Recall that in a first-order theory $T$, a hyperimaginary is an equivalence class of some type-definable equivalence relation $E(x,y)$ (with $x$ a possibly infinite tuple of parameters). The $E$-equivalence class of a tuple $a$ is written $a_E$.
A hyperimaginary is finitary if it comes from an equivalence relation on finite tuples and infinitary otherwise. $T$ has elimination of hyperimaginaries if for any type-definable equivalence relation $E(x,y)$, there is a family $(\varepsilon_i(x,y))_{i \in I}$ of definable equivalence relations such that $E(x,y)$ is logically equivalent to $\bigwedge_{i\in I} \varepsilon_i(x,y)$.
Let $\mathbb{M}$ be the monster model of $T$. Given a set of hyperimaginaries $A$, $\mathrm{Aut}(\mathbb{M}/A)$ is the group of automorphisms $\sigma$ of $\mathbb{M}$ satisfying that $E(\sigma a,a)$ for each $a_E \in A$. We say that $b_F$ is definable over $A$ if $b_F$ is fixed by every $\sigma \in \mathrm{Aut}(\mathbb{M}/A)$. We write $\mathrm{dcl}^{heq}(A)$ for the class of hyperimaginaries that are definable over $A$. We say that $A$ and $B$ are interdefinable if $\mathrm{dcl}^{heq}(A) = \mathrm{dcl}^{heq}(B)$.
An important observation is that given a type-definable equivalence relation $E(x,y)$, there is a family $(F_i(x,y))_{i\in I}$ of countably type-definable equivalence relations such that $E(x,y)$ is logically equivalent to $\bigwedge_{i \in I} F_i(x,y)$. This implies that any hyperimaginary $a_E$ is definable over some set of hyperimaginaries that are equivalence classes of countably type-definable equivalence relations.
It is a slightly surprising fact that a theory as simple as $\mathsf{DLO}$ actually fails to eliminate (infinitary) hyperimaginaries: Let $x$ and $y$ be tuples of variables indexed by $\mathbb{Q}$. Let $E(x,y)$ be $\bigwedge_{i<j}(x_i<y_j) \wedge (y_i<x_j)$. It is not too hard to see that this is in fact an equivalence relation, and it is not that much harder to see that it cannot be written as a conjunction of definable equivalence relations. (Something similar can be done in any theory with the strict order property. See here.)
I'm curious if in $\mathsf{DLO}$ this is in some sense the only hyperimaginary that cannot be eliminated. You can modify this definition by using larger dense linear order than $\mathbb{Q}$ to index your variables, but such equivalence relations are clearly conjunctions of instances of $E$.
| Question: Is it true that for any set $A$ of hyperimaginaries (in $\mathsf{DLO}$), there is a set $B$ of real elements and a set $C$ of hyperimaginaries of the form $c_E$ such that $A$ and $BC$ are interdefinable?
If not, is there an easy to describe family of type-definable equivalence relations that is sufficient to describe all hyperimaginaries up to interdefinability?
