Orthogonality and domination in o-minimal expansions of ordered groups
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arXiv
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| Format: | Preprint |
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2025
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| author | Guerrero, Pablo Andújar Eleftheriou, Pantelis E. Mennuni, Rosario |
| author_facet | Guerrero, Pablo Andújar Eleftheriou, Pantelis E. Mennuni, Rosario |
| contents | We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orthogonality and domination in o-minimal expansions of ordered groups Guerrero, Pablo Andújar Eleftheriou, Pantelis E. Mennuni, Rosario Logic 03C64 (Primary) 03C45 (Secondary) We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not. |
| title | Orthogonality and domination in o-minimal expansions of ordered groups |
| topic | Logic 03C64 (Primary) 03C45 (Secondary) |
| url | https://arxiv.org/abs/2503.17120 |