\({\mathcal M}\)-gap conjecture and \(m\)-normal theories
Basic definitions and properties, connections with investigations in other papers, and the construction of small weakly minimal groups with an isolated generic type of arbitrarily large finite \({\mathcal M}\)-rank or of \({\mathcal M}\)-rank \(\infty\) are introduced. Of importance is the formulation of the \({\mathcal M}\)-gap conjecture: In a small superstable theory there are no types \(p\) such that \(\omega\leq{\mathcal M}(p)< \infty\). An \(m\)-normal theory satisfies the \({\mathcal M}\)-gap conjecture. So-called \(*\)-finite tuples play a similar role for the \(m\)-independence, as imaginaries do for the forking independence. Such special objects allow the author to define the notion of an \(m\)-normal theory more smoothly. Elements which are \(*\)-algebraic over a finite set of \(*\)-finite tuples are important for the description of \(m\)-independence. The proof of the fact that a 1-based superstable theory with few countable models is \(m\)-normal involves an analysis of traces of types under the few models assumption. Traces of complete types are traces of some formulas. The question arises whether any superstable theory with few countable models is \(m\)-normal. The examples considered in the introduction are groups. The additional properties of \({\mathcal M}\)-rank on generic types of a group are considered. Finally, a notion of independence induced by measure is considered. It is compared with the notion of \(m\)-independence.
scientific article; zbMATH DE number 1216498
scientific article