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Permutation modules for Ramsey structures

David M. Evans

Abstract

Suppose $R$ is a commutative ring and $G$ is a group acting on a set $W$. We consider the $RG$-module $RW$ in the case where $G$ is the automorphism group of an $ω$-categorical structure $M$ and $W$ is, for example, $M^n$ (for $n \in \mathbb{N}$). We develop methods which may provide information about two questions in the case where $R$ is a field $F$: whether $FW$ has a.c.c. on submodules; and in the case where $M$ is finitely homogeneous, whether $FW$ is of finite composition length. In the case where $M$ is a Ramsey structure and so $G$ is extremely amenable, we give a simple `decision procedure' for membership in a submodule of $RW$ specified by a given generating set. If $F$ is a field, we show that there is a duality between submodules of $FW$ and the topological $FG$-module of definable functions from $W$ to $F$.

Permutation modules for Ramsey structures

Abstract

Suppose is a commutative ring and is a group acting on a set . We consider the -module in the case where is the automorphism group of an -categorical structure and is, for example, (for ). We develop methods which may provide information about two questions in the case where is a field : whether has a.c.c. on submodules; and in the case where is finitely homogeneous, whether is of finite composition length. In the case where is a Ramsey structure and so is extremely amenable, we give a simple `decision procedure' for membership in a submodule of specified by a given generating set. If is a field, we show that there is a duality between submodules of and the topological -module of definable functions from to .

Paper Structure

This paper contains 13 sections, 16 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Background and Questions
  3. Ramsey structures and a decision procedure
  4. Background definitions and notation
  5. Permutation groups
  6. Model theory
  7. Ramsey classes and extreme amenability
  8. Dualities
  9. Vector space duality
  10. Proof of the Main Results
  11. Further results
  12. The definable dual
  13. A (possible) reduction for a.c.c.

Key Result

Theorem 1.4

Suppose $R$ is a commutative ring, $M$ is an $\omega$-categorical Ramsey structure and $W$ is a sort in $M^{eq}$. Let $x, v_1,\ldots, v_t$, $S$, $\Omega_S$ be as defined above. Then

Theorems & Definitions (22)

  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • ...and 12 more