On definable J-sets

TL;DR

This paper investigates the properties of definable J-sets within definable groups, examining their relationship with weakly generic sets and establishing model-theoretic invariance, with positive results for certain stable groups and examples illustrating noncoincidence.

Contribution

It introduces the concept of definable J-sets, compares them with weakly generic sets, and demonstrates their invariance across saturated models, providing new insights in model theory.

Findings

J-sets coincide with weakly generic sets in certain stable groups

The property is invariant across sufficiently saturated models

Examples show cases where J-sets and weakly generic sets do not coincide

Abstract

We study definable J-sets for definable groups and compare them with weakly generic sets. We show that the property that J-sets coincide with weakly generic sets is invariant on enough saturated models, and hence a model-theoretical property. We have positive results for superstable commutative groups and some easy examples in pCF. We also give an example for noncoincidence.