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Hyperimaginaries and automorphism groups

  • D. Lascar (a1) and A. Pillay (a2)

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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 Meq 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].

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[1]Hart, B., Kim, B., and Pillay, A., Coordinatization and canonical bases in simple theories, this Journal, vol. 65 (2000), pp. 293309.
[2]Hrushovski, E., Simplicity and the lascar group, preprint 1997.
[3]Kim, B., A note on Lascar strong types in simple theories, this Journal, vol. 63 (1998), pp. 926936.
[4]Kim, B. and Pillay, A., Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 149164.
[5]Lascar, D., On the category of models of a complete theory, this Journal, vol. 47 (1982), pp. 249266.
[6]Lascar, D., Autour de la propriété du petit indice, Proceedings of the London Mathematical Society, vol. 62 (1991), pp. 2553.
[7]Lascar, D. and Poizat, B., An introduction to forking, this Journal, vol. 44 (1979), pp. 330350.
[8]Lascar, D. and Shelah, S., Uncountable saturated structures have the small index property. The Bulletin of the London Mathematical Society, vol. 25 (1993), pp. 125131.
[9]Pillay, A. and Poizat, B., Pas d'imaginaires dans l'infini, this Journal, vol. 52 (1987), pp. 400403.
[10]Weil, A., Groupes topologiques, Hermann, Paris, 1940.

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