The Journal of Symbolic Logic

Research Article

Hyperimaginaries and automorphism groups

D. Lascara1 and A. Pillaya2

a1 UFR de Math. Upresa 7056, CNRS, Universite Paris VII, 2. Place Jussieu. Cace 7012, 75251 Paris Cedex 05., France, E-mail: lascar@logique.jussieu.fr

a2 Dept Math., Altgeld Hall, University of Illinois and Msri, 1409 W. Green St. Urbana, IL 61801., USA, E-mail: pillay@math.uiuc.edu

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], 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 4 we study a certain group introduced in [5], which we call the Galois group of T, develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

(Received April 05 1998)

(Revised July 19 1999)