Hyperimaginaries and Automorphism Groups

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [?], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M). In section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In section ∗Partially supported by an NSF grant This work was begun during a visit of the two authors to the Centre de Recerca Matemèmatica, Institut d’Estudis Catalans. The authors wish to express their gratitude for its support and hospitability