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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2401.00550 |
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| _version_ | 1866909217897578496 |
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| author | Kurilić, Miloš S. |
| author_facet | Kurilić, Miloš S. |
| contents | The poset of copies of a relational structure ${\mathbb X}$ is the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X})=\{ Y\subset X: {\mathbb Y} \cong {\mathbb X}\}$. Investigating the classification of structures related to isomorphism of the Boolean completions ${\mathbb B}_{\mathbb X} ={\mathop{\rm ro}\nolimits}({\mathop{\rm sq}\nolimits} ({\mathbb P} ({\mathbb X}) ))$ we extend the results concerning linear orders to the class of structures definable in linear orders by first-order $Σ_0$-formulas (monomorphic structures). So, ${\mathbb B}_{\mathbb X} \cong {\mathbb B}_{\mathbb L}$ holds for some linear order ${\mathbb L}$, if ${\mathbb X}$ is definable in a $σ$-scattered (in particular, countable) or additively indecomposable linear order. For example, ${\mathbb B}_{\mathbb X} \cong {\mathop{\rm ro}\nolimits}({\mathbb S} )$, where ${\mathbb S}$ is the Sacks forcing, whenever ${\mathbb X}$ is a non-constant structure chainable by a real order type containing a perfect set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Copies of Monomorphic Structures Kurilić, Miloš S. Logic 06A05, 06A10, 03E40, 03E35 The poset of copies of a relational structure ${\mathbb X}$ is the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X})=\{ Y\subset X: {\mathbb Y} \cong {\mathbb X}\}$. Investigating the classification of structures related to isomorphism of the Boolean completions ${\mathbb B}_{\mathbb X} ={\mathop{\rm ro}\nolimits}({\mathop{\rm sq}\nolimits} ({\mathbb P} ({\mathbb X}) ))$ we extend the results concerning linear orders to the class of structures definable in linear orders by first-order $Σ_0$-formulas (monomorphic structures). So, ${\mathbb B}_{\mathbb X} \cong {\mathbb B}_{\mathbb L}$ holds for some linear order ${\mathbb L}$, if ${\mathbb X}$ is definable in a $σ$-scattered (in particular, countable) or additively indecomposable linear order. For example, ${\mathbb B}_{\mathbb X} \cong {\mathop{\rm ro}\nolimits}({\mathbb S} )$, where ${\mathbb S}$ is the Sacks forcing, whenever ${\mathbb X}$ is a non-constant structure chainable by a real order type containing a perfect set. |
| title | Copies of Monomorphic Structures |
| topic | Logic 06A05, 06A10, 03E40, 03E35 |
| url | https://arxiv.org/abs/2401.00550 |