Equivalents of NOTOP
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908553563865088 |
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| author | Laskowski, Michael C. Ulrich, Danielle S. |
| author_facet | Laskowski, Michael C. Ulrich, Danielle S. |
| contents | Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP.
In particular, NOTOP is equivalent to V-DI, the assertion that any type $V$-dominated by an independent triple is isolated over the triple.
If $T$ has NOTOP, then every model $N$ is atomic over an independent tree of countable, elementary substructures, and hence is
determined up to back-and-forth equivalence over such a tree.
We also verify Shelah's assertion from Chapter XII of \cite{Shc} that NOTOP implies PMOP (without using NDOP). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivalents of NOTOP Laskowski, Michael C. Ulrich, Danielle S. Logic 03C45 Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type $V$-dominated by an independent triple is isolated over the triple. If $T$ has NOTOP, then every model $N$ is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah's assertion from Chapter XII of \cite{Shc} that NOTOP implies PMOP (without using NDOP). |
| title | Equivalents of NOTOP |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2505.00844 |