This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.

本调查旨在概述组合代数中最古老和最著名的开放问题之一：单关系幺半群的单词问题。我们提供了从1914 年开始的问题历史，并详细概述了中心结果的证明，尤其是 Adian 和他的学生 Oganesian 的证明。在展示了如何将问题减少到左取消案例之后，调查的后半部分重点介绍了解决这个家庭中部分案例的各种方法。我们以与这个问题有关的一些现代和最近的结果结束，包括到 Collat​​z 猜想的链接。在此过程中，我们强调并解决了多年来文献中出现的一些不正确和不准确的陈述。我们还填补了将特殊逆幺半群与单关系幺半群联系起来的定理证明中的一个空白，并稍微加强了这个定理的陈述。