Semigroups of non-negative integral matrices

Abstract

In [1], Pall proved an interesting result on a certain class of 2 × 2 integral matrices. He showed that the semigroup of 2 × 2 matrices of determinant 1 and non-negative entries contains exactly 2 primes , and every other non-unit is expressible uniquely as products of these primes. Before formally stating this result, we need some notation. Let G n denote the semigroup of n × n matrices with determinant 1 and nonnegative integral entries, I n the n × n identity matrix, the n × n matrix with a 1 as its ( i, j ) element and zeros elsewhere, and let . When the dimension is clear, we shall drop the superscripts.