Monadic non-definability and gain-graphic matroids

We present an analogue of a Myhill-Nerode characterisation which will allow us to prove that classes of hypergraphs cannot be defined by sentences in the counting monadic second-order logic of hypergraphs. We apply this to classes of gain-graphic matroids, and show that if the group $Γ$ is not uniformly locally finite, then the class of $Γ$-gain-graphic matroids is not monadically definable. (A group is uniformly locally finite if and only if there is a maximum size amongst subgroups generated by at most $k$ elements, for every $k$.) 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 $Γ$-gain-graphic matroids is not monadically definable. This will be useful in future constructions.