Automorphism Group of the Rank-decreasing Graph Over the Semigroup of Upper Triangular Matrices

Abstract

Let Fq be a finite field with q elements, n ≥ 2 a positive integer, and T(n, q) the semigroup of all n × n upper triangular matrices over Fq. The rank-decreasing graph 𝕋 of T(n, q) is a directed graph which has T(n, q) as vertex set, and there is a directed edge from A ∈ T(n, q) to B ∈ T(n, q) if and only if r(AB) < r(B). The zero-divisor graph 𝒯 of T(n, q), with vertex set of all nonzero zero-divisors of T(n, q) and there is a directed edge from a vertex A to a vertex B if and only if AB = 0, can be viewed as a subgraph of 𝕋. In [16], L. Wang has determined the automorphisms of the zero-divisor graph 𝒯 of T(n, q). In this article, by applying the main result of [17] we determine the automorphisms of the rank-decreasing graph 𝕋 of T(n, q).