We introduce the notion of first order amenability, as a property of a first order theory T: every complete type over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∅</mi> </math></EquationSource> </InlineEquation>, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of T follows from amenability of the (topological) group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${{\,\mathrm{{Aut}}\,}}(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mi mathvariant="normal">Aut</mi> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all sufficiently large <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\aleph _{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℵ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from Selecta Math. (N.S.) 28, 16 (2022), we prove that if T is amenable, then T is G-compact, namely Lascar strong types and Kim-Pillay strong types over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∅</mi> </math></EquationSource> </InlineEquation> coincide. This extends and essentially generalizes a similar result proved via different methods for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-categorical theories in Adv. Math. 345, 1253–1299 (2019). In the special case when amenability is witnessed by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∅</mi> </math></EquationSource> </InlineEquation>-definable global Keisler measures (which is for example the case for amenable <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-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.

On first order amenability

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2026年2月25日

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Ehud Hrushovski,
Krzysztof Krupiński,
Anand Pillay

摘要

We introduce the notion of first order amenability, as a property of a first order theory T: every complete type over $\emptyset $ $\emptyset$ , 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 ${{\,\mathrm{{Aut}}\,}}(M)$ $Aut (M)$ for all sufficiently large $\aleph _{0}$ $ℵ_{0}$ -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 Selecta Math. (N.S.) 28, 16 (2022), we prove that if T is amenable, then T is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset $ $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $\omega $ $ω$ -categorical theories in Adv. Math. 345, 1253–1299 (2019). In the special case when amenability is witnessed by $\emptyset $ $\emptyset$ -definable global Keisler measures (which is for example the case for amenable $\omega $ $ω$ -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.