Graph homomorphisms on rectangular matrices over division rings II

Abstract

Let D m×n be the set of m × n matrices over a division ring D. Two matrices A, B ∈ D m×n are adjacent if rank(A − B) = 1. By the adjacency, D m×n is a connected graph. Suppose D, D ′ are division rings and m, n, m ′ , n ′ ≥ 2 are integers. We determine additive graph homomorphisms from D m×n to D ′m ′ ×n ′ . When |D| ≥ 4, we characterize the graph homomorphism ϕ : D n×n → D ′m ′ ×n ′ if ϕ(0) = 0 and there exists A 0 ∈ D n×n such that rank(ϕ(A 0 )) = n. We also discuss properties and ranges on degenerate graph homomorphisms. If f : D m×n → D ′m ′ ×n ′ (where min{m, n} = 2) is a degenerate graph homomorphism, we prove that the image of f is contained in a union of two maximal adjacent sets of different types. For the case of finite fields, we obtain two better results on degenerate graph homomorphisms.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 D m×n . Then there are exactly two maximal sets M and M ′ containing A and B. Moreover, M and M ′ are of different types. Corollary 2.3 (cf. [9, 12, 26]) If M and N are two distinct maximal sets of the same type [resp. different types] in D m×n with M ∩ N ∅, then |M ∩ N| = 1 [resp. |M ∩ N| ≥ 2].Lemma 2.4 (cf. [14, Lemma 3.4]) Suppose that M, N are two distinct maximal sets in D m×n with M ∩ N ∅. Then: (i) if M, N are of different types, then for any A ∈ M ∩ N, there are P ∈ GL m (D) and Q ∈ GL n (D) such that M = PM 1 Q + A = PM 1 + A and N = PN 1 Q + A = N 1 Q + A;