Isomorphism relations on classes of c.e. algebras

TL;DR

The paper develops a systematic framework for the computability-theoretic analysis of isomorphism problems for classes of $n$-generated and finitely generated c.e. algebras presented by c.e. congruences. By leveraging computable reducibility and benchmark relations such as $=^{ce}$, $E_0^{ce}$, and $E_{ ext{min}}(\\alpha)$, it connects algebraic structure (e.g., ACC on congruence lattices) to upper bounds on isomorphism complexity, and provides precise classifications for several natural classes (e.g., AG, UF$^{1}$, CS, CM). Notably, the work shows that when ACC holds, the isomorphism relations lie below $=^{ce}$ (often equating to $E_{ ext{min}}$-degree structures such as $E_{ ext{min}}(\omega\cdot n)$), while certain higher-arity or non-ACC settings yield $\,\Sigma^0_3$-complete isomorphism relations (e.g., $\mathcal{UF}^{2}_{1}$). The results illuminate how algebraic restrictions dictate classification complexity and lay groundwork for a broader hierarchy of isomorphism problems across computable algebra. The framework offers a roadmap for further exploration of when isomorphism relations are strictly below or above standard benchmarks, with potential implications for structural classification in computable algebraic varieties.

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(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.

Figures (4)

• Figure 1: An example of an algebra in ${\mathcal{UF}^{1}_{f.g.}}$. There are two components; the one on the left is finite and the one on the right is infinite. In the infinite component, the part that is not colored red (i.e. everything below and including $f(x_6)$) is the finite graph that characterizes the component.
• Figure 2: A visualization of how the invariant is calculated for a particular $S(\mathcal{A})$ of a $2$-generated presentation $\mathcal{A}$. The dots represent elements in $A_0$ and the lines represent elements in $A_1$. From their cardinalities, we get $\gamma(A) = \omega\cdot 7 + 22$.
• Figure 3: The isomorphism relations investigated in this paper, compared to well-understood equivalence relations. A question mark indicates that we do not know whether or not the reducibility is strict. Apart from strict reducibilities, we omitted non-reducibility results.
• Figure 4: The algebra in Theorem \ref{['thm:uf21-complete']}

Theorems & Definitions (100)

• Theorem 2.1: Third Isomorphism Theorem
• Definition 2.2: delle2020word
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Lemma 2.7
• proof
• Lemma 2.8
• proof