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

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 $\Gamma$ is not uniformly locally finite, then the class of $\Gamma$\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 $\Gamma$ has an infinite conviviality graph, then the class of $\Gamma$\dash gain-graphic matroids is not monadically definable. This will be useful in future constructions.