On first order amenability
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866909903916892160 |
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| author | Hrushovski, Ehud Krupiński, Krzysztof Pillay, Anand |
| author_facet | Hrushovski, Ehud Krupiński, Krzysztof Pillay, Anand |
| contents | We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive.
First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $ω$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $ω$-categorical theories), we also give a different proof, based on stability in continuous logic.
Parallel (but easier) results hold for the notion of extreme amenability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_08306 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On first order amenability Hrushovski, Ehud Krupiński, Krzysztof Pillay, Anand Logic 03C45, 43A07 We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $ω$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $ω$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability. |
| title | On first order amenability |
| topic | Logic 03C45, 43A07 |
| url | https://arxiv.org/abs/2004.08306 |