3\leq k\leq m$ and

[TABLE]

We prove \scriptsize\varphi\left(\begin{array}[]{c}\mathbb{D}^{k\times n}\\ 0\\ \end{array}\right)\subseteq(\mathcal{M}+R)\cup(\mathcal{N}+R) as follows.

Let \scriptsize A=\left(\begin{array}[]{c}A_{1}\\ 0\\ \end{array}\right)\in\mathbb{D}^{m\times n}, where \scriptsize A_{1}=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{k}\\ \end{array}\right)\in\mathbb{D}^{k\times n} with $\alpha_{k}\neq 0$. Put \scriptsize B_{1}=\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{k-1}\\ \end{array}\right). Then

[TABLE]

Let

[TABLE]

[TABLE]

Then $\ell_{1},\ell_{2},\ell_{d}^{\prime}$ ($d\in\mathbb{D}^{*}$) are $|\mathbb{D}|+1$ distinct lines in affine geometries on $|\mathbb{D}|+1$ maximal sets in $\mathbb{D}^{m\times n}$. Moreover, \ell_{1}\cap\ell_{2}=\ell_{i}\cap\ell_{d}^{\prime}=\ell_{d_{1}}^{\prime}\cap\ell_{d_{2}}^{\prime}=\scriptsize\left(\begin{array}[]{c}B_{1}\\ 0\\ \end{array}\right) for $i=1,2$ and $d,d_{1},d_{2}\in\mathbb{D}^{*}$ with $d_{1}\neq d_{2}$. Clearly, $\varphi(\ell_{i})$ or $\ell_{d}^{\prime}$ is contained in a maximal set in ${\mathbb{D}^{\prime}}^{m^{\prime}\times n^{\prime}}$, $i=1,2$, $d\in\mathbb{D}^{*}$. Since \scriptsize\ell_{i},\ell_{d}^{\prime}\subseteq\left(\begin{array}[]{c}\mathbb{D}^{(k-1)\times n}\\ 0\\ \end{array}\right), it follows from (78) that $\varphi(\ell_{i})\subseteq\mathcal{M}+R$ or $\varphi(\ell_{i})\subseteq\mathcal{N}+R$, $i=1,2$; $\varphi(\ell_{d}^{\prime})\subseteq\mathcal{M}+R$ or $\varphi(\ell_{d}^{\prime})\subseteq\mathcal{N}+R$, $d\in\mathbb{D}^{*}$.

Let either $\mathcal{M}^{\prime}=\mathcal{M}+R$ or $\mathcal{M}^{\prime}=\mathcal{N}+R$. Suppose that there is some $k$ or $d$ such that $\varphi(\ell_{k})$ or $\ell_{d}^{\prime}$ is not contained in a line in $AG(\mathcal{M}^{\prime})$. Then, from (79) $\varphi(A)$ is adjacent with three noncollinear points in $AG(\mathcal{M}^{\prime})$. By Lemma 2.8, we get $\varphi(A)\in\mathcal{M}^{\prime}$. Hence $\varphi(A)\in(\mathcal{M}+R)\cup(\mathcal{N}+R)$.

Now, we assume that $\varphi(\ell_{k})$ [resp. $\ell_{d}^{\prime}$] is contained in a line in $AG(\mathcal{M}^{\prime})$ for $k=1,2$ [resp. all $d\in\mathbb{D}^{*}$]. Let $X,Y\in\ell_{1}\cup\ell_{2}\cup_{d\in\mathbb{D}^{*}}\ell_{d}^{\prime}$ with $X\neq Y$. Then $X\sim Y$ and hence $\varphi(X)\sim\varphi(Y)$ and $\varphi(X)\neq\varphi(Y)$. By (79), we have $\varphi(A)\sim\varphi(X)$ and $\varphi(A)\sim\varphi(Y)$. Clearly, $\mathcal{M}+R$ or $\mathcal{N}+R$ contains $\lceil(|\mathbb{D}|+1)/2\rceil$ elements of the set $\left\{\varphi(\ell_{1}),\varphi(\ell_{2}),\varphi(\ell_{d}^{\prime}),d\in\mathbb{D}^{*}\right\}$. Without loss of generality, we assume that $\mathcal{M}+R$ contains $\lceil(|\mathbb{D}|+1)/2\rceil$ elements of the set $\left\{\varphi(\ell_{1}),\varphi(\ell_{2}),\varphi(\ell_{d}^{\prime}),d\in\mathbb{D}^{*}\right\}$. Then, $\mathcal{M}+R$ contains $(|\mathbb{D}|-1)\lceil(|\mathbb{D}|+1)/2\rceil+1$ distinct points in $\varphi(\ell_{1})\cup\varphi(\ell_{2})\cup_{d\in\mathbb{D}^{*}}\varphi(\ell_{d}^{\prime})$. Since $|\mathbb{D}^{\prime}|\leq(|\mathbb{D}|-1)\lceil(|\mathbb{D}|+1)/2\rceil$, every line in $AG(\mathcal{M}+R)$ contains at most $(|\mathbb{D}|-1)\lceil(|\mathbb{D}|+1)/2\rceil$ points. Thus $AG(\mathcal{M}+R)$ contains at least three noncollinear points. It follows from (79) that $\varphi(A)$ are adjacent with three noncollinear points in $AG(\mathcal{M}+R)$. Applying Lemma 2.8, we obtain $\varphi(A)\in\mathcal{M}+R$.

Therefore, we always have $\varphi(A)\in(\mathcal{M}+R)\cup(\mathcal{N}+R)$ for any \scriptsize A\in\left(\begin{array}[]{c}\mathbb{D}^{k\times n}\\ 0\\ \end{array}\right). Then

[TABLE]

Taking $k=m$, we get (77). $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 5.10

Let $n,m,p,q\geq 2$ be integers. By [23, Theorems 1.2 and 1.3], it is easy to see that every degenerate graph homomorphism $\varphi:\mathbb{F}_{q}^{m\times n}\rightarrow\mathbb{F}_{q}^{p\times q}$ is a (vertex) colouring. When $\mathbb{D}=\mathbb{D}^{\prime}=\mathbb{F}_{2}$ or $\mathbb{F}_{3}$, Both Theorem 4.1 and Corollary 5.7 still hold (cf. [23, 13, 17]). However, when $q\leq 3$ with $q<q^{\prime}$, it is an open problem to characterize the non-degenerate graph homomorphisms from $\mathbb{F}_{q}^{n\times n}$ to $\mathbb{F}_{q^{\prime}}^{p\times q}$.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Project 11371072), and the Scientific Research Fund of Hunan Provincial Education Department 16C0037.