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Monadic non-definability and gain-graphic matroids

Daryl Funk, Angus Matthews, Dillon Mayhew

TL;DR

This paper investigates the definability of Gamma-gain-graphic matroids within counting monadic second-order logic for hypergraphs ($CMSO_{1}$). It develops a Myhill–Nerode–style framework for hypergraphs, including coloured systems, coloured sums, clefts, and registries, to bound the index of CMSO$_1$-definable classes. The authors prove that if a group $\Gamma$ is not uniformly locally finite, then the class of $\Gamma$-gain-graphic matroids is not $CMSO_{1}$-definable, and they further show that an infinite $F$-conviviality graph (for some finite subgroup $F$) yields the same non-definability result. The work leverages ultrapower techniques to transfer local finiteness properties and uses matroid amalgams and gain-graphs to build the necessary counterexamples. Together, these results delineate the limitations of CMSO$_1$ in capturing broad classes of gain-graphic matroids and provide structural tools for future constructions.

Abstract

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.

Monadic non-definability and gain-graphic matroids

TL;DR

This paper investigates the definability of Gamma-gain-graphic matroids within counting monadic second-order logic for hypergraphs (). It develops a Myhill–Nerode–style framework for hypergraphs, including coloured systems, coloured sums, clefts, and registries, to bound the index of CMSO-definable classes. The authors prove that if a group is not uniformly locally finite, then the class of -gain-graphic matroids is not -definable, and they further show that an infinite -conviviality graph (for some finite subgroup ) yields the same non-definability result. The work leverages ultrapower techniques to transfer local finiteness properties and uses matroid amalgams and gain-graphs to build the necessary counterexamples. Together, these results delineate the limitations of CMSO in capturing broad classes of gain-graphic matroids and provide structural tools for future constructions.

Abstract

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 is not uniformly locally finite, then the class of -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 elements, for every .) In addition, we define the conviviality graph of a group, and show that if the group has an infinite conviviality graph, then the class of -gain-graphic matroids is not monadically definable. This will be useful in future constructions.

Paper Structure

This paper contains 11 sections, 18 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraphs
  4. Monadic second-order logic
  5. Gain-graphic matroids
  6. Ultrapowers
  7. A Myhill-Nerode analogue
  8. Amalgams
  9. Uniformly locally finite groups
  10. The conviviality graph
  11. Acknowledgements

Key Result

Proposition 2.1

Let $\Omega=(G,\mathcal{B})$ be a biased graph. Assume that the element $e$ is contained in two distinct long lines of $F(\Omega)$. Then $e$ is a loop edge of $G$.

Figures (4)

  • Figure 1: The graph $H_{n,N}^{*}$.
  • Figure 2: The graph $H_{n,N} \setminus H_{n,N}^{*}$.
  • Figure 3: The graph $\Lambda^{*}_{\Gamma_1,\Gamma_2}$
  • Figure 4: The graph $\Lambda_{\Gamma_1,\Gamma_2} \setminus \Lambda^{*}_{\Gamma_1,\Gamma_2}$

Theorems & Definitions (66)

  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 56 more