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Coordinatisation and canonical bases in simple theories

Published online by Cambridge University Press:  12 March 2014

Bradd Hart ,
Byunghan Kim  and
Anand Pillay
Bradd Hart
Affiliation:
Department of Mathematics and Statistics, Mcmaster University, Hamilton, ON, Canada, E-mail: [email protected]
Byunghan Kim
Affiliation:
Department of Mathematics, Massachusets Institute of Technology, Cambridge, MA, USA, E-mail: [email protected]
Anand Pillay
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL, USA, E-mail: [email protected]
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 293 - 309
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

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