Scott Complexity of Reduced Abelian $p$-Groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913423299706880 |
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| author | Alvir, Rachael Csima, Barbara F. MacLean, Luke |
| author_facet | Alvir, Rachael Csima, Barbara F. MacLean, Luke |
| contents | Given a reduced abelian $p$-group, we give an upper bound on the Scott complexity of the group in terms of its Ulm invariants. For limit ordinals, we show that this upper bound is tight. This gives an explicit sequence of such groups with arbitrarily high Scott complexity below $ω_1$. Along the way, we give a largely algebraic characterization of the back-and-forth relations on reduced abelian $p$-groups, making progress on an open problem from the book by Ash and Knight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scott Complexity of Reduced Abelian $p$-Groups Alvir, Rachael Csima, Barbara F. MacLean, Luke Logic Group Theory 03D45, 20K10, 03C75, 03E15 Given a reduced abelian $p$-group, we give an upper bound on the Scott complexity of the group in terms of its Ulm invariants. For limit ordinals, we show that this upper bound is tight. This gives an explicit sequence of such groups with arbitrarily high Scott complexity below $ω_1$. Along the way, we give a largely algebraic characterization of the back-and-forth relations on reduced abelian $p$-groups, making progress on an open problem from the book by Ash and Knight. |
| title | Scott Complexity of Reduced Abelian $p$-Groups |
| topic | Logic Group Theory 03D45, 20K10, 03C75, 03E15 |
| url | https://arxiv.org/abs/2407.06940 |