## The Journal of Symbolic Logic

### Coordinatisation and canonical bases in simple theories

a1 Department of Mathematics and Statistics, Mcmaster University, Hamilton, ON, Canada, E-mail: hartb@mcmaster.ca

a2 Department of Mathematics, Massachusets Institute of Technology, Cambridge, MA, USA, E-mail: bkim@math.mit.edu

a3 Department of Mathematics, University of Illinois, Urbana, IL, USA, E-mail: pillay@math.uiuc.edu

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a/E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2].

Throughout this paper we will work in a large, saturated model M of a complete theory T. All types, sets and sequences will have size smaller than the size of M. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

(Received February 07 1998)

(Revised July 19 1998)