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 $\omega$-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 $F$.
From: David Evans [view email]
[v1]
Tue, 31 Mar 2026 11:22:55 UTC (18 KB)
[v2]
Wed, 9 Sep 2026 14:48:23 UTC (21 KB)
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