Normal hyperimaginaries

TL;DR

This paper introduces the concept of normal hyperimaginaries, develops their theory, and applies them to provide a new proof of a key theorem on bounded hyperimaginaries, linking model theory and topological group structures.

Contribution

It defines normal hyperimaginaries, explores their properties, and uses them to give a new proof of Lascar-Pillay's theorem, connecting hyperimaginary theory with compact group structures.

Findings

Normal hyperimaginaries are characterized and their basic properties are established.

A new proof of Lascar-Pillay's theorem on bounded hyperimaginaries is provided.

All closed sets in Kim-Pillay spaces are shown to be equivalent to hyperimaginaries.

Abstract

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of Lascar-Pillay's theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of Peter-Weyl's theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of $\varphi$-types for bounded hyperimaginaries.