Taking model-complete cores
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918263414325248 |
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| author | Bodirsky, Manuel Bodor, Bertalan Marimon, Paolo |
| author_facet | Bodirsky, Manuel Bodor, Bertalan Marimon, Paolo |
| contents | A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; the core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, they always exist for $ω$-categorical theories. We show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved by moving to the core companion of a theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $ω$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Taking model-complete cores Bodirsky, Manuel Bodor, Bertalan Marimon, Paolo Logic 03C10, 03C40, 03C45, 20B27 A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; the core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, they always exist for $ω$-categorical theories. We show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved by moving to the core companion of a theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $ω$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal. |
| title | Taking model-complete cores |
| topic | Logic 03C10, 03C40, 03C45, 20B27 |
| url | https://arxiv.org/abs/2512.21278 |