Isomorphism relations on classes of c.e. algebras

TL;DR

This paper studies the complexity of isomorphism relations among classes of finitely generated computably enumerable algebras, developing a framework to compare their complexity using computable reducibility and analyzing specific algebraic classes.

Contribution

It introduces a systematic computability-theoretic framework for analyzing isomorphism problems of c.e. algebras and applies it to various algebraic classes.

Findings

If algebras satisfy the ascending chain condition, their isomorphism relation reduces to =^{ce}

The framework compares complexity of isomorphism relations to canonical c.e. benchmarks

Analyzes isomorphism complexity for finitely generated semigroups, monoids, and groups

Abstract

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(\alpha). 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.