On Rank Not Only in Nsop Theories

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (φ (x,y),q(y)), where φ (x,y) is a formula and $q(y)$ is a global type. We prove that in any NSOP ₁ theory these ranks satisfy some desirable properties; in particular, DQ(x=x)<ω for any finite tuple of variables x and any Q , if q⊇ p is a Kim-forking extension of types, then DQ(q)<DQ(p) for some Q , and if q⊇ p is a Kim-non-forking extension, then DQ(q)=DQ(p) for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP ₁ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory T_∞ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in T_∞ , strengthening an earlier observation that T_∞ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP ₁ to find out that our local ranks are bounded by the well-known ranks: the inp-rank ( burden ), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP ₁ and NTP ₂ theories.