Motivated by the study of word problems of monoids, we explore two ways of viewing binary relations on $X^{⁎}$ as languages. We exhibit a hierarchy of classes of binary relations on $X^{⁎}$, according to the class of languages the relation belongs to and the chosen viewpoint. We give examples of word problems of monoids distinguishing the various classes. Aside from the algebraic interest, these examples demonstrate that the hierarchy still holds when restricted to equivalence relations.

中文翻译：

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二进制关系的语言层次

受研究词元组问题的启发，我们探索了两种查看二元关系的方式 $X^{⁎}$作为语言。我们展示了关于$X^{⁎}$，根据关系所属的语言类别和所选的观点。我们给出了区分各个类的类词的单词问题示例。除了代数兴趣之外，这些示例还表明，当仅限于等价关系时，层次结构仍然成立。