Groups Interpretable in Simple Theories

Ziv Shami

Type-definable abelian groups or even algebraically closed fields arise naturally in the general stable context due to an interaction between types, more precisely due to non-orthogonality of regular types. A strong version of interaction which always appears as consequence of non-orthogonality is internality. In this case the automorphism group of one type (more precisely strong type) over the other is known to be type-definable (see [H]). The latest phenomenon is the basic idea for proving the existence of typedefinable abelian groups or algebraically closed fields that mentioned above. We discuss an extension of this phenomenon for simple theories as well as possible analogs (see [S]).

[H] Ehud Hrushovski, Unidimensional theories are superstable, Annals of Pure and Applied Logic 50, 1990, pgs 117-138.

[S] Ziv Shami, Internality and type-definablity of the automorphism group in simple theories, preprint.

Address: Department of Mathematics, University of Notre Dame, South Bend, Indiana, USA

E-mail: Zshami@darwin.helios.nd.edu