TY - JOUR

T1 - Borel equivalence relations and lascar strong types

AU - Krupiński, Krzysztof

AU - Pillay, Anand

AU - Solecki, Sławomir

N1 - Funding Information:
The research of first author was supported by the Polish Government grant N N201 545938. The second author was supported by EPSRC grant EP/I002294/1. The third author was supported by NSF grant DMS-1001623.

PY - 2013/12

Y1 - 2013/12

N2 - The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group GalL(T) of T, have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic 1 (2001) 305-319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math. 231 (2012) 605-623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint (2012), arXiv:1201.5221v1].

AB - The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group GalL(T) of T, have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic 1 (2001) 305-319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math. 231 (2012) 605-623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint (2012), arXiv:1201.5221v1].

KW - Borel reducibility

KW - Lascar strong types

UR - http://www.scopus.com/inward/record.url?scp=84887435576&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887435576&partnerID=8YFLogxK

U2 - 10.1142/S0219061313500086

DO - 10.1142/S0219061313500086

M3 - Article

AN - SCOPUS:84887435576

VL - 13

JO - Journal of Mathematical Logic

JF - Journal of Mathematical Logic

SN - 0219-0613

IS - 2

M1 - 1350008-1

ER -