On rank not only in NSOP1 theories

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.


Introduction
In the past years, we observed a rapid development of geometric tools and techniques related to model-theoretic stability theory.After a successful use of these techniques in the context of stable theories and remarkable applications in algebraic geometry, studies went beyond the class of stable theories.One of the main tools of a geometric nature in model theory is the notion of an independence relation (cf.[2]), which plays a key role in the description of simple theories (cf.[24]).Another important geometric tool in model theory is the notion of a rank, which also can be used to characterize dividing lines in the stability hierarchy.For example a theory is simple if and only if the local rank D(x = x, ϕ, k) is finite for every choice of a formula ϕ and every natural number k (cf.Proposition 3.13 in [4]).Actually, in the case of simple theories there is an elegant connection between the local rank D and forking independence | ⌣ , in short: the rank decreases in an extension of types if and only if this extension is a forking extension (Proposition 5.22 in [4]).On top of that, the local rank in simple theories was used to develop the theory of generics there ( [25]).
Independence relations and ranks behave less nicely in the case of non-simple NSOP 1 theories.The NSOP 1 theories were defined in [17], then studied more intensively in [11] and in [20], where also the ideas from Kim's talk ( [23]) came to the picture (roughly speaking: Kim proposed a notion of independence corresponding to non-dividing along Morley sequences).Further studies on NSOP 1 focused on proving desired properties of the notion of independence related to the notion of Kim-forking as defined in [20] (where Kim-dividing and Kim-forking were defined with the use of global invariant types), e.g.[21] and [22].The problem with this approach is that sometimes there are no invariant global types extending a given type (however, everything is fine if we work only over models).Then, in [15], the authors redefined the notions of Kim-dividing, Kim-forking and Kim-independence to avoid this obstacle and worked with definitions more in the spirit of [23].However, they needed an extra assumption, i.e. they were working in NSOP 1 theories enjoying the existence axiom for forking independence.The NSOP 1 theories enjoying the existence axiom were also studied in [10], where e.g.transitivity of Kim-independence (as defined in [15]) was obtained over arbitrary sets.The important question, whether every NSOP 1 theory automatically enjoys the existence axiom for forking independence remains open.
Similarly as forking independence in the case of simple theories, Kim-independence was used to describe the class of NSOP 1 theories ( [20]).Therefore one could expect that there should also exist a good notion of a rank, which, similarly to the situation in simple theories, is related to Kim-independence in the context of NSOP 1 theories and which also describes the class of NSOP 1 theories (i.e. the rank is finite if and only if the theory is NSOP 1 ).Some attempts to define such a rank for NSOP 1 theories were made in [10], however they were not fully successful in relating the rank to Kim-independence (see Question 4.9 in [10]).On the other hand, the rank defined in [10] is finite provided T is NSOP 1 with existence and, in a private communication, Byunghan Kim informed us that SOP 1 implies that this rank is not finite (for some formula ϕ, natural number k and some type q).Thus finiteness of the rank from [10] characterizes the class of NSOP 1 theories.
Let us mention here that also the situation with generics in NSOP 1 groups is more difficult than in groups with simple theory.For example, the theory of vector spaces with a generic bilinear form with values in an algebraically closed field, does not have Kim-forking generics for the additive group of vector space (see [14]).The theory of generics in NSOP 1 groups is currently under development and a suitable notion of rank could be very useful in that context.
To summarize, for us, there were three main properties expected from the new notion of rank: being finite if and only if the theory is NSOP 1 , being related to Kim-independence, and having a prospective use in the development of generics in NSOP 1 groups.Here is what we managed to obtain so far.Our notion of rank (Definition 3.1) is local and depends on pairs consisting of a formula and a global type.It has all the usual properties of a rank and it is finite, provided the theory T is NSOP 1 .Nevertheless, it is also finite outside the class of NSOP 1 theories (e.g. in DLO, see Example 3.6), which was not expected, but makes the rank more interesting outside of the class of NSOP 1 theories (more on that in Section 6).To obtain a rank which is related to the notion of Kim-independence, we follow some ideas from the doctoral thesis of Hans Adler (see Section 2.4 in [2]).More precisely, our rank is not a foundation rank (i.e.defined recursively), but a rank which is witnessed by our account on dividing patterns ( [2]) related to Kim-dividing given as in [20].In Section 3.2, we explain the connection between our rank and Kim-independence, which depends on some notions of stationarity.
After noticing that our notion of rank might be finite outside of NSOP 1 , we investigated behaviour of the rank in the case of NTP 2 theories.It turned out that a slight modification of the main definition (compare Definition 3.1 and Definition 6.10) results in finiteness of the rank in any theory of finite dp-rank.Moreover, the modified rank is bounded by the inp-rank and hence by the dp-rank.On top of that, the aforementioned modification does not affect the notion of our rank in the case of NSOP 1 theories.Therefore, in this paper, we provide a notion of local rank which shares finiteness in two opposite corners of the stability hierarchy, which seems to be quite intriguing.Perhaps, a good notion of rank will be more suitable to work across different dividing lines in the neo-stability hierarchy than a notion of independence (like, for example, thorn-independence).On top of that, to provide the definition of ranks in Section 6, we introduce semi-global types and show Kim's lemma for Kim-dividing witnessed via sequences in semi-global types.
In Section 4 we study forking in the theory T ∞ of infinite-dimensional vector spaces over an algebraically closed field with a generic bilinear form, which is one of the main algebraic examples of a non-simple NSOP 1 theory.We describe forking of formulae in T ∞ , answering in particular a question from [18] about equivalence of dividing and dividing finitely.
The paper is organized as follows.In Section 2, we recall definitions and several facts needed later.Section 3 contains the definition and some basic properties of the new rank; then the new rank is related to the notion of Kim-independence.Section 4 is focused on forking in the theory T ∞ and verifies auxiliary notions introduced in the previous section.In Section 5, we collect more examples of NSOP 1 theories and we discuss the three relevant notions of independence.Finally, in Section 6, we go with our rank beyond the class of NSOP 1 theories.
We thank Zoé Chatzidakis, Itay Kaplan, Byughan Kim and Tomasz Rzepecki for helpful discussions and comments.

Basics about NSOP 1
As usual, we work with a complete L-theory T and with a monster model C of T , i.e. a κ-strongly homogeneous and κ-saturated model of T for some big cardinal κ.By a small tuple/subset/substructure we mean some tuple/subset/substructure of size strictly smaller than κ.Unless stated otherwise, all considered tuples/subsets/substructures will be small.For example A ⊆ C tacitly implies that |A| < κ.In short, we follow conventions being standard in model theory, e.g.outlined in [4] and [27].
At the beginning, we need to evoke several definitions and facts about Kimdividing and NSOP 1 theories.A reader unfamiliar with the subject may consult e.g.[20], [15] and [10].
Definition 2.2.We say that T enjoys the existence axiom for forking independence if for each set A and each tuple b we have that b | ⌣A A. If T is NSOP 1 and enjoys the existence axiom for forking independence, then we say that T is NSOP 1 with existence.
Definition 2.3 (Morley sequence in a type).Let A ⊆ C, p(y) ∈ S(A), and let (I, <) be a linearly ordered set.We call a sequence Definition 2.4 (Morley sequence in an invariant global type).Assume that q(y) ∈ S(C) (i.e.q is a global type) is A-invariant and let (I, <) be a linearly ordered set.By a Morley sequence in q over A (of order type I) we understand a sequence b = (b i ) i∈I such that b i |= q| Ab<i for each i ∈ I.By q ⊗I we indicate a global Therefore, if q(y) ∈ S(C) is A-invariant, we have two possible notions of a Morley sequence: a Morley sequence in q| A and a Morley sequence in q over A. Of course, a Morley sequence in q over A is a Morley sequence in q| A .The converse does not hold in general.Definition 2.5.Let q(y) be an A-invariant global type.We say that a formula ϕ(x; y) q-divides over A if for some (equivalently: any) Morley sequence (b i ) i<ω in q over A, the set {ϕ(x, b i ) | i < ω} is inconsistent.
We have the following two notions of Kim-dividing, (A) appears in [15] and (B) appear in [20].For us notion (A) is the one which we will use here.However, Theorem 7.7 from [20] implies that (A) and (B) coincide over a model (i.e. if A = M C) provided T is NSOP 1 , and therefore we will switch very often to notion (B) if the situation is placed over a model.
We say that ϕ(x; b) Kim-divides over A if there exists an A-invariant global type q(y) ⊇ tp(b/A) such that ϕ(x; y) q-divides over A. Equivalently, we say that ϕ(x; b) Kim-divides over A if there exists an A-invariant global type q(y) ⊇ tp(b/A) and a Morley sequence Fact 2.7 (Kim's lemma over models).Assume that T has NSOP 1 and let M C.
The following are equivalent: (1) ϕ(x; b) Kim-divides over M , (2) for any M -invariant global type q(y) ⊇ tp(b/M ) and any Morley sequence Later on, we will define a local rank related to the notion of Kim-dividing.Our rank will focus on Kim-dividing over models, so one could ask how much of the picture is lost if we restrict our attention only to Kim-dividing over models.First, let us make an observation easily following from the definition: if ϕ(x, b) Kim-divides over A with respect to definition (B) and A ⊆ B then there exists c ≡ A b such that c | ⌣A B and such that ϕ(x, c) Kim-divides over B with respect to definition (B).This means that passing to Kim-dividing over models is not so harmful if we decide to work with the definition (B).The following lemma shows the same for the notion of Kim-dividing from the definition (A).The proof is not a surprise in any meaning, but let us follow the argument for a little warm-up.Proof.By the definition of k-Kim-dividing, there exists an A-indiscernible sequence (a i ) i<ω such that a 0 = a, for each i < ω we have a i | ⌣A Aa <i and {ϕ(x, a i ) | i < ω} is k-inconsistent.We will use several properties of | ⌣ , which hold in any theory T , these are listed e.g. in Remark 5.3 in [4].
Step 1: increase the length of (a i ) i<ω to a big cardinal λ (big enough for the use of Erdős-Rado theorem for B-indiscernibility, see e.g.Propostion 1.6 in [4]).To do it, consider: Invariance and finite character of | ⌣ imply that also b α | ⌣A Ab <α for each α < λ.
Step 2: force | ⌣ -independence over B. We define recursively partial elementary over A maps f α : dcl(Ab α ) → C, where α < λ, such that ⌣A Bf α (b <α ), for each α < λ.We start with obtaining f 0 .Since b 0 | ⌣A A, tp(b 0 /A) does not fork over A and so there exists a non-forking extension: There exists an extension . We move on to the limit ordinal step: α<β , so, again, we can find an extension . After the recursion is done, we take f ∈ Aut(C) which extends α<λ f α and set (b ′ α ) α<λ = (f (b α )) α<λ .
For some α 0 < λ we have that ⌣ satisfies monotonicity and invariance, we obtain that also c 0 | ⌣A B.
(1) A partial type π(x) Kim-forks over A if for some n < ω and ψ j (x; b j ) Kim-divides over A for each j n.
(2) Let a be a tuple from C and let A, B ⊆ C. We say that a is Kim-independent from B over A, denoted by a | ⌣ K A B, if tp(a/AB) does not Kim-fork over A.
One could redefine the notions of Kim-forking and Kim-independence using Kimdividing with respect to definition (B).In such a case, we will always indicate that we work with Kim-forking with respect to definition (B) or use | ⌣ K,q to denote Kim-independence defined with Kim-dividing with respect to definition (B).
The previous lemma easily leads to the following.
The following characterization of NSOP 1 proved much more useful in studying Kim-independence than the original definition of NSOP 1 introduced in [17].
(2) Kim's lemma for Kim-dividing: For any M C and any ϕ(x; b), if ϕ(x; y) q-divides over M for some M -invariant q(y) ∈ S(C) with tp(b/M ) ⊆ q(y), then ϕ(x; y) q-divides over M for any M -invariant q(y) ∈ S(C) with tp(b/M ) ⊆ q(y).
Fact 2.7 has a generalization to Kim-dividing over arbitrary sets: Theorem 2.12 (Theorem 3.5 in [15]).Let T be NSOP 1 with existence.Then T satisfies Kim's lemma for Kim-dividing over arbitrary sets: if a formula ϕ(x, b) Kim-divides over A with respect to some Morley sequence in tp(b/A) then the formula ϕ(x, b) Kim-divides over A with respect to any Morley sequence in tp(b/A).
As we will notice in a moment, Kim's lemma for Kim-dividing (over models) will be the main reason behind the fact that our local rank is finite in the context of NSOP 1 theories.

Rank
3.1.Definition and basic properties.In this section, we are interested in defining a local rank depending on pairs consisting of an L-formula and a global type.We will prove several properties of this new rank.Our idea for the rank was in some way motivated by Hans Adler's doctoral dissertation ( [2]).More precisely, in Section 2 of his dissertation, Adler defines so called dividing patterns and then defines a local rank measuring the length of a maximal dividing pattern.In our case, we could not simply reuse his idea, since we are trying to "domesticate" Kim-dividing, and instead of that we propose our own variation on Kim-dividing patterns.
Let us explain a little bit the concept behind this rank.For simplicity we assume that Q = ((ϕ, q)).Then the witnesses from the definition of D Q (π) λ, (M α , b α i ) α<λ,i<ω , may be used to draw the following tree: Each horizontal sequence of b α i 's is a Morley sequence in some global M α -invariant type and so witnesses Kim-dividing of ϕ(x; b α 0 ) over M α .This is nothing new.The first new ingredient in our rank is that we require that also the leftmost branch in our tree forms a Morley sequence, this time in the previously fixed global type q (which is M 0 -invariant).In other words, we focus only on Morley sequences -and that is in accordance with the intuition that all the essential data in a NSOP 1 theory is coded by Morley sequences.
The second new ingredient in our rank is that we allow "jumps" in the extension of the base parameters between levels.More precisely, instead of the sequence dom(π) ⊆ M 0 M 1 . .., we could consider a more standard sequence dom(π) ⊆ dom(π)b 0 0 ⊆ dom(π)b 0 0 b 1 0 ⊆ . ... However, let us recall that | ⌣ K does not necessarily satisfy the base monotonicity axiom, thus we allow in our rank some freedom in choosing the parameters over which each next level Kim-divides.
Remark 3.3.Because in NSOP 1 theories the both notions of Kim-dividing ( (A) and (B) from Definition 2.6) coincide over a model and in our rank we consider only Kim-dividing over models, our rank is suitable to work with both notions of Kim-dividing in the NSOP 1 environment.
In the following lines, we will make use of our intuition and show several nice properties of the Q-rank, we also refine the definition of Q-rank, so it will become more technical, but it will allow us to proof a few more facts.We start with something completely trivial.( Proof.All the three items follow by the definition.

Corollary 3.5. There exists finite
As we will see in a moment, the Q-rank is finite in the case of NSOP 1 theories, which is a desired property of our rank.It also happens that outside of NSOP 1 the rank may be finite, e.g. in the case of T being DLO.In the following example we work with Definition 3.1, however a more general result on finiteness of our rank in the context of NIP theories is provided in Section 6, where we work with a slightly modified definition of Q-rank (see Definition 6.10).
Example 3.6.Let T be the theory of dense linear orders without endpoints, DLO.First, we will show that D Q (x = x) ≤ 1 for Q = (φ(x, yz), q), where q is an arbitrary invariant type and φ(x, yz) = (y 0 ), hence by the consistency condition we must have a 0 0 < a 1 0 < c 1 0 < c 0 0 (*).But then an automorphism over M 0 moving b 0 0 to b 0 1 will move φ(x; a 1 0 , c 1 0 ) to a formula inconsistent with it.This contradicts M 0 -invariance of the restriction q ′ of q to the first variable as a 1 0 |= q ′ |M 1 .Note also that if φ(x, y) is of the form x > y or x < y or x = x, then D (φ,q) (x = x) = 0, as in that case no instance of φ(x, y) Kim-divides over any set.Also, it is easy to see that if φ(x, y) = (x = y) then D (φ,q) (x = x) = 1.Now let Q = (φ(x, y), q(y)) with x = x 0 . . .x n−1 be a variable of length n, φ(x, y) a quantifier-free formula and q(y) a global invariant type.By quantifier elimination and completeness of q(y) we have that q(y) ∧ φ(x, y) ≡ q(y) ∧ i<k φ i (x, y) where each φ i (x, y) defines a product of intervals with ends in the set y ∪ {+∞, −∞}.We claim that D Q (x = x) ≤ nk.Suppose for a contradiction that D Q (x = x) > nk.
Then by pigeonhole principle one easily gets that and let ψ(x) be a formula equivalent to qf tp(c/∅).Wlog φ(x, y) ⊢ ψ(x).Hence φ(x, y) is of the form ψ(x) ∧ j<n χ j (x j , y), where each χ j (x j , y) defines an interval with ends in y ∪ {+∞, −∞}.Again by pigeonhole principle we must have that D Q ′′ (x = x) ≥ 2 with Q ′′ = (ψ(x) ∧ χ j (x j , y))) for some j < n.As ψ(x) is over ∅ and is consistent, clearly we get that any family of instances of ψ(x) ∧ χ j (x j , y)) is consistent if and only if the corresponding family of instances of

To see this we introduce auxiliary N
C which contains M and N , and a Morley sequence This remark says that C ϕ,q is in some sense a uniform bound and it can not happen that for each M C and each corresponding Morley sequence (a From now on (if not stated otherwise), we assume that T has NSOP 1 .Lemma 3.9.Then D Q (π) λ if and only if there exists η ∈ n λ and (b Proof.The implication right-to-left is straightforward and holds even without the assumption about NSOP 1 .Let us show the implication left-to-right.By Kim's lemma (Fact 2.7), we can replace the condition (7) from Definition 3.1, by: for each α < λ there exists cα = (c α i Proof.Let us deal first with the case when Q = ((ϕ(x; y), q(y))).Assume that D Q (π) λ > 0 and let (b α , M α ) α<λ be as in Definition 3.1 (η is constant, so we skip it).
We switch to the general case where C ϕj ,kj < ω for each j < n.Fix some j < n and assume that |η −1 [j]| > 0. Let α < λ be the first index such that η(α) = j.Repeating the first part of this proof for q j we obtain what we need.
q 0 , . . ., q n−1 are M -invariant, Proof.The left-to-right implication is straightforward: by Lemma 3.9 there are proper η ∈ n λ and (b α i ) i<ω , M α α<λ , we set M := M 0 and reuse (b α i ) α<λ,i<ω .Let us take care of the right-to-left implication.We will recursively define a sequence ((e α i ) i<ω , M α ) α<λ satisfying all the six conditions from Lemma 3.9.Because we do not want to get lost in a notational madness, instead of introducing new subscripts, we will sketch a few first steps.
We set M 0 := M and take M 1 C which is a |M 0 | + -saturated, and which contains M 0 and (b 0 i ) i<ω .Consider a Morley sequence and let us choose one more Morley sequence and (e 0 i ) i<ω := (b 0 i ) i<ω .Continuing this process we will obtain a sequence of models M 0 M 1 . . .M λ−1 and Morley sequences ( Therefore all the conditions of Lemma 3.9 are satisfied for ((e α i ) i<ω , M α ) α<λ .Example 3.13.Let us provide an example of a situation when the rank of the home sort is finite, but strictly bigger than 1.Let k be any natural number greater than 1 and let p be equal to zero or to a prime number distinct from 2. Consider the 2-sorted theory T m of m-dimensional vector spaces equipped with a nondegenerate symmetric bilinear form (see [14,Chapter 10]).Let x and y be single vector variables, φ(x, y) = (x ⊥ y ∧ x = 0), and let q(y) be the generic type in the vector sort V (so q is ∅-invariant).Put Q = {(φ(x, y), q(y))}.We will show that Let (v i ) i<ω be a Morley sequence in q(y).Then in particular v 0 . . .v k−1 are linearly independent, which easily implies that i<k φ(x, v i ) is inconsistent.Thus For the other inequality, put M i = acl(a ik , a ik+1 , . . ., a ik+k−1 ) for i < k and note that b , and each M i is an elementary submodel by quantifier elimination.Also, the sequence (v (i+1)k , v (i+1)k+1 , . . . ) is Morley over M i , and it witnesses that φ(x, b i ) Kim-divides over M i for each i < k.Finally, i<k−1 φ(x, b i ) is consistent, as there is a non-zero vector orthogonal to b 0 , . . ., b i−2 .This shows that Remark 3.14.Note that if T is an NSOP 1 theory (as we assume here) and for some k < ω, a formula ϕ(x, y), M C, some M -invariant q(y) ∈ S(C) and some Example 3.13).To see this, consider a linear order I being (k − 1)-many copies of ω (one after another one) and (c i ) i∈I |= q ⊗I | M and use Lemma 3.12.In other words, for π(x) := {x = x} the situation is quite simple and either Let N C contain M and dom(π ′ ), and let Hence it is enough to show that Thus there is i 0 m such that π ∪ {ψ i0 } ∪ {ϕ η(α) (x; b α ) | α < λ} is consistent.Because dom(π∪{ψ i0 }) ⊆ dom(π∪{ j ψ j }) ⊆ M 0 , we have that D Q (π∪{ψ i0 }) λ and so also max j D Q (π ∪ {ψ j }) λ.
Let us recall that we are working in a theory T which is NSOP 1 .
then by Lemma 3.17 there exists a finite subset Thus ¬ψ(x) ∈ π(x) ⊆ p(x) and we got a contradiction with ψ(x) ∈ p(x).
On the other hand, if T is NSOP 1 but not simple, then there must be a formula ϕ and some k 0 < ω such that D({x = x}, {ϕ}, k 0 ) ω. Thus for every K < ω there exists k K (e.g.k = K + k 0 ) such that D({x = x}, {ϕ}, k) ω but, D ((ϕ,q)) ({x = x}) < ω for any choice of q ∈ S(C).Therefore sharp inequality in Remark 3.19 happens outside of the class of simple theories.One could ask about equality under the assumption on simplicity.The following counterexample, which is even stable, leaves no doubt.
Example 3.20.Let T be the theory of an equivalence relation E with infinitely many classes all of which are infinite.It is well-known that T is ω-stable of Morley rank 2. Let ϕ 0 (x, y) = E(x, y) and ϕ 1 (x, y) = (x = y).Then it is easy to see that for any k > 1 we have that D({x = x}, {ϕ 0 (x, y), ϕ 1 (x, y)}, k) = 2. Now, fix two arbitrary invariant global types q 0 (x), q 1 (x) and put Q = {(ϕ 0 , q 0 ), (ϕ 1 , q 1 )}.We claim that D 2 witnessed by M 0 , M 1 , b 0 , b 1 and η : 2 → 2. The case where η(0) = 1 can be excluded immediately, so assume η(0) = 0. Observe that |= E(b 0 , b 1 ), as otherwise ϕ η(0) (x, b 0 ) ∧ ϕ η(1) (x, b 1 ) would be inconsistent (note ϕ i (x, y) ⊢ E(x, y) for i = 0, 1).On the other hand, as b 1 |= q η(1) | M 0 b 0 , we have in particular that b 1 | ⌣M 0 b 0 , so E(b 0 , M 0 ) = ∅, which contradicts that ϕ 0 (x, b 0 ) divides over M 0 .3.2.Rank vs Kim-independence.We know that Kim-generics do not exist in the theory of infinite dimensional vector spaces with a bilinear form (see Proposition 8.15 in [14]), which is NSOP 1 .Since in the case of simple theories, a notion of finite local rank, which is compatible with forking and somehow invariant under shifts by elements of a definable group (e.g.Fact 3.7 in [26]), would lead to the existence of forking generics (see Lemma 3.8 in [26]), we probably should not expect that a notion of finite local rank in the case of NSOP 1 theories will be compatible with Kim-forking and invariant under shifts by group elements (otherwise one could try to prove existence of Kim-generics, which does always hold).Anyway, it seems that our notion of rank does not behave well under shifts by elements of some definable group, so does not immediately exclude compatibility of the rank with Kim-forking.
Here, we study this problem and relate our results to an important question from [10].
Remark 3.25.Let p(x) be a complete type over M ⊆ C with x possibly infinite.
(1) Let T be simple, then p(x) is stationary if and only if p| x0 is stationary for every finite subtuple x 0 of x. (2) (T being NSOP 1 ) Let M C. We have that p(x) Kim-stationary if and only if p| x0 is Kim-stationary for every finite subtuple x 0 of x.
Proof.We start with the proof of (1).The implication from left to right follows from Fact 3.24.Assume p| x0 is stationary for every finite subtuple x 0 of x and let q 0 , q 1 be global non-forking extensions of p. Then for every finite subtuple x 0 of x we have that q 0 | x0 and q 1 | x0 are non-forking extensions of p| x0 , so by stationarity of p| x0 they are equal, hence q 0 = q 1 and p is stationary.
The argument for Kim-stationarity is exactly the same.
Lemma 3.26.Let A ⊆ C, let (I, <) be an infinite linear ordering without a maximal element, and let q(y) ∈ S(C) be an A-invariant type.
(1) Let T be simple.If q| A is stationary then also q ⊗ω | A is stationary.
(2) (any T ) If q ⊗ω | A is stationary then q ⊗I | A is stationary.
(3) (any T ) If q ⊗ω | A is Kim-stationary then q ⊗I | A is Kim-stationary.
Proof.We start with the proof of (1).By Remark 3.25, it is enough to show that for each n < ω, the type q ⊗n | A is stationary.This can be shown inductively by the use of the following claim: The proof of ( 3) is similar to the proof of (2), as the only property of | ⌣ used was monotonicity, which also holds for | ⌣ K .
(1) Assume that N is |M | + -saturated and q| M is strongly stationary, then (2) Assume that T is simple and q| M is stationary, then (3) If q| M is strongly Kim-stationary, then It is worth to compare ( 2) and (3) in the above corollary with Theorem 4.7(2) from [10].We are aware that we introduced a different notion of rank, but because of that, we were able to drop the assumption on simplicity from [10][Theorem 4.7.(2)] at the cost of assuming strong stationarity of the type q and so to provide partial answer to the counterpart to Question 4.9 from [10] for our notion of rank.
In Example 4.1, we will observe that the stationarity assumption in Lemma 3.22 cannot be removed.Moreover, in Proposition 4.2, we will see that in the case of vector spaces with a bilinear form, being strongly Kim-stationary is equivalent to being Kim-stationary.

Forking in T ∞
In this section we describe forking in the theory T ∞ of vector spaces with a generic bilinear form, answering in particular a question about equivalence of dividing and dividing finitely stated in the first paragraph of Section 12.5 in [18].Before that let us recall the basic definitions and provide some facts related to the previous sections.
Let T ∞ be the theory of two-sorted vector spaces over an algebraically closed field with a sort V for vectors and a sort K for scalars, equipped with a non-degenerate symmetric (or alternating) bilinear form, as studied in [18].Then T is NSOP 1 by [11,Corollary 6.1] and it has existence by [14,Proposition 8.1].Fix a monster model [14,Corollary 8.13], for any sets A, B, C we have that where dcl( * ) K means (dcl( * )) K (for an algebraic description of dcl( * ) see [18, Proposition 9.5.1]).For a discussion about forking and dividing in T ∞ see [18,Subsection 12.3] and the rest of this section below.
and let q 0 be the unique global | ⌣ Γ -generic type in V .Then q 0 | M (x) ∪ {x ⊥ v} does not Kim-fork over M so it has a realisation w with w | ⌣ K M N .Let ϕ(x, y) express that x ∈ y \ {0} with x and y being single vector variables, and Q = {(ϕ(x, y), q 0 (y))}.We claim that D Q (w/N ) = 0 and D Q (w/M ) ≥ 1, which shows that the stationarity assumption in Lemma 3.22 cannot be removed.Indeed, note that if N ≺ N 0 and b |= q 0 | N0 then b is not orthogonal to v, so tp(w/N ) ∪ ϕ(x, b) is inconsistent, hence D Q (w/N ) = 0. On the other hand, tp(w/M )∪ϕ(x, b) is consistent and ϕ(x, b) q 0 -forks over N 0 (as the sets b i \ {0} are pairwise disjoint for a Morley sequence (b i ) i<ω in q 0 over N 0 , as such a sequence needs to be linearly independent since q 0 is generic).Thus D Q (w/M ) ≥ 1 (and it is actually easy to see that D Q (w/M ) = 1).Proposition 4.2.Let p(x) = tp(a/M ) be a complete type over M |= T ∞ .Then p(x) is Kim-stationary iff a ⊆ K(C) ∪ V (M ) .Hence p(x) is Kim-stationary if and only if it is strongly Kim-stationary.
Proof.By Remark 3.25, we may assume x is a finite tuple of variables.Assume the right-hand side first.Then by compactness p(x) and definable over M in the pure field K, we can view p(x) as a complete type over V 0 (M ) ∪ K(M ) in the finite Morley rank structure V 0 ∪ K(C).Hence Kimstationarity of p(x) follows as in V 0 ∪K(C) Kim-independence coincides with forking independence and V 0 (M ) ∪ K(M ) ≺ V 0 ∪ K(C).Now assume that p(x) is stationary and let p ′ (x 0 ) be its restriction to a single vector variable x 0 .By Fact 3.24, p ′ (x 0 ) is Kim-stationary.Let a 0 |= p ′ (x 0 ) be the coordinate of a corresponding to x 0 .Suppose for a contradiction that a 0 / ∈ M .Let v ∈ V(C) be orthogonal to M with [v, v] = 1 and let N ≻ M be such that v ∈ N .Let r(x 0 ) be the unique Γ-independent extension of p ′ (x 0 ) over N and let [20,Definition 9.35]), so in particular [b, v] = 0. Hence, as b and v are linearly independent over , so tp(b/N ) and tp(b ′ /N ) are two distinct Kim-independent extensions of p ′ (x 0 ), which contradicts Kim-stationarity of p ′ (x 0 ).
From now on, we assume the bilinear form is symmetric; the arguments in the alternating case are analogous.For m < ω, T m denotes the theory of m-dimensional vector spaces over an algebraically closed field equipped with a non-degenerate symmetric bilinear form.For the definitions of dividing finitely and ΓMS-dividing see Definitions 12.3.8and 12.3.11 in [18].Note that, by Kim's Lemma, ΓMSdividing is the same as Kim dividing.
By ω * = {i * : i < ω} we will denote a copy of ω with the reversed order.Next, we will describe dividing of formulae in T ∞ in terms of dividing in the ωstable theories T m (which are interpretable in ACF ).We will need to recall some definitions and facts about approximating sequences.
Conversely, suppose ϕ(x, b) k-divides in M , and consider arbitrary l < ω and a finite subset p 0 ⊆ tp(b/A).Then there is a sequence (b i ) i<l of tuples in M such that for any i

Further examples
In this section, we collect more examples of NSOP 1 theories with existence.In each of them, there exist at least three related notions of independence: | ⌣ , | ⌣ K and | ⌣ K,q (recall that | ⌣ K,q denotes Kim-independence with respect to definition (B) of Kim-dividing).We would like to better understand what are the relations between these three notions of independence.Obviously: The first natural question is: when if and only if for every C and every p(x) ∈ S(C) we have that p(x) extends to a global C-invariant type.The implication "⇒" holds for arbitrary T .
Proof."⇐": We only need to show that a | ⌣ Remark 5.2.
(1) Note in particular that if there exists C with acl(C) = dcl(C) then for any c ∈ acl(C) \ dcl(C) the type tp(c/C) does not extend to a global C-invariant type, so (2) By exactly the same argument as above, if T is NSOP 1 with existence, then | ⌣ K,q coincides with | ⌣ K over algebraically closed sets if and only if for any algebraically closed C any p(x) ∈ S(C) we have that p(x) extends to a global C-invariant type.In particular, if T = T eq is stable then | ⌣ K,q coincides with | ⌣ K = | ⌣ over algebraically closed sets.
Example 5.3.We are working in T ∞ .As there are sets C with dcl(C) = acl(C) (for example in the sort K(C), on which the induced structure is just that of a pure algebraically closed field), we have that over algebraically closed sets which however relies on Question 5.4 below: by the proof of [14, Proposition 8.1], any complete type over a finite set C extends to a global type p(x) which is invariant over acl eq (C).By compactness, the same is true for arbitrary C, as the existence of a global acl eq (C)-invariant extension of p(x) is equivalent to consistency of the type p(x) ∪ {ϕ(x, d) ↔ ϕ(x, d ′ ) : ϕ(x, y) ∈ L, d, d ′ ∈ C eq , d ≡ acl eq (C) d ′ }.So, if dcl eq (acl(C)) = acl eq (C) then p(x) extends to a global acl(C) -invariant type in T eq ∞ , which restricts to a global acl(C)-invariant type in T ∞ .Question 5.4.Is it true in T ∞ that for any set C we have dcl eq (acl(C)) = acl eq (C)?
Example 5.5.Assume that T is the theory of ω-free PAC fields, let F * be a monster model of T and let F := (F * ) sep (so F is a model of SCF and in this example, if we refer to SCF, we actually refer to Th( F )). Assume that A = acl(A), B = acl(B), C = acl(C) (by [7], we know that acl( * ) is obtained by closing under λ-functions and then taking the field-theoretic algebraic closure) and C ⊆ A ∩ B. By [6], we have A | ⌣C B if and only if ).So we see that the existence axiom for forking independence holds over algebraically closed sets.Actually, the existence axiom for forking independence holds over arbitrary sets (cf.Remark 2.15 and Remark 2.16 in [15]).
Let us recall what is | ⌣ K and | ⌣ K,q in the case of ω-free PAC fields.Up to our knowledge, there is no description of | ⌣ K or | ⌣ K,q over arbitrary sets, so we need to pick-up a model F F * .Assume that A = acl(A) and B = acl(B), and F ⊆ A∩B.
F B (as both notions of Kim-dividing coincide over models in NSOP 1 ), if and only if , where (2) is considered in the so-called sorted system, which is a first order structure build from all the quotients by open normal subgroups of the absolute Galois group (e.g.see [8], [5], [19]).In fact, as the absolute Galois group is the free profinite group on ω-many generators, this sorted system is stable (by [5]) and hence the second dot above can be replaced by "SG(A) | ⌣SG(F ) SG(B)", i.e. the forking independence relation, which is described in Proposition 4.1 in [5].
Example 5.6.Consider the theory ACFG, exposed in [13] and put it in the place of T .More precisely, fix some prime number p > 0 and let L G be the language of rings extended by a symbol for a unary predicate G. Models of the L G -theory ACF G are algebraically closed fields of characteristic p with an additive subgroup under the predicate G. Now, the L G -theory ACFG is the model companion of ACF G ; let (K, G) be its monster model.
Therefore ACFG, which is a NSOP Example 5.7.In [12], another example of an NSOP 1 theory with existence was studied.More precisely, T m,n denotes the theory of existentially closed incidence structures omitting the complete incidence structure K m,n .In other words, T m,n is the theory of a generic K m,n -free bipartite graph.Theorem 4.11 from [12] gives us that T m,n is NSOP 1 and Corollary 4.24 from [12] implies that T m,n satisfies the existence axiom for forking independence.Actually, the aforementioned corollary gives us also a nice description of forking independence, thus let us evoke it: for any A, B, C ⊆ C |= T m,n we have and | ⌣ I is a ternary relation coinciding with | ⌣ K,q over models.Thus, again, forking independence is induced by forcing base monotonicity on a ternary relation related to | ⌣ K,q .
The above examples motivate asking the following question: We know that in any theory T (not necessarily an NSOP 1 theory), for each a and A it holds that a | ⌣ K,q A A 1 .Assuming that Question 5.8 has an affirmative answer and assuming that | ⌣ K coincide with | ⌣ K,q over algebraically closed sets, we would obtain the forking existence axiom over algebraically closed sets.

Beyond NSOP 1
Now, let us consider arbitrary theory T (not necessarily NSOP 1 ).Instead of Definition 3.1, one could define the rank by the conditions from Lemma 3.12.Such a rank would be always finite and satisfy the standard properties (Lemma 3.15, Lemma 3.16, Lemma 3.17, Corollary 3.18).This is some strategy, however it seems that we require too much here.Thus, let us derive yet another notion of rank and show its finiteness in an important class of theories.The new rank is a slight modification of Q-rank and, as we will see in a moment, the new rank coincides with the Q-rank in NSOP 1 theories. 1We proved this fact under assumption of NSOP 1 with the use of our rank, but after a talk on the topic, Itay Kaplan, in a private communication, shared with us a simpler proof of this fact, which does not need the NSOP 1 assumption.
6.1.Refining notions.We start with introducing a refined notion of an invariant type and a notion of a surrogate of a global type.These notions are used in Definition 6.10 and came out from studying examples similar to the one in Subsection 6.3.Definition 6.1.Let A, B be small subsets of C and let C ⊆ C. A type q(y) ∈ S(C) is B/A-invariant if f (q) = q for every f ∈ Aut(C/A) such that f (B) = B. (This implies some restrictions on the set of parameters C.) Remark 6.2.Let A ⊆ B be small subsets of C, let C ⊆ C and let q(y) ∈ S(C).We have: q is A-invariant ⇒ q is B/A-invariant ⇒ q is B-invariant.Definition 6.3.A type q(y) is a semi-global type over A if there exists π(y) ∈ S(A) such that q(y) ∈ S(A ∪ π(C)) and π ⊆ q.Definition 6.4 (Morley in semi-global type).Let q be an A-invariant semi-global type over A and let (I, <) be a linearly ordered set.By a Morley sequence in q over A over A (of order type I) we understand a sequence B = (b i ) i∈I such that b i |= q| Ab<i .Remark 6.5.Assume that A ⊆ B are small subsets of C, and q is an B/A-invariant semi-global type over B and (I, <) is linearly ordered set. ( , and so we do not need to add new parameters to the domain of q (to satisfy B/A-invariance of q) and being a B/A-invariant semi-global type over B is well-defined.(1) T is NSOP 1 .

6.3.
Example where rank is infinite.Let B be a Boolean algebra considered in language L BA = {∧, ∨, c , 0, 1} and (P, , 0 P , 1 P ) a linearly ordered set with minimal element 0 P and maximal element 1 P .We call a function v : B → P a valuation, if v(x) = 0 P iff x = 0, V (1) = 1 P , and v(x ∨ y) = max(v(x), x(y)) for all x, y ∈ B. Now let L V BA = L BA ∪ {v, 0 P , 1 P } be a two-sorted language on sorts B and P , where v is a symbol of a unary function from B to P .Let DV BA 0 be the L V BA -theory expressing that (B, ∨, ∧, c , 0, 1) is an atomless Boolean algebra, v : B → P is a valuation, and for all x y in B (here, " " indicates the canonical ordering in Boolean algebras) and any p ∈ P such that v(x) < p < v(y) there exists z ∈ B with x z y and v(z) = p (density) and there are disjoint y 1 , y 2 y with v(y 1 ) = v(y 2 ) = v(y) (no valuation-atoms).
It is easy to see by compactness that DV BA 0 is consistent.By Example 4.33 from [3], DV BA 0 has ATP, so we intuitively expect that the Q-rank is infinite in DV BA 0 .Let us argue on that.Proposition 6.23.DV BA 0 is complete and admits QE in L V BA .

Lemma 2 . 8
(any T ).Let A ⊆ B, ϕ(x; y) ∈ L(A).If ϕ(x, a) k-Kim-divides over A, then there exists c ≡ A a such that c | ⌣A B and ϕ(x, c) k-Kim-divides over B.

Corollary 2 . 10 . 1 ) 2 ) 3 )
Let A ⊆ B, a, d ∈ C. (If ϕ(x, a) Kim-divides over A, then there exists c ≡ A a such that c | ⌣A B and ϕ(x, c) Kim-divides over B. (If for all c ≡ A a such that c | ⌣A B, the formula ϕ(x, c) does not Kim-divide over B, then ϕ(x, a) does not Kim-divide over A. (If for all d ′ ≡ A d and all c ≡ A a such that c | ⌣A B we have

Step 1 .
Let d |= tp(a/M ) ∪ {ϕ(x; b α 0 ) | α < λ} and let N ′′ C be such that aN ≡ M dN ′′ .By the existence axiom for forking independence, there exists N ′ ≡ Md N ′′ such that M <λ b <λ <ω | ⌣ Md N ′ .Because a | ⌣M N , we have that d | ⌣M N ′′ and then also that d | ⌣M N ′ .As N ′ is |M | + -saturated, Proposition 5.4 from [4] assures us that d | ⌣M N ′ and M <λ b <λ <ω | ⌣Md N ′ combine into dM <λ b <λ <ω | ⌣M N ′ .Then monotonicity of | ⌣ gives us b <λ <ω and let a ′ b ′ |= q ⊗ p| B .Our goal is ab |= q ⊗ p| B .As ab | ⌣A B, we have b | ⌣A B. Since b ′ |= p| B and p is A-invariant, also b ′ | ⌣A B. Stationarity of p| A implies that there exists f ∈ Aut(C/B) such that f (b) = b ′ , thus b |= p| B .Because a |= q| Ab , we have that f (a) |= q| Ab ′ .Since q is A-invariant, we obtain that f (a) | ⌣A b ′ .On the other hand, ab | ⌣A B changes via f into f (a)b ′ | ⌣A B, then symmetry, base monotonicity, normality and monotonicity of | ⌣ give us f (a) | ⌣Ab ′ Bb ′ .By transitivity of | ⌣ , f (a) | ⌣A b ′ and f (a) | ⌣Ab ′ Bb ′ imply that f (a) | ⌣A Bb ′ , which by the fact that a ′ |= q| Bb ′ (so a ′ | ⌣A Bb ′ ) and by stationarity of q|A implies f (a) ≡ Bb ′ a ′ .As f (a) |= q| Bb ′ , we have that a |= q| Bb , thus ab |= q ⊗ p| B and we end the proof of the claim.Now, we are moving to the proof of (2).Assume that ā = (a i ) i∈I |= q ⊗I | A and b = (b i ) i∈I |= q ⊗I | A are such that ā | ⌣A B and b | ⌣A B for some B ⊇ A. We need to show that tp(ā/B) = tp( b/B), which holds if and only if for each n < ω and each i 1 , . . ., i n ∈ I, such that i 1 < . . .< i n , we have tp(a in . . .a i1 /B) = tp(b in . . .b i1 /B).Consider such i 1 , . . ., i n ∈ I and choose any infinite sequence I 0 ⊆ I starting with (i 1 , . . ., i n ) which has the order type of (ω, <).Let a I0 := (a i ) i∈I0 and b I0 := (b i ) i∈I0 .Then a I0 |= q ⊗ω | A , b I0 |= q ⊗ω | A .Monotonicity of | ⌣ gives us that a I0 | ⌣A B and b I0 | ⌣A B. As q ⊗ω | A is stationary we have that a I0 ≡ B b I0 , so in particular tp(a in . . .a i1 /B) = tp(b in . . .b i1 /B).
1) is exactly the content of Lemma 3.22.For (2), we use Lemma 3.26 and repeat the proof of Lemma 3.22, where in Step 1, instead of the saturation assumption we use transitivity, which naturally holds in simple theories.(3) also uses a modification of the proof of Lemma 3.22.More precisely, in Step 1, we use that | ⌣ implies | ⌣ K and that transitivity (over arbitrary sets) holds in NSOP 1 with existence.We obtain dM <λ b <λ <ω | ⌣ K M N ′ and then b <λ <ω | ⌣ K M N ′ .Then, Step 3 follows by the definition of strong Kim-stationarity.The rest (Step 2 and Step 4) remains the same.

Proposition 4 . 3 .
Let A be a countable set of parameters and ϕ(x, b) a formula which divides over A. Then ϕ(x, b) divides finitely over A (so it divides uniformly over A by[18, Lemma 12.3.9]).Moreover, if k is such that ϕ(x, b) k-divides over A, then there is an A-indiscernible sequence witnessing this contained in a model of T 2k|V (b)| .Proof.Let k be such that ϕ(x, b) k-divides over A. By compactness we can find an A-indiscernible sequence (b i ) i<ω⌢ω * with b 0 * = b such that the set {ϕ(x, b i ) : i < ω ⌢ ω * } is k-inconsistent.Note that for any i the type tp(b i * /Ab <ω b (<i) * ) is finitely satisfiable in b <ω , hence in particular, putting C := Ab <ω we get that the sequence I := (b i * ) i<ω is Morley over C. Hence ϕ(x, b) Kim-divides over C, so (by Kim's Lemma), it ΓMS-divides over C. Thus, by [18, Lemma 12.3.12],ϕ(x, b) divides finitely over C so, as A ⊆ C, it divides finitely over A. Moreover, the proof [18, Lemma 12.3.12]gives that a C-indiscernible sequence witnessing k-dividing of ϕ(x, b) over C can be found in an R-dimensional subspace of V (C) provided that there is a model N R |= T R containing k elements of a Γ-Morley sequence witnessing ΓMS-dividing of ϕ(x, b) over C.But such k elements contain at most k|V (b)| vectors, so there is a model of T 2k|V (b)| containing all of them (see e.g.[14, Fact 3.2]).This gives the 'moreover' clause.By [18, Remark 12.3.4]we conclude: Corollary 4.4.In T ∞ , forking and dividing coincide: for any small set A and any formula ϕ(x, b) we have that ϕ(x, b) forks over A if and only if ϕ(x, b) divides over A.

Proposition
M and b i |= p 0 for every i < l.Then, as b i0 . . .b i k−1 contains at most k|V (b)| vectors and m ≥ 2k|V (b)|, we get by Fact 4.5 that j<k ϕ(x, b ij ) is inconsistent in T ∞ as well.So, by compactness, ϕ(x, b) k-divides in T ∞ .
for arbitrary a, B, C. By assumption tp(B/C) extends to a global A-invariant type r(x).If a | ⌣ K,q C , then there is an aC-indiscernible Morley sequence in r(x) over A starting with C. As I is in particular | ⌣ -Morley over A, by Kim's Lemma we get that a | ⌣ K C B. "⇒": Suppose there exist B and C such that tp(B/C) does not extend to a global C-invariant type.Then for any D we have that tp(BD/C) does not extend to a global C-invariant type, so, vacuously, E | ⌣ K,q C BD for any D, E. But we can choose (assuming T has infinite models) D and E such that E ⊆ acl(D) \ acl(C), in which case E | ⌣ forking independence in SCF) and • acl(AB) ∩ (AB 0 ) sep B sep = acl(AB 0 )B for each B 0 = acl(B 0 ) ⊆ B. Let A ′ be a small subset of F * and let B = acl(B).By the general properties of forking independence (e.g.Remark 5.3 in [4]), A ′ | ⌣B B holds if and only if A | ⌣B B, where A := acl(A ′ B).By the above description of forking independence in the theory of ω-free PAC fields, we have A ′ | ⌣B B (and so also A ′ | ⌣ K B B and For A ⊆ K, let Ā denote the field-theoretic algebraic closure of A in K. Take A, B, C ⊆ K and recall that the weak independence relation, A | ⌣ w C B, is given as follows: • A | ⌣ ACF C B and • G(AC + BC) = G(AC) + G(BC), where | ⌣ ACF denotes the forking independence relation in ACF.By Corollary 3.16 from [13], we have that | ⌣ is | ⌣ w after forcing base monotonicity:

Question 5 . 8 .
Assume that T is NSOP 1 with existence.Is it true that | ⌣ is | ⌣ K after forcing base monotonicity, i.e.:
[13]eory, enjoys the existence axiom for forking independence.In a private communication with Christian DElbée, it was suggested to us to use Corollary 1.7 from[13]and Theorem 2.12, to show that |