Strict orders prohibit elimination of hyperimaginaries

A theory with the strict order property does not eliminate hyperimaginaries. Hence a theory without the independence property eliminates hyperimaginaries if and only if it is stable. A type definable equivalence relation is an equivalence relation on tuples of a certain length (possibly infinite) which is defined by a partial type E(x̄; ȳ) over ∅. A hyperimaginary is an equivalence class ā/E. If E(x̄; ȳ), where x̄ = (xi)i∈I , ȳ = (yi)i∈I , defines an equivalence relation only on the realisations of a partial type p(x̄) over ∅, then the partial type Ep(x̄; ȳ) = E(x̄; ȳ) ∪ { (φ(x̄) ∧ φ(ȳ)) ∨ xi = yi ∣∣ φ(x̄) ∈ p(x̄), i ∈ I} defines an equivalence relation on all tuples of the appropriate length, which coincides with E on the realisations of p and is equality on the other tuples. Thus even in this more general case ā/E is a hyperimaginary [2]. A hyperimaginary ā/E is said to be eliminable if there is a set of imaginaries A such that an automorphism of the monster model fixes A pointwise if and only if it fixes ā/E (i.e. maps ā to a tuple ā′ such that |= E(ā; ā′). It is a well-known and easy fact that a hyperimaginary ā/E is eliminable if and only if the equivalence Ep(x̄; ȳ) ≡ { (x̄; ȳ) ∣∣ (x̄; ȳ) definable equivalence relation over ∅, and E(x̄; ȳ) ` (x̄; ȳ)} holds, where p = tp(ā) [3]. Lemma 1. Let ā = (ai)i∈Q be an indiscernible sequence which is ordered by a formula φ(x̄; ȳ) without parameters, i.e. |= φ(āi; āj) ⇐⇒ i < j. Then the relation defined on the realisations of p(x̄) = tp(ā) by the partial type E(x̄; ȳ) = { (φ(x̄i; ȳj) ∧ φ(ȳi; x̄j) ∣∣ i, j ∈ Q, i < j} is clearly reflexive and symmetric. If it is also transitive, then the hyperimaginary ā/E is not eliminable. Proof. To simplify notation we will write the tuples x̄, ȳ as elements and the formula φ as <. So there is an indiscernible strictly <-ascending chain ā = (ai)i∈Q. Let p(x̄) = tp(ā). The partial type E(x̄; ȳ) = { (xi < yj) ∧ (yi < xj) ∣∣ i, j ∈ Q, i < j} is clearly reflexive, symmetric and transitive for realisations of p, so it defines a type-definable equivalence relation on p. Therefore ā/E is a hyperimaginary. We will show that it is not eliminable. Suppose (x̄; ȳ) is a definable equivalence relation such that Ep(x̄; ȳ) ` (x̄; ȳ). Now let A = { b̄ = (bi)i∈Q ∣∣ bi = af(i), where f : Q→ Q is order-preserving} be the set of all realisations of p that are actually subsequences of ā (which have been re-ordered in an order-preserving way). Of course A is not type-definable, but we will examine on A anyway. In fact, we will show that is trivial on A.