Meager forking
Let \(T\) be a stable theory. Recall that if \(A\) is a finite set and \(\varphi(x)\) is a formula, then the set of all strong types over \(A\) consistent with \(\varphi(x)\) is always closed, and open when \(\varphi(x)\) does not fork over \(A\). If \(P\) is a closed set of strong types over \(A\), then forking is said to be meager on \(P\) if and only if for every formula \(\varphi(x)\) forking over \(A\), the set of \(r\in P\) consistent with \(\varphi(x)\) is nowhere dense in \(P\). Correspondingly a notion of meager regular type is proposed. First it is shown that a meager regular type is locally modular. Recall that if \(T\) is superstable and \(p\) is a non- trivial locally modular regular type, then over some finite set \(A\) there is a definable Abelian group \(G\) with locally modular generics non- orthogonal to \(p\). If \(p\) is meager, then also the generics of \(G\) are meager. Now assume that \(T\) has \(< 2^{\aleph_ 0}\) countable models and \(G\) is a definable Abelian group with locally modular regular generics. Under these conditions, a counterpart of Saffe's conjecture is proved. More precisely, it is shown that, if \(A\) is finite, \(a\in G\) is generic over \(A\) and \(q= \text{tp}(a/A)\), then one of the following conditions holds. (1) There is \(b\in G\) such that \(b\) is generic over \(A\), \(b\in a+ S\) and \(\text{tp}(b/A)\) is isolated. (2) There are finitely many realizations \(a_ 0, \dots, a_{n-1}\) of \(q\) such that, for every \(b\models q\), there is \(i< n\) such that \(\text{stp}(b)\) is realized in \(a_ i+ S\). Using these results, the conjecture of vanishing multiplicities is proved for a superstable \(T\), saying that if \(T\) has \(< 2^{\aleph_ 0}\) countable models, then every complete type is multiplically stable. As a final application, a question concerning topological stability of pseudo-types is treated.
definable Abelian group with locally modular regular generics
scientific article; zbMATH DE number 706878
scientific article