Borel equivalence relations and Lascar strong types

TL;DR

This paper investigates the Borel complexity of spaces of Lascar strong types and related structures in model theory, providing classifications, computations, and exploring definable maps to understand their properties and implications for G-compactness.

Contribution

It establishes Borel cardinalities for Lascar strong types and related spaces, computes these for various examples, and explores definable maps and isomorphisms between these structures.

Findings

Spaces of Lascar strong types have well-defined Borel cardinalities

Computed Borel cardinalities for known and new examples

Explored definable maps and isomorphisms between quotient objects

Abstract

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via definable groups, of new examples of non G-compact first order theories.