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Monadic transductions and definable classes of matroids

Susan Jowett, Dillon Mayhew, Songbao Mo, Christopher Tuffley

Abstract

A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-contained presentation of the theory of transductions for the monadic second-order language of matroids. This includes a proof of the matroid version of the Backwards Translation Theorem, which lifts any formula applied to the images of the transduction into a formula which we can apply to the pre-images. Applications include proofs that the class of lattice-path matroids and the class of spike-minors can be defined by sentences in monadic second-order logic.

Monadic transductions and definable classes of matroids

Abstract

A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-contained presentation of the theory of transductions for the monadic second-order language of matroids. This includes a proof of the matroid version of the Backwards Translation Theorem, which lifts any formula applied to the images of the transduction into a formula which we can apply to the pre-images. Applications include proofs that the class of lattice-path matroids and the class of spike-minors can be defined by sentences in monadic second-order logic.
Paper Structure (15 sections, 24 theorems, 53 equations)

This paper contains 15 sections, 24 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shorthands
  4. Interpretations
  5. Matroid formulas.
  6. Transductions
  7. The definition
  8. Examples
  9. Backwards Translation Theorem
  10. Operations on definable classes
  11. Sums
  12. Extensions and coextensions
  13. Definable classes
  14. Lattice-path matroids
  15. Spikes

Key Result

Proposition 2.3

There exists an $\mathit{MS}_{0}$- sentence $\text{ Matroid}$ that is satisfied by a set-system $M=(E,\mathcal{I})$ if and only if $\mathcal{I}$ is the collection of independent sets of a matroid on the ground set $E$.

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • ...and 40 more