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arXiv · 2510.00139

Myhill-Nerode for hypergraphs and an application to gain-graphic matroids

Daryl Funk·Angus Matthews·Dillon Mayhew
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Abstract

We present a Myhill-Nerode theorem for hypergraphs. The theorem involves an operation which takes two input structures and produces a hypergraph as output. Using this operation, we define a Myhill-Nerode-type equivalence relation and show that if a class of hypergraphs is definable in the counting monadic second-order logic of hypergraphs, then the equivalence relation has finite index. We apply this tool to classes of gain-graphic matroids, and show that if the group $Γ$ is not uniformly locally finite, then the class of $Γ$\dash gain-graphic matroids is not monadically definable. (A group is uniformly locally finite if, for every $k$, there is a maximum size amongst subgroups generated by at most $k$ elements.) In addition, we define the conviviality graph of a group, and show that if the group $Γ$ has an infinite conviviality graph, then the class of $Γ$\dash gain-graphic matroids is not monadically definable. This will be useful in future constructions.

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Daryl Funk, Angus Matthews, Dillon Mayhew. 2026-04-10. Myhill-Nerode for hypergraphs and an application to gain-graphic matroids. https://arxiv.org/abs/2510.00139

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