Ellis groups in model theory and strongly generic sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916078139998208 |
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| author | Malinowski, Adam |
| author_facet | Malinowski, Adam |
| contents | Assume $G$ is a group and $\mathcal{A}$ is an algebra of subsets of $G$ closed under left translation. We study various ways to understand the Ellis group of the $G$-flow $S(\mathcal{A})$ (the Stone space of $\mathcal{A}$), with particular interest in the model-theoretic setting where $G$ is definable in a first order structure $M$ and $\mathcal{A}$ consists of externally definable subsets of $G$.
In one part of the thesis we explore strongly generic sets. Maximal algebras of such sets are shown to carry enough information to retrieve the Ellis group. A subset of $G$ is strongly generic if each non-empty Boolean combination of its translates is generic. Trivial examples include what we call *periodic* sets, which are unions of cosets of finite index subgroups of $G$. We give several characterizations of strongly generic sets, in particular, we relate them to almost periodic points of the flow $2^G$. For groups without a smallest finite index subgroup we show how to construct non-periodic strongly generic subsets in a systematic way. When $G$ is definable in a model $M$, a definable, strongly generic subset of $G$ will remain as such in any elementary extension of $M$ only if it is strongly generic in $G$ in an adequately uniform way. Sets satisfying this condition are called *uniformly strongly generic*. We analyse a few examples of these sets in different groups.
In the second part we assume that $G$ is a topological group and consider a particular algebra of its subsets denoted $\mathcal{SBP}$. It consists of subsets of $G$ that have the *strong Baire property*, meaning nowhere dense boundary. We explicitly describe the Ellis group of $S(\mathcal{A})$ for an arbitrary subalgebra $\mathcal{A}$ of $\mathcal{SBP}$ under varying assumptions on the group $G$, including the case when $G$ is a compact topological group. [...]
(Full abstract in the article) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00327 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ellis groups in model theory and strongly generic sets Malinowski, Adam Logic General Topology 03C, 37B (Primary) 54H11 (Secondary) Assume $G$ is a group and $\mathcal{A}$ is an algebra of subsets of $G$ closed under left translation. We study various ways to understand the Ellis group of the $G$-flow $S(\mathcal{A})$ (the Stone space of $\mathcal{A}$), with particular interest in the model-theoretic setting where $G$ is definable in a first order structure $M$ and $\mathcal{A}$ consists of externally definable subsets of $G$. In one part of the thesis we explore strongly generic sets. Maximal algebras of such sets are shown to carry enough information to retrieve the Ellis group. A subset of $G$ is strongly generic if each non-empty Boolean combination of its translates is generic. Trivial examples include what we call *periodic* sets, which are unions of cosets of finite index subgroups of $G$. We give several characterizations of strongly generic sets, in particular, we relate them to almost periodic points of the flow $2^G$. For groups without a smallest finite index subgroup we show how to construct non-periodic strongly generic subsets in a systematic way. When $G$ is definable in a model $M$, a definable, strongly generic subset of $G$ will remain as such in any elementary extension of $M$ only if it is strongly generic in $G$ in an adequately uniform way. Sets satisfying this condition are called *uniformly strongly generic*. We analyse a few examples of these sets in different groups. In the second part we assume that $G$ is a topological group and consider a particular algebra of its subsets denoted $\mathcal{SBP}$. It consists of subsets of $G$ that have the *strong Baire property*, meaning nowhere dense boundary. We explicitly describe the Ellis group of $S(\mathcal{A})$ for an arbitrary subalgebra $\mathcal{A}$ of $\mathcal{SBP}$ under varying assumptions on the group $G$, including the case when $G$ is a compact topological group. [...] (Full abstract in the article) |
| title | Ellis groups in model theory and strongly generic sets |
| topic | Logic General Topology 03C, 37B (Primary) 54H11 (Secondary) |
| url | https://arxiv.org/abs/2401.00327 |