Copies of Monomorphic Structures (2401.00550v2)

Abstract: 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 $\Sigma _0$-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 $\sigma$-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.