Relations in the semigroup of 2 × 2 upper-triangular matrices

Abstract

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.