Weak dividing, chain conditions, and simplicity
In the late 70s, S.~Shelah introduced simple theories, generalizing stable theories, and characterized them as those where forking has local character. In his PhD Thesis of 1996 B.~Kim showed that forking in simple theories enjoys some nice properties of stable theories, and, in particular, that dividing is the same as forking, and forking is symmetric. In his work the focus had been on dividing in simple theories. Shelah introduced and used in his work on simple theories the notion of weak dividing. A complete type \(p(x)\) over \(A\) weakly divides over \(B\subseteq A\) if there exists a formula \(\theta(x_1,\dots,x_n)\) with parameters from \(B\) such that there are realizations \(c_1,\dots,c_n\) of \(p\upharpoonright B\) with \(\models\theta(c_1,\dots,c_n)\) but there are no realizations \(c_1,\dots,c_n\) of \(p\) with \(\models\theta(c_1,\dots,c_n)\). The author studies weak dividing in the context of simplicity and shows that simple theories can be characterized using weak dividing in various ways. Most of the work can be summarized as follows (where some of the results are not new but are mentioned because the author provides new proofs using weak dividing). The following are equivalent for a theory \(T\): (1)~\(T\) is simple; (2)~weak dividing has a chain condition; (3)~\(T\) admits smearing of non-weak-dividing extensions; (4)~dividing implies weak dividing; (5)~\(T\) has a notion of independence which is both local and has a chain condition; (6)~dividing is symmetric; (7)~heirs over sets do not divide. Also, for stable theories weak dividing is characterized in terms of dividing.
scientific article; zbMATH DE number 2102099