On definable J-sets
Zhentao Zhang
TL;DR
The paper investigates when definable $J$-sets in definable groups align with weakly generic sets, introducing the $J$-property as an invariance feature across $\aleph_0$-saturated models and analyzing it via the extern definable type space $S^{ext}_G(M)$. It proves that superstable commutative groups have the $J$-property and provides $pCF$-theory examples where the property holds, illustrating that superstability is not necessary. It also presents a concrete example of a commutative expansion of $(\mathbb{Z},+)$ lacking the $J$-property, showing non-coincidence between $J$-sets and weakly generic sets in some theories. Overall, the work connects $J$-set phenomena to stability/classification theory and demonstrates both positive results and intrinsic limitations in broader settings.
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.