On rank not only in NSOP1 theories

Abstract

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is a global type. We prove that in any NSOP1 theory these ranks satisfy some desirable properties; in particular, DQ(x=x)<\omega for any finite variable x and any Q, if q\supseteq p is a Kim-forking extension of types, then DQ(q)<DQ(p) for some Q, and if q\supseteq p is a Kim-non-forking extension, then DQ(q)=DQ(p) for every Q that involves only invariant types whose Morley powers are \ind^K-stationary. We give natural examples of families of invariant types satisfying this property in some NSOP1 theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory T_\infty of vector spaces with a generic bilinear form. We conclude that forking equals dividing in T_\infty, strengthening an earlier observation that T_\infty satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP1 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 ranks identifies a non-trivial class of theories containing all NSOP1 and NTP2 theories.