Permutation modules for Ramsey structures
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914434507603968 |
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| author | Evans, David M. |
| author_facet | Evans, David M. |
| contents | 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 $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Permutation modules for Ramsey structures Evans, David M. Group Theory Combinatorics Logic Primary 20B27, Secondary 20B07, 03C15 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 $F$. |
| title | Permutation modules for Ramsey structures |
| topic | Group Theory Combinatorics Logic Primary 20B27, Secondary 20B07, 03C15 |
| url | https://arxiv.org/abs/2603.29606 |