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.

中文翻译：

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2 × 2 上三角矩阵半群中的关系

令 [Formula: see text] 和 [Formula: see text] 成为 [Formula: see text] 有理项的上三角矩阵。在这些矩阵生成的乘法半群中，我们检查是否存在 [Formula: see text] 形式的关系，其中 [Formula: see text] [Formula: see text] 和 [Formula: see text] 我们给出找到关系的算法以前的形式。我们的结果是 Charlier 和 Honkala 在 [The freeness problem over matrix semigroups and bounded languages, Inf. 中获得的一些定理的扩展。计算。237 (2014) 243–256]。我们的论文处于代数、数论和理论计算机科学之间的界面。虽然主要结果涉及可判定性和半群论，但获得它们的方法来自数论。