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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.00550 |
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Table of 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.