Copies of Monomorphic Structures

Abstract

The poset of copies of a relational structure $\mathbb{X}$ is the partial order $⟨\mathbb{P}(\mathbb{X}),\subset ⟩$, 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 ${\mathbb{B}}_{\mathbb{X}}=\mathrm{ro}(\mathrm{sq}(\mathbb{P}(\mathbb{X})))$ 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, ${\mathbb{B}}_{\mathbb{X}}\cong \mathrm{ro}(\mathbb{S})$, 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.

18 pages