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This is a Quantitative Study study.

October 20, 2025Open Access

Graphings of arithmetical equivalence relations

Key Points

The study demonstrates that some arithmetical equivalence relations can be graphically represented, enhancing understanding of structure connections.
Key findings show that the computable isomorphism relation on natural numbers can be expressed using a simpler graph structure.
An analysis is performed on the graphings of Friedman-Stanley jumps, establishing an arithmetical construction method for these graphs.
This research may enable further exploration of complex equivalence relations through more accessible graph representations.

Abstract

This paper studies when an arithmetical equivalence relation E can be realized as the connectedness relation of a graph G which is simpler to define than E. Several examples of such equivalence relations are established. In particular, it is proved that the ⁰₃ relation of computable isomorphism of structures on in a computable first-order language is ⁰₂-graphable, i. e. , is the connectedness relation of a ⁰₂ graph. Graphings of Friedman-Stanley jumps are studied, including an arithmetical construction of a graphing of the Friedman-Stanley jump of E from a graphing of E.

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Tyler Arant (Tue,) studied this question.

synapsesocial.com/papers/68f5a78aab63786de5b460b7 https://doi.org/https://doi.org/10.48550/arxiv.2505.14920

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