Copies of monomorphic structures

Abstract The poset of copies of a relational structure 𝕏 is the partial order 〈ℙ(𝕏), ⊂〉, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">P</m:mi> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">X</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mo fence="false" stretchy="false">{</m:mo> <m:mi>Y</m:mi> <m:mo>⊂</m:mo> <m:mi>X</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Y</m:mi> </m:mrow> <m:mo>≅</m:mo> <m:mrow> <m:mi mathvariant="double-struck">X</m:mi> </m:mrow> <m:mo fence="false" stretchy="false">}</m:mo> <m:mo>.</m:mo> </m:math> $\mathbb{P}(\mathbb{X})=\{Y \subset X: \mathbb{Y} \cong \mathbb{X}\}.$ Investigating the classification of structures related to isomorphism of the Boolean completions 𝔹 𝕏 = ro(sq(ℙ(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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">X</m:mi> </m:mrow> </m:mrow> </m:msub> <m:mo>≅</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">L</m:mi> </m:mrow> </m:mrow> </m:msub> </m:math> $\mathbb{B}_{\mathbb{X}} \cong \mathbb{B}_{\mathbb{L}}$ holds for some linear order 𝕃, if 𝕏 is definable in a σ -scattered (in particular, countable) or additively indecomposable linear order. For example, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">X</m:mi> </m:mrow> </m:mrow> </m:msub> <m:mo>≅</m:mo> <m:mo form="prefix">ro</m:mo> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">S</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:math> $\mathbb{B}_{\mathbb{X}} \cong \operatorname{ro}(\mathbb{S}),$ where 𝕊 is the Sacks forcing, whenever 𝕏 is a non-constant structure chainable by a real order type containing a perfect set.