This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.

中文翻译：

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Word 问题和 ceers

本注释解决了通过可计算可枚举（或简称为 ce）结构（例如 ce 半群、群和环）的单词问题可以实现哪些 ceer 的问题，其中实现意味着落入相同的可还原度（在等价关系的约简概念下，通常称为“可计算约简”），或在相同的同构类型中（具有由可计算函数引起的同构），或在相同的强同构类型中（具有由可计算的置换引起的同构）自然数）。例如，我们观察到每个 ceer 与某个 ce 半群的单词问题同构，但是（回答 Gao 和 Gerdes 的问题）并非每个 ceer 都与某个有限表示的半群的单词问题具有相同的可约度，也不在某个非周期半群的同一可约度上。我们还表明，Peano Arithmetic 的可证明等价提供的 ceer 与某些非交换和非布尔 ce 环的单词问题属于同一强同构类型。