G-Compactness and Groups

TL;DR

This paper explores the structure of G-compactness in model theory, generalizing known results about equivalence relations and constructing a new example involving a definable group to investigate non-G-compact theories.

Contribution

It generalizes Lascar's decomposition of E_KP to other equivalence relations and introduces a new structure M_0 with a definable group to study G-compactness.

Findings

Lascar group of M_0 is a semi-direct product of that of M and G/G_L

Relationship between G-compactness of M and M_0 is analyzed

Potential new examples of non-G-compact theories are proposed

Abstract

Lascar described E_KP as a composition of E_L and the topological closure of EL. We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M_0 consisting of M and X as two sorts, where X is an affine copy of G and in M_0 we have the structure of M and the action of G on X. We prove that the Lascar group of M_0 is a semi-direct product of the Lascar group of M and G/G_L. We discuss the relationship between G-compactness of M and M_0. This example may yield new examples of non-G-compact theories.