Beautiful pairs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911448586780672 |
|---|---|
| author | Kovacsics, Pablo Cubides Hils, Martin Ye, Jinhe |
| author_facet | Kovacsics, Pablo Cubides Hils, Martin Ye, Jinhe |
| contents | We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called \textit{beautiful pairs}, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with properties analogous to properties in Fraïssé classes.
After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically closed, $p$-adically closed and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces and our main result specializes to their strict pro-definability. Most notably, we exhibit the strict pro-definability of a natural space of types associated to Huber's analytification. In this way, we also recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields, which corresponds to Berkovich's analytification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00651 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Beautiful pairs Kovacsics, Pablo Cubides Hils, Martin Ye, Jinhe Logic 03C45, 03C10 (Primary) 03C64, 12J10, 12L12 (Secondary) We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called \textit{beautiful pairs}, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with properties analogous to properties in Fraïssé classes. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically closed, $p$-adically closed and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces and our main result specializes to their strict pro-definability. Most notably, we exhibit the strict pro-definability of a natural space of types associated to Huber's analytification. In this way, we also recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields, which corresponds to Berkovich's analytification. |
| title | Beautiful pairs |
| topic | Logic 03C45, 03C10 (Primary) 03C64, 12J10, 12L12 (Secondary) |
| url | https://arxiv.org/abs/2112.00651 |