The Lascar Group and the Strong Types of Hyperimaginaries

This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries, and make some new observations on the strong types over those. In particular we show that in a simple theory, Ltp ≡ stp in real context implies that for hyperimaginary context. The Lascar group introduced by Lascar [8], and related subjects have been studied by many authors ([9][1][7][6][3] and more). Notably in [9][1], a new look on the Lascar group is given, and using compact Lie group theory Lascar and Pillay proved that any bounded hyperimaginary is interdefinable with a sequence of finitary bounded hyperimaginaries. Good summaries on the Lascar group are written in [10][11]. While this is another short expository note stating known results in [8][9][1][7], we supply a couple of new observations. We study the Lascar group in slightly more general context namely over hyperimaginaries. The notion of strong types over hyperimaginaries is somewhat subtler even at the level of the definition (see Example 3.5). As a by-product we show that in a simple theory if Ltp(a/A) ≡ stp(a/A) for real tuples a and A, then the same holds for hyperimaginaries. A question remains whether this holds in any theory. We work with an arbitrary complete theory T in L, and a fixed large saturated model M |= T of size κ̄, as usual. We recall some of definitions. Unless said otherwise a tuple can have an infinite size (< κ̄). By a hyperimaginary we mean an equivalence class of a typedefinable equivalence relation over ∅. So a hyperimaginary has the form a/E = aE where a is a tuple fromM and E(x, y) is the ∅-type-definable equivalence relation on M|x|. We call aE an E-hyperimaginary. We say the hyperimaginary is finitary if a is a finite tuple. In general we put |aE| := |a|. In the note arity means an arity of a real tuple. From now on a, b, c, ..., A,B, ... denote hyperimaginaries, but M,N, .. denote elementary small submodels of M. Clearly any tuple from This is a note submitted for the proceedings of the conference: Recent developments in model theory, Oléron, France, June 2011, where the author gave a talk on different subjects in [4],[5]. This work is supported by an NRF grant 2010-0016044.