可计算枚举代数类上的同构关系
Isomorphism relations on classes of c.e. algebras
AI总结：
研究可计算枚举代数类的同构关系复杂性，通过可计算约简分析其算法复杂性与代数结构的相互作用。
摘要：
我们研究了有限生成和n生成的可计算枚举(可计算枚举)代数类的同构关系的复杂性，这些代数通过可计算枚举呈现——即作为由可判定生成集上的术语代数商去可计算同余关系。我们的目标是建立一个系统框架，从计算理论的角度分析此类同构问题。为了比较其复杂性，我们采用可计算约简的概念，将这些关系与可计算集上的标准基准进行比较，如=^{ce}、E_0^{ce}以及由序数索引的E_min(α)家族。我们工作的核心洞察是代数结构与算法复杂性之间的相互作用：我们证明如果一个类中的每个代数在其同余格上满足升链条件，则相应的同构关系可计算约简到=^{ce}。我们还应用这一框架到一系列具体的案例中。特别是，我们分析了有限生成交换半群、单体和群的同构关系，并将它们置于更广泛的分类问题背景下。
英文摘要：
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets of generators by c.e. congruences. Our goal is to develop a systematic framework for analyzing such isomorphism problems from a computability-theoretic perspective. To compare their complexity, we employ the notion of computable reducibility, measuring these relations against canonical benchmarks on c.e. sets, such as =^{ce}, E_0^{ce}, and the ordinal-indexed family E_min(α). A central insight of our work is the interplay between the algebraic structure and the algorithmic complexity: we show that if every algebra in a class satisfies the ascending chain condition on its congruence lattice, then the corresponding isomorphism relation is computably reducible to =^{ce}. We also apply this framework to a range of concrete cases. In particular, we analyze the isomorphism relations for finitely generated commutative semigroups, monoids, and groups, positioning them within the broader landscape of classification problems.
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