1 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. [Reference Adler2]), which plays a key role in the description of simple theories (cf. [Reference Kim and Pillay24]). 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
$D(x=x,\varphi ,k)$
is finite for every choice of a formula
$\varphi $
and every natural number k (cf. Proposition 3.13 in [Reference Casanovas4]). 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 [Reference Casanovas4]). On top of that, the local rank in simple theories was used to develop the theory of generics there (see [Reference Pillay25]).
Independence relations and ranks behave less nicely in the case of non-simple NSOP
$_1$
theories. The NSOP
$_1$
theories were defined in [Reference Džamonja and Shelah17], then studied more intensively in [Reference Chernikov and Ramsey11] and in [Reference Kaplan and Ramsey20], where also the ideas from Kim’s talk (see [Reference Kim23]) 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 [Reference Kaplan and Ramsey20] (where Kim-dividing and Kim-forking were defined with the use of global invariant types), e.g., [Reference Kaplan and Ramsey21, Reference Kaplan, Ramsey and Shelah22]. A limitation of 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 [Reference Dobrowolski, Kim and Ramsey15], 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 [Reference Kim23]. 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 [Reference Chernikov, Byungham and Ramsey10], where, e.g., transitivity of Kim-independence (as defined in [Reference Dobrowolski, Kim and Ramsey15]) 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 (see [Reference Kaplan and Ramsey20]). Therefore one could expect that there should also exists 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 [Reference Chernikov, Byungham and Ramsey10], however they were not fully successful in relating the rank to Kim-independence (see Question 4.9 in [Reference Chernikov, Byungham and Ramsey10]). On the other hand, the rank defined in [Reference Chernikov, Byungham and Ramsey10] 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
$\varphi $
, natural number k and some type q). Thus finiteness of the rank from [Reference Chernikov, Byungham and Ramsey10] 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 [Reference Dobrowolski14]). 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 [Reference Adler2]). 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 (see [Reference Adler2]) related to Kim-dividing given as in [Reference Kaplan and Ramsey20]. 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 6.11) 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_\infty $
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_\infty $
, answering in particular a question from [Reference Granger18] about equivalence of dividing and dividing finitely, and yielding, to the best of our knowledge, first example of a non-simple NSOP
$_1$
theory in which forking and dividing coincide for formulae.
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
$T_{\infty }$
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.
2 Basics about NSOP
$_1$
As usual, we work with a complete
$\mathcal {L}$
-theory T and with a monster model
$\mathfrak {C}$
of T, i.e., a
$\kappa $
-strongly homogeneous and
$\kappa $
-saturated model of T for some big cardinal
$\kappa $
. By a small tuple/subset/substructure we mean some tuple/subset/substructure of size strictly smaller than
$\kappa $
. Unless stated otherwise, all considered tuples/subsets/substructures will be small. For example,
$A\subseteq \mathfrak {C}$
tacitly implies that
$|A|<\kappa $
. In short, we follow conventions being standard in model theory, e.g., outlined in [Reference Casanovas4, Reference Tent and Ziegler26].
At the beginning, we need to evoke several definitions and facts about Kim-dividing and NSOP
$_1$
theories. A reader unfamiliar with the subject may consult, e.g., [Reference Chernikov, Byungham and Ramsey10, Reference Dobrowolski, Kim and Ramsey15, Reference Kaplan and Ramsey20].
Definition 2.1. A formula
$\varphi (x;y)$
has SOP
$_1$
(Strict Order Property of the first kind) if there exists a collection of tuples
$(a_{\eta })_{\eta \in 2^{<\omega }}$
such that:
-
(1)
$\{\varphi (x;a_{\eta |_{\alpha }})\;|\;\alpha <\omega \}$ is consistent for every
$\eta \in 2^\omega $ ,
-
(2) if
$\eta \in 2^{<\omega }$ and
$\nu \trianglerighteq \eta \smallfrown \langle 0\rangle $ , then
$\{\varphi (x;a_{\nu }),\,\varphi (x;a_{\eta \smallfrown \langle 1\rangle })\}$ is inconsistent,
where
$\trianglelefteq $
denotes the tree partial order on
$2^{<\omega }$
. The theory T has SOP
$_1$
if there is a formula which has SOP
$_1$
. Otherwise we say that T is NSOP
$_1$
.
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 . 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
$A\subseteq \mathfrak {C}$
,
$p(y)\in S(A)$
, and let
$(I,<)$
be a linearly ordered set. We call a sequence
$(b_i)_{i\in I}$
a Morley sequence in p if:
-
(1)
for each
$i\in I$ ,
-
(2)
$(b_i)_{i\in I}$ is A-indiscernible,
-
(3)
$b_i\models p$ for each
$i\in I$ .
Definition 2.4 (Morley sequence in an invariant global type).
Assume that
$q(y)\in S(\mathfrak {C})$
(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
$\bar {b}=(b_i)_{i\in I}$
such that
$b_i\models q|_{Ab_{<i}}$
for each
$i\in I$
. By
$q^{\otimes I}$
we indicate the global A-invariant type in variables
$(x_i)_{i\in I}$
such that for any
$B\supseteq A$
, if
$\bar {b}\models q^{\otimes I}|_B$
then then
$b_i\models q|_{Bb_{<i}}$
for all
$i\in I$
.
Therefore, if
$q(y)\in S(\mathfrak {C})$
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
$\varphi (x;y)\ q$
-divides over A if for some (equivalently: any) Morley sequence
$(b_i)_{i<\omega }$
in q over A, the set
$\{\varphi (x,b_i)\;|\;i<\omega \}$
is inconsistent.
We have the following two notions of Kim-dividing, (A) appears in [Reference Dobrowolski, Kim and Ramsey15] and (B) appear in [Reference Kaplan and Ramsey20]. For us notion (A) is the one which we will use here. However, Theorem 7.7 from [Reference Kaplan and Ramsey20] implies that (A) and (B) coincide over a model (i.e., if
$A=M\preceq \mathfrak {C}$
) provided T is NSOP
$_1$
, and therefore we will switch very often to notion (B) if the situation is placed over a model.
Definition 2.6. Let
$A\subseteq \mathfrak {C}$
.
-
(A) Assume that
$k\in \mathbb {N}\setminus \{0\}$ . We say that
$\varphi (x,b)\ k$ -Kim-divides over A if there exists a Morley sequence
$(b_i)_{i<\omega }$ in
$\operatorname {tp}(b/A)$ such that
$\{\varphi (x,b_i)\;|\;i<\omega \}$ is k-inconsistent. We say that
$\varphi (x,b)$ Kim-divides over A if there exists
$k\in \mathbb {N}\setminus \{0\}$ such that
$\varphi (x,b)\ k$ -Kim-divides over A.
-
(B) We say that
$\varphi (x;b)$ Kim-divides over A if there exists an A-invariant global type
$q(y)\supseteq \operatorname {tp}(b/A)$ such that
$\varphi (x;y)\ q$ -divides over A. Equivalently, we say that
$\varphi (x;b)$ Kim-divides over A if there exists an A-invariant global type
$q(y)\supseteq \operatorname {tp}(b/A)$ and a Morley sequence
$(b_i)_{i<\omega }\models q^{\otimes \omega }|_A$ such that
$b_0=b$ and the set
$\{\varphi (x,b_i)\;|\;i<\omega \}$ is inconsistent.
Fact 2.7 (Kim’s lemma over models).
Assume that T has NSOP
$_1$
and let
$M\preceq \mathfrak {C}$
. The following are equivalent:
-
(1)
$\varphi (x;b)$ Kim-divides over M,
-
(2) for any M-invariant global type
$q(y)\supseteq \operatorname {tp}(b/M)$ and any Morley sequence
$(b_i)_{i<\omega }\models q^{\otimes \omega }|_M$ we have that the set
$\{\varphi (x,b_i)\;|\;i<\omega \}$ is inconsistent.
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
$\varphi (x,b)$
Kim-divides over A with respect to definition (B) and
$A\subseteq B$
then there exists
$c\equiv _A b$
such that
and such that
$\varphi (x,c)$
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.
Lemma 2.8 (any T).
Let
$A\subseteq B$
,
$\varphi (x;y)\in \mathcal {L}(A)$
. If
$\varphi (x,a)\ k$
-Kim-divides over A, then there exists
$c\equiv _A a$
such that
and
$\varphi (x,c)\ k$
-Kim-divides over B.
Proof By the definition of k-Kim-dividing, there exists an A-indiscernible sequence
$(a_i)_{i<\omega }$
such that
$a_0=a$
, for each
$i<\omega $
we have
and
$\{\varphi (x,a_i)\;|\;i<\omega \}$
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 [Reference Casanovas4].
Step 1: Increase the length of
$(a_i)_{i<\omega }$
to a big cardinal
$\lambda $
(big enough for the use of Erdős–Rado theorem for B-indiscernibility, see, e.g., Proposition 1.6 in [Reference Casanovas4]). To do it, consider:

Let
$\bar {b}=(b_{\alpha })_{\alpha <\lambda }\models p(\bar {x})$
, then
$\bar {b}$
is A-indiscernible,
$b_0=a_0=a$
,
$\{\varphi (x,b_{\alpha })\;|\;\alpha <\lambda \}$
is k-inconsistent. Invariance and finite character of
imply that also
for each
$\alpha <\lambda $
.
Step 2: Force -independence over B. We define recursively partial elementary over A maps
$f_{\alpha }:\operatorname {dcl}(Ab_{\leqslant \alpha })\to \mathfrak {C}$
, where
$\alpha <\lambda $
, such that:
-
•
$f_{\alpha +1}|_{\operatorname {dcl}(Ab_{\leqslant \alpha })}=f_{\alpha }|_{\operatorname {dcl}({Ab_{\leqslant \alpha })}}$ ,
-
•
,
for each
$\alpha <\lambda $
. We start with obtaining
$f_0$
. Since
,
$\operatorname {tp}(b_0/A)$
does not fork over A and so there exists a non-forking extension:

Let
$f_0$
be determined by
$f_0(b_0)=b_0'$
and
$f_0|_A=\operatorname {id}_A$
. Now, we deal with the successor step
$\alpha \rightsquigarrow \alpha +1$
. Let
$f_{\alpha }'\in \operatorname {Aut}(\mathfrak {C})$
extend
$f_{\alpha }$
. Because
, we have that
. There exists an extension

which does not fork over A, i.e., . Let
$f_{\alpha +1}$
be determined by
$f_{\alpha +1}(b_{\alpha +1})=b_{\alpha +1}'$
and
$f_{\alpha +1}|_{\operatorname {dcl}(Ab_{\leqslant \alpha })}=f_{\alpha }|_{\operatorname {dcl}(Ab_{\leqslant \alpha })}$
. We move on to the limit ordinal step:
$\beta <\lambda $
and
$\beta \in \operatorname {Lim}$
. Let
$f^-_{\beta }\in \operatorname {Aut}(\mathfrak {C})$
extend
$\bigcup _{\alpha <\beta }f_{\alpha }$
. From
we obtain
, so, again, we can find an extension

which does not fork over A, i.e., . Let
$f_{\beta }$
be determined by
$f_{\beta }(b_{\beta })=b^{\prime }_{\beta }$
and
$f_{\beta }|_{\operatorname {dcl}(Ab_{<\beta })}=f^-_{\beta }|_{\operatorname {dcl}(Ab_{<\beta })}$
. After the recursion is done, we take
$f\in \operatorname {Aut}(\mathfrak {C})$
which extends
$\bigcup _{\alpha <\lambda }f_{\alpha }$
and set
$(b^{\prime }_{\alpha })_{\alpha <\lambda }=(f(b_{\alpha }))_{\alpha <\lambda }$
.
Step 3: Erdős–Rado. By Erdős–Rado theorem (e.g., Proposition 1.6 in [Reference Casanovas4]), there exists a B-indiscernible sequence
$(c_i)_{i<\omega }$
such that for each
$n<\omega $
there exist
$\alpha _0<\dots <\alpha _n<\lambda $
satisfying

Thus:
-
•
$c_0\equiv _B b^{\prime }_{\alpha _0}\equiv _A b^{\prime }_0\equiv _A b_0=a$ ,
-
•
$\{\varphi (x,c_i)\;|\;i<\omega \}$ is k-inconsistent (since
$\bar {b}'\equiv _A\bar {b}$ and
$\{\varphi (x,b_{\alpha })\;|\;\alpha <\lambda \}$ is k-inconsistent),
-
•
(because
$c_0\ldots c_n\equiv _B b^{\prime }_{\alpha _0}\ldots b^{\prime }_{\alpha _n}$ and
, and
satisfies monotonicity, base-monotonicity, and invariance).
Therefore
$\varphi (x,c_0)\ k$
-Kim-divides over B and
$c_0\equiv _A a$
.
For some
$\alpha _0<\lambda $
we have that
$c_0\equiv _B b^{\prime }_{\alpha _0}$
. Then, because
and because
satisfies monotonicity and invariance, we obtain that also
.
Definition 2.9.
-
(1) A partial type
$\pi (x)$ Kim-forks over A if
$$ \begin{align*}\pi(x)\vdash\bigvee\limits_{j\leqslant n}\psi_j(x;b_j)\end{align*} $$
$n<\omega $ and
$\psi _j(x;b_j)$ Kim-divides over A for each
$j\leqslant n$ .
-
(2) Let a be a tuple from
$\mathfrak {C}$ and let
$A,B\subseteq \mathfrak {C}$ . We say that a is Kim-independent from B over A, denoted by
, if
$\operatorname {tp}(a/AB)$ does not Kim-fork over A.
One could redefine the notions of Kim-forking and Kim-independence using Kim-dividing 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 to denote Kim-independence defined with Kim-dividing with respect to definition (B).
The previous lemma easily leads to the following.
Corollary 2.10. Let
$A\subseteq B$
,
$a,d\in \mathfrak {C}$
.
-
(1) If
$\varphi (x,a)$ Kim-divides over A, then there exists
$c\equiv _A a$ such that
and
$\varphi (x,c)$ Kim-divides over B.
-
(2) If for all
$c\equiv _A a$ such that
, the formula
$\varphi (x,c)$ does not Kim-divide over B, then
$\varphi (x,a)$ does not Kim-divide over A.
-
(3) If for all
$d'\equiv _A d$ and all
$c\equiv _A a$ such that
we have
then
.
The following characterization of NSOP
$_1$
proved much more useful in studying Kim-independence than the original definition of NSOP
$_1$
introduced in [Reference Džamonja and Shelah17].
Theorem 2.11 (Theorem 8.1 in [Reference Kaplan and Ramsey20]).
The following are equivalent:
-
(1) T is NSOP
$_1$ .
-
(2) Kim’s lemma for Kim-dividing: For any
$M\preceq \mathfrak {C}$ and any
$\varphi (x;b)$ , if
$\varphi (x;y)\ q$ -divides over M for some M-invariant
$q(y)\in S(\mathfrak {C})$ with
$\operatorname {tp}(b/M)\subseteq q(y)$ , then
$\varphi (x;y)\ q$ -divides over M for any M-invariant
$q(y)\in S(\mathfrak {C})$ with
$\operatorname {tp}(b/M)\subseteq q(y)$ .
Fact 2.7 has a generalization to Kim-dividing over arbitrary sets:
Theorem 2.12 (Theorem 3.5 in [Reference Dobrowolski, Kim and Ramsey15]).
Let T be NSOP
$_1$
with existence. Then T satisfies Kim’s lemma for Kim-dividing over arbitrary sets: if a formula
$\varphi (x,b)$
Kim-divides over A with respect to some Morley sequence in
$\operatorname {tp}(b/A)$
then the formula
$\varphi (x,b)$
Kim-divides over A with respect to any Morley sequence in
$\operatorname {tp}(b/A)$
.
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.
3 Rank
3.1 Definition and basic properties
In this section, we are interested in defining a local rank depending on pairs consisting of an
$\mathcal {L}$
-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 (see [Reference Adler2]). 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
$Q:=\big ((\varphi _0(x;y_0),q_0(y_0)),\ldots ,(\varphi _{n-1}(x;y_{n-1}),q_{n-1}(y_{n-1}) )\big )$
, where
$\varphi _0,\ldots $
$\dots ,\varphi _{n-1}\in \mathcal {L}$
and
$q_0,\ldots ,q_{n-1}$
are global types.
Definition 3.1. We define a local rank, called Q-rank,

For any set of
$\mathcal {L}$
-formulae
$\pi (x)$
we have
$D_Q(\pi (x))\geqslant \lambda $
if and only if there exists
$\eta \in n^{\lambda }$
and
$(b^{\alpha },M^{\alpha })_{\alpha <\lambda }$
such that:
-
(1)
$\operatorname {dom}(\pi (x))\subseteq M^0$ ,
-
(2)
$q_0,\ldots ,q_{n-1}$ are
$M^0$ -invariant,
-
(3)
$M^{\alpha }\preceq \mathfrak {C}$ for each
$\alpha <\lambda $ and
$(M^{\alpha })_{\alpha <\lambda }$ is increasing and continuous,
-
(4)
$b^{\alpha }M^{\alpha }\subseteq M^{\alpha +1}$ for each
$\alpha +1<\lambda $ ,
-
(5)
$b^{\alpha }\models q_{\eta (\alpha )}|_{M^{\alpha }}$ for each
$\alpha <\lambda $ ,
-
(6)
$\pi (x)\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha })\;|\;\alpha <\lambda \}$ is consistent,
-
(7) each
$\varphi _{\eta (\alpha )}(x;b^{\alpha })$ Kim-divides over
$M^{\alpha }$ with respect to definition (B), i.e., for each
$\alpha <\lambda $ there exists an
$M^{\alpha }$ -invariant global type
$r_{\alpha }(y_{\eta (\alpha )})$ extending
$\operatorname {tp}(b^{\alpha }/M^{\alpha })$ and
$\bar {b}^{\alpha }=(b^{\alpha }_i)_{i<\omega }\models r_{\alpha }^{\otimes \omega }|_{M^{\alpha }}$ such that
$b^{\alpha }_0=b^{\alpha }$ and
$\{\varphi _{\eta (\alpha )}(x;b^{\alpha }_i)\;|\;i<\omega \}$ is inconsistent.
If
$D_Q(\pi )\geqslant \lambda $
for each
$\lambda \in \operatorname {Ord}$
, then we set
$D_Q(\pi )=\infty $
. Otherwise
$D_Q(\pi )$
is the maximal
$\lambda \in \operatorname {Ord}$
such that
$D_Q(\pi )\geqslant \lambda $
.
Remark 3.2. If T is NSOP
$_1$
, then Definition 3.1 is equivalent to the same definition but with the condition (3) replaced by
-
(3*)
$M^{\alpha }\preceq \mathfrak {C}$ for each
$\alpha <\lambda $ ,
$(M^{\alpha })_{\alpha <\lambda }$ is continuous, and each
$M^{\alpha +1}$ is
$|M^{\alpha }|^+$ -saturated,
or even by
-
(3**)
$M^{\alpha }\preceq \mathfrak {C}$ for each
$\alpha <\lambda $ ,
$(M^{\alpha })_{\alpha <\lambda }$ is continuous, and each
$M^{\alpha +1}$ is
$|M^{\alpha }|^+$ -saturated and strongly
$|M^{\alpha }|^+$ -homogeneous.
The proof is a quite standard and long recursion, which uses transitivity and symmetry of over models. Thus we omit it. If we refer to Definition 3.1 in the case of T being NSOP
$_1$
, we usually have in mind its equivalent formulation with the condition (3
$^\ast $
).
Let us explain a little bit the concept behind this rank. For simplicity we assume that
$Q=( (\varphi ,q) )$
. Then the witnesses from the definition of
$D_Q(\pi )\geqslant \lambda $
,
$(M^{\alpha },b^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
, may be used to draw the following tree:

Each horizontal sequence of
$b^{\alpha }_i$
’s is a Morley sequence in some global
$M^{\alpha }$
-invariant type and so witnesses Kim-dividing of
$\varphi (x;b^{\alpha }_0)$
over
$M^{\alpha }$
. 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
$\operatorname {dom}(\pi )\subseteq M^0\preceq M^1\preceq \ldots $
, we could consider a more standard sequence
$\operatorname {dom}(\pi )\subseteq \operatorname {dom}(\pi )b^0_0\subseteq \operatorname {dom}(\pi )b^0_0b^1_0\subseteq \ldots $
. However, let us recall that
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 prove a few more facts. We start with something completely trivial.
Fact 3.4.
-
(1)
$D_Q(\pi )\geqslant \lambda $ ,
$f\in \operatorname {Aut}(\mathfrak {C})\ \Rightarrow \ D_{f(Q)}(f(\pi ))\geqslant \lambda $ .
-
(2) If
$\pi '\subseteq \pi $ then
$D_Q(\pi )\leqslant D_{Q}(\pi ')$ .
-
(3)
$D_Q(\pi )\leqslant \oplus _{j<n}D_{((\varphi _j,q_j))}(\pi )$ , where
$\oplus $ denotes the Cantor sum.
Proof Items (1) and (2) follow by the definition. Item (3) follows from the following fact, which can be proven by a straightforward induction on
$\alpha $
: for any ordinal
$\alpha =A_0\dot \cup A_1\dot \cup \cdots \dot \cup A_{n-1}$
, if
$\alpha _i$
is the order type of
$A_i$
for
$i<n$
, then
$\alpha \leqslant \alpha _0\oplus \alpha _1\oplus \dots \oplus \alpha _{n-1}$
.
Corollary 3.5. There exists finite Q such that
$D_Q(x=x)\geqslant \omega $
if and only if there exists Q such that
$|Q|=1$
and
$D_Q(x=x)\geqslant \omega $
.
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 6.11).
Example 3.6. Let T be the theory of dense linear orders without endpoints, DLO. First, we will show that
$D_Q(x=x)\leq 1$
for
$Q=(\phi (x,yz),q)$
, where q is an arbitrary invariant type and
$ \phi (x,yz)=(y<x<z)$
. Suppose
$M^i, b^i_j, i< 2$
,
$j<\omega $
with
$b^i_j=(a^i_j,c^i_j)$
witness that
$D_Q(x=x)\geq 2$
. As
$\phi (x;a^1_0,c^1_0)$
divides over
$a^0_0c^0_0$
(i.e., over
$b^0_0$
),
$a^0_0,c^0_0$
cannot lie in
$(a^1_0,c^1_0)$
, 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
$\phi (x;a^1_0,c^1_0)$
to a formula inconsistent with it. This contradicts
$M_0$
-invariance of the restriction
$q'$
of q to the first variable as
$a^1_0\models q'|M_1$
.
Note also that if
$\phi (x,y)$
is of the form
$x>y$
or
$x<y$
or
$x=x$
, then
$D_{(\phi ,q)}(x=x)=0$
, as in that case no instance of
$\phi (x,y)$
Kim-divides over any set. Also, it is easy to see that if
$\phi (x,y)=(x=y)$
then
$D_{(\phi ,q)}(x=x)=1$
.
Now let
$Q=(\phi (x,y),q(y))$
with
$x=x_0\dots x_{n-1}$
be a variable of length n,
$\phi (x,y)$
a formula and
$q(y)$
a global invariant type. By quantifier elimination and completeness of
$q(y)$
we have that
$q(y)\wedge \phi (x,y)\equiv q(y)\wedge \bigvee _{i<k} \phi _i(x,y)$
where each
$\phi _i(x,y)$
defines a product of intervals with endpoints in the set
$y\cup \{+\infty , -\infty \}$
.
We claim that
$D_Q(x=x)\leq nk$
. Suppose for a contradiction that
$D_Q(x=x)> nk$
. Then by pigeonhole principle one easily gets that
$D_{Q'}(x=x)> n$
where
$Q'=(\phi _l,q)$
for some
$l<k$
. Let
$(b^i,M^i)_{i<n+1}$
witness that
$D_{Q'}(x=x)\geq n+1$
. Choose
$c\models \{\phi (x,b^i):i<N\}$
and let
$\psi (x)$
be a formula equivalent to
$\operatorname {qftp}(c/\emptyset )$
. By replacing
$\phi _l(x,y)$
with
$\psi (x)\wedge \phi _l(x,y)$
we may assume
$\phi _l(x,y)$
is of the form
$\psi (x)\wedge \bigwedge _{j<n} \chi _j(x_j,y)$
, where each
$\chi _j(x_j,y)$
defines an interval with endpoints in
$y\cup \{+\infty , -\infty \}$
. Again by pigeonhole principle we must have that
$D_{Q"}(x=x)\geq 2$
with
$Q"=(\psi (x)\wedge \chi _j(x_j,y))$
for some
$j<n$
. As
$\psi (x)$
is over
$\emptyset $
and is consistent, we get that, for any j, any family of instances of
$\psi (x)\wedge \chi _j(x_j,y)$
is consistent if and only if the corresponding family of instances of
$\chi _j(x_j,y)$
is consistent (any realisation of the latter can be extended to a realisation of
$\psi (x)$
, since there is a unique 1-type in DLO). Hence
$D_{Q"}(x=x)=D_{(\chi _j(x_j,y),q)}(x_j=x_j)$
. But
$D_{(\chi _j(x_j,y),q)}(x_j=x_j)\leq 1$
by the first two paragraphs, a contradiction.
Definition 3.7. Let
$\varphi (x;y)\in \mathcal {L}$
and let
$q(y)\in S(\mathfrak {C})$
. We define
$C_{\varphi ,q}\leqslant \omega $
as

Remark 3.8. Let
$M,N\preceq \mathfrak {C}$
. Assume that:
-
• q is M-invariant,
$(a_i)_{i<\omega }\models q^{\otimes \omega }|_M$ and
$k_M$ is the maximal number (or
$\omega $ ) such that
$\{\varphi (x;a_i)\;|\;i<k_M\}$ is consistent.
-
• q is N-invariant,
$(b_i)_{i<\omega }\models q^{\otimes \omega }|_N$ and
$k_N$ is the maximal number (or
$\omega $ ) such that
$\{\varphi (x;b_i)\;|\;i<k_M\}$ is consistent.
Then
$k_N=k_M$
. To see this we introduce auxiliary
$\bar {N}\preceq \mathfrak {C}$
which contains M and N, and a Morley sequence
$(c_i)_{i<\omega }\models q^{\otimes \omega }|_{\bar {N}}$
. We have that
$(a_i)_{i<\omega }\equiv _M (c_i)_{i<\omega }\equiv _N (b_i)_{i<\omega }$
. This remark says that
$C_{\varphi ,q}$
is in some sense a uniform bound and it can not happen that for each
$M\preceq \mathfrak {C}$
and each corresponding Morley sequence
$(a_i)_{i<\omega }$
,
$\max \{k\;|\;\{\varphi (x;a_i)\;|\;i<k\}\}$
is finite, but
$C_{\varphi ,q}=\omega $
.
From now on (if not stated otherwise), we assume that T has NSOP
$_1$
.
Lemma 3.9. Then
$D_Q(\pi )\geqslant \lambda $
if and only if there exist
$\eta \in n^\lambda $
and
$\big ((b^{\alpha }_i)_{i<\omega },M^{\alpha }\big )_{\alpha <\lambda }$
such that:
-
(1)
$\operatorname {dom}(\pi (x))\subseteq M^0$ ,
-
(2)
$q_0,\ldots ,q_{n-1}$ are
$M^0$ -invariant,
-
(3)
$M^{\alpha }\preceq \mathfrak {C}$ for each
$\alpha <\lambda $ ,
$(M^{\alpha })_{\alpha <\lambda }$ is continuous, and each
$M^{\alpha +1}$ is
$|M^{\alpha }|^+$ -saturated,
-
(4)
$(b^{\alpha }_i)_{i<\omega } M^{\alpha }\subseteq M^{\alpha +1}$ for each
$\alpha +1<\lambda $ ,
-
(5)
$\pi (x)\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha }_0)\;|\;\alpha <\lambda \}$ is consistent,
-
(6) for each
$\alpha <\lambda $ , we have
$(b^{\alpha }_i)_{i<\omega }\models q^{\otimes \omega }_{\eta (\alpha )}|_{M^{\alpha }}$ and
$\{\varphi _{\eta (\alpha )}(x;b^{\alpha }_i)\;|\;i<\omega \}$ is inconsistent.
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 Remark 3.2, there is a configuration satisfying conditions (1)–(7) from Definition 3.1 such that each
$M^{\alpha +1}$
is
$|M^{\alpha }|^+$
-saturated. By Kim’s lemma (Fact 2.7), we can replace the condition (7) from Definition 3.1, by: for each
$\alpha <\lambda $
there exists
$\bar {c}^{\alpha }=(c^{\alpha }_i)_{i<\omega }\models q^{\otimes \omega }_{\eta (\alpha )}|_{M^{\alpha }}$
such that
$c^{\alpha }_0=b^{\alpha }$
and
$\{\varphi _{\eta (\alpha )}(x;c^{\alpha }_i)\;|\;i<\omega \}$
is inconsistent. By saturation of
$M^{\alpha +1}$
, we find
$(b^{\alpha }_i)_{i<\omega }\subseteq M^{\alpha +1}$
such that
$(b^{\alpha }_i)_{i<\omega }\equiv _{M^{\alpha }b^{\alpha }}(c^{\alpha }_i)_{i<\omega }$
which is the desired sequence to finish the proof.
Lemma 3.10.
$D_Q(\{x=x\})\leqslant \sum \limits _{j\in J}C_{\varphi _j,q_j}<\omega $
, where
$J=\{j< n\;|\;C_{\varphi _j,q_j}<\omega \}$
.
Proof Let us deal first with the case when
$Q=((\varphi (x;y),q(y)))$
. Assume that
$D_Q(\pi )\geqslant \lambda>0$
and let
$(b^{\alpha },M^{\alpha })_{\alpha <\lambda }$
be as in Definition 3.1 (
$\eta $
is constant, so we skip it).
We have that
$b^{\alpha }\models q|_{M^{\alpha }}$
and since
$b^{\alpha }M^{\alpha }\subseteq M^{\alpha +1}$
, it is
$b^{\alpha }\models q|_{M^0b^{<\alpha }}$
. We know also that q is
$M^0$
-invariant and that
$\{\varphi (x;b^{\alpha })\;|\;\alpha <\lambda \}$
is consistent. On the other hand, by Lemma 3.9, there exists a Morley sequence
$(b^0_i)_{i<\omega }\models q^{\otimes \omega }|_{M^0}$
such that
$b^0_0=b^0$
and
$\{\varphi (x;b^0_i)\;|\;i<\omega \}$
is inconsistent. Therefore
$C_{\varphi ,q}<\omega $
and so
$\lambda \leqslant C_{\varphi ,q}<\omega $
.
We switch to the general case where

Now, the function
$\eta \in n^{\lambda }$
becomes important. We have that
$\lambda =\bigcup _{j< n}\eta ^{-1}[j]$
, so we are done if we can show that
$|\eta ^{-1}[j]|=0$
or
$|\eta ^{-1}[j]|\leqslant C_{\varphi _j,k_j}<\omega $
for each
$j< n$
. Fix some
$j< n$
and assume that
$|\eta ^{-1}[j]|>0$
. Let
$\alpha <\lambda $
be the first index such that
$\eta (\alpha )=j$
. Repeating the first part of this proof for
$q_j$
we obtain what we need.
Corollary 3.11.
$D_Q(\pi )\leqslant \sum \limits _{j\in J}C_{\varphi _j,q_j}<\omega $
, where
$J=\{j< n\;|\;C_{\varphi _j,q_j}<\omega \}$
.
Lemma 3.12.
$D_Q(\pi )\geqslant \lambda $
if and only there exist
$\eta \in n^{\lambda }$
,
$(b^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
, and
$M\preceq \mathfrak {C}$
such that
-
(1)
$\operatorname {dom}(\pi )\subseteq M$ ,
-
(2)
$q_0,\ldots ,q_{n-1}$ are M-invariant,
-
(3)
$\pi (x)\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha }_0)\;|\;\alpha <\lambda \}$ is consistent,
-
(4)
$\{\varphi _{\eta (\alpha )}(x;b^{\alpha }_i)\;|\;i<\omega \}$ is inconsistent for each
$\alpha <\lambda $ ,
-
(5)
$$ \begin{align*}(b^{\lambda-1}_i)_{i<\omega}^\smallfrown\ldots^\smallfrown(b^0_i)_{i<\omega}\models q_{\eta(\lambda-1)}^{\otimes\omega}\otimes\cdots\otimes q_{\eta(0)}^{\otimes\omega}|_{M}.\end{align*} $$
Proof The left-to-right implication is straightforward: by Lemma 3.9 there are proper
$\eta \in n^\lambda $
and
$\big ((b^{\alpha }_i)_{i<\omega },M^{\alpha }\big )_{\alpha <\lambda }$
, we set
$M:=M^0$
and reuse
$(b^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
.
Let us take care of the right-to-left implication. We will recursively define a sequence
$((e^{\alpha }_i)_{i<\omega },M^{\alpha })_{\alpha <\lambda }$
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
$M^1\preceq \mathfrak {C}$
which is a
$|M^0|^+$
-saturated, and which contains
$M^0$
and
$(b^0_i)_{i<\omega }$
. Consider a Morley sequence

We have

Now, let
$M^2\preceq \mathfrak {C}$
be
$|M^1|^+$
-saturated such that
$M^1(c^1_i)_{i<\omega }\subseteq M^2$
and let us choose one more Morley sequence

We see that

and so

Set
$(e^2_i)_{i<\omega }:=(d^2_i)_{i<\omega }$
,
$(e^1_i)_{i<\omega }:=(c^1_i)_{i<\omega }$
and
$(e^0_i)_{i<\omega }:=(b^0_i)_{i<\omega }$
.
Continuing this process we will obtain a sequence of models
$M^0\preceq M^1\preceq \cdots \preceq M^{\lambda -1}$
and Morley sequences
$(e^{\alpha }_i)_{i<\omega }\models q_{\eta (\alpha )}^{\otimes \omega }|_{M^{\alpha }}$
, where
$\alpha <\lambda $
, such that
$M^{\alpha }(e^{\alpha }_i)_{i<\omega }\subseteq M^{\alpha +1}$
and

Since
$\operatorname {dom}(\pi (x))\subseteq M=M^0$
, we have also that
$\pi (x)\,\cup \,\{\varphi _{\eta (\alpha )}(x;e^{\alpha }_0)\;|\;\alpha <\lambda \}$
is consistent and that
$\{\varphi _{\eta (\alpha )}(x;e^{\alpha }_i)\;|\;i<\omega \}$
is inconsistent for each
$\alpha <\lambda $
. Therefore all the conditions of Lemma 3.9 are satisfied for
$((e^{\alpha }_i)_{i<\omega },M^{\alpha })_{\alpha <\lambda }$
.
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
$T_k$
of k-dimensional vector spaces over an algebraically closed field of characteristic p, equipped with a non-degenerate symmetric bilinear form (see [Reference Dobrowolski14, Chapter 10]). Let x and y be single vector variables,
$\phi (x,y)=(x\perp y\wedge x\neq 0)$
, and let
$q(y)$
be the generic type in the vector sort V (so q is
$\emptyset $
-invariant). Put
$Q=((\phi (x,y),q(y)))$
. We will show that
$D_Q(\{x=x\})=k-1$
.
Let
$(v_i)_{i<\omega }$
be a Morley sequence in
$q(y)$
. Then in particular
$v_0,\dots ,v_{k-1}$
are linearly independent, which easily implies that
$\bigwedge _{i<k}\phi (x,v_i)$
is inconsistent. Thus
$D_Q(\{x=x\})\leq k-1$
. For the other inequality, put
$M_i=\operatorname {acl}(a_{0},a_{1},\dots ,a_{k-1+i})$
for
$i<k$
and note that
$b^i:=v_{k+i}\models q|_{M_i}$
(as
and
$b^i\models q|_{\emptyset }$
), and each
$M_i$
is an elementary submodel by quantifier elimination. Also, the sequence
$(v_{k+i},v_{k+i+1},\dots )$
is Morley over
$M_i$
, and it witnesses that
$\phi (x,b^i)$
Kim-divides over
$M_i$
for each
$i<k$
. Finally,
$\bigwedge _{i<k-1} \phi (x,b^i)$
is consistent, as there is a non-zero vector orthogonal to
$b^0,\dots ,b^{i-2}$
. This shows that
$D_Q(\{x=x\})=k-1$
.
Remark 3.14. Note that if T is an NSOP
$_1$
theory (as we assume here) and for some
$k<\omega $
, a formula
$\varphi (x,y)$
,
$M\preceq \mathfrak {C}$
, some M-invariant
$q(y)\in S(\mathfrak {C})$
and some
$(b_i)_{i<\omega }\models q^{\otimes \omega }|_M$
, the set
$\{\varphi (x,b_i)\;|\;i<\omega \}$
is k-inconsistent but not
$(k-1)$
-inconsistent, then
$D_{((\varphi ,q))}(x=x)=k-1$
(as in Example 3.13). To see this, consider a linear order I being
$(k-1)$
-many copies of
$\omega $
(one after another one) and
$(c_i)_{i\in I}\models q^{\otimes I}|_M$
and use Lemma 3.12. In other words, for
$\pi (x):=\{x=x\}$
the situation is quite simple and either
$D_{((\varphi ,q))}(\pi )=0$
or
$D_{((\varphi ,q))}(\pi )=C_{\varphi ,q}$
provided
$C_{\varphi ,q}<\omega $
.
Lemma 3.15. If
$\pi \vdash \pi '$
then

Proof Assume that
$D_Q(\pi )\geqslant \lambda \in \mathbb {N}_{>0}$
, then by Lemma 3.12 there exist
$\eta \in n^{\lambda }$
,
$(b^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
, and
$M\preceq \mathfrak {C}$
such that:
-
(1)
$\operatorname {dom}(\pi )\subseteq M$ ,
-
(2)
$q_0,\ldots ,q_{n-1}$ are M-invariant,
-
(3)
$\pi (x)\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha }_0)\;|\;\alpha <\lambda \}$ is consistent,
-
(4)
$\{\varphi _{\eta (\alpha )}(x;b^{\alpha }_i)\;|\;i<\omega \}$ is inconsistent for each
$\alpha <\lambda $ ,
-
(5)
$$ \begin{align*}(b^{\lambda-1}_i)_{i<\omega}^\smallfrown\ldots^\smallfrown(b^0_i)_{i<\omega}\models q_{\eta(\lambda-1)}^{\otimes\omega}\otimes\dots\otimes q_{\eta(0)}^{\otimes\omega}|_{M}.\end{align*} $$
Let
$N\preceq \mathfrak {C}$
contain M and
$\operatorname {dom}(\pi ')$
, and let

Then naturally
$\operatorname {dom}(\pi ')\subseteq N$
and
$q_1,\ldots ,q_n$
are N-invariant. Because

we have also that
$\{\varphi _{\eta (\alpha )}(x;c^{\alpha }_i)\;|\;i<\omega \}$
is inconsistent for each
$\alpha <\lambda $
, and that
$\pi (x)\cup \{\varphi _{\eta (\alpha )}(x;c^{\alpha }_0)\;|\;\alpha <\lambda \}$
is consistent. Moreover,
$\pi \vdash \pi '$
implies that
$\pi '(x)\cup \{\varphi _{\eta (\alpha )}(x;c^{\alpha }_0)\;|\;\alpha <\lambda \}$
is consistent. Hence, Lemma 3.12 gives us
$D_Q(\pi ')\geqslant \lambda $
.
Lemma 3.16.
$D_Q\Big(\pi \cup \Big\{\bigvee \limits _{j\leqslant m}\psi _j\Big\}\Big)=\max \limits _{j\leqslant m} D_Q(\pi \cup \{\psi _j\})$
.
Proof Because
$\pi \cup \{\psi _i\}\vdash \pi \cup \{\bigvee _{j}\psi _j\}$
for each
$i\leqslant m$
, Lemma 3.15 gives us that

Hence it is enough to show that

Let
$D_Q(\pi \cup \{\bigvee _{j}\psi _j\})\geqslant \lambda $
, i.e., there exists
$\eta \in n^\lambda $
and
$(b^{\alpha },M^{\alpha })_{\alpha <\lambda }$
as in Definition 3.1, in particular,
-
•
$\operatorname {dom}(\pi \cup \{\bigvee _{j}\psi _j\})\subseteq M^0$ ,
-
•
$\pi \cup \{\bigvee _j\psi _j\}\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha })\;|\;\alpha <\lambda \}$ is consistent.
Thus there is
$i_0\leqslant m$
such that
$\pi \cup \{\psi _{i_0}\}\cup \{\varphi _{\eta (\alpha )}(x;b^{\alpha })\;|\;\alpha <\lambda \}$
is consistent. Because
$\operatorname {dom}(\pi \cup \{\psi _{i_0}\})\subseteq \operatorname {dom}(\pi \cup \{\bigvee _j\psi _j\})\subseteq M^0$
, we have that
$D_Q(\pi \cup \{\psi _{i_0}\})\geqslant \lambda $
and so also
$\max _{j} D_Q(\pi \cup \{\psi _j\})\geqslant \lambda $
.
Let us recall that we are working in a theory T which is NSOP
$_1$
.
Lemma 3.17. Assume that
$q_0(y_0)=\dots =q_{n-1}(y_{n-1})=q(y)$
(in Q).
-
(1) Let
$\{\pi _{\beta }\}$ lists all finite subsets of
$\pi $ . If for each
$\beta $ we have that
$D_Q(\pi _{\beta })\geqslant \lambda <\omega $ , then
$D_Q(\pi )\geqslant \lambda $ .
-
(2) We can always find a finite
$\pi _0\subseteq \pi $ such that
$D_Q(\pi _0)=D_Q(\pi )$ .
Proof Because for each
$\beta $
we have
$D_Q(\pi _{\beta })\geqslant \lambda $
, by Lemma 3.12, for each
$\beta $
there exists a function
$\eta _{\beta }\in n^{\lambda }$
, a sequence of sequences
$(b^{\beta ,\alpha }_i)_{\alpha <\lambda ,i<\omega }$
and a model
$M^{\beta }\preceq \mathfrak {C}$
such that:
-
(1)
$\operatorname {dom}(\pi _{\beta })\subseteq M^{\beta }$ ,
-
(2) q is
$M^{\beta }$ -invariant,
-
(3)
$\pi _{\beta }(x)\cup \{\varphi _{\eta _{\beta }(\alpha )}(x;b^{\beta ,\alpha }_0)\;|\;\alpha <\lambda \}$ is consistent,
-
(4)
$\{\varphi _{\eta _{\beta }(\alpha )}(x;b^{\beta ,\alpha }_i)\;|\;i<\omega \}$ is inconsistent for each
$\alpha <\lambda $ ,
-
(5)
$$ \begin{align*}(b^{\beta,\lambda-1}_i)_{i<\omega}^\smallfrown\ldots^\smallfrown(b^{\beta,0}_i)_{i<\omega}\models q^{\otimes\omega}\otimes\dots\otimes q^{\otimes\omega}|_{M^{\beta}}.\end{align*} $$
Let
$N\preceq \mathfrak {C}$
be such that
$\bigcup \limits _{\beta }M^{\beta }\subseteq N$
. Moreover, let us pick up the following Morley sequences:

and, without loss of generality, let
$\{\varphi _{\eta _{\beta }(\alpha )}(x;y)\;|\;\beta ,\alpha <\lambda \}=\{\varphi _0(x;y),\ldots ,\varphi _{r-1} (x;y)\}$
for some
$r\leqslant n$
. We introduce
$\psi (x;y):=\bigvee \limits _{i<r}\varphi _i(x;y)$
, and note that

is consistent (otherwise, by compactness,
$\pi _{\beta }(x)\,\cup \,\{\psi (x;c^{\alpha }_0)\;|\;\alpha <\lambda \}$
is inconsistent for some
$\beta $
, which is impossible, since

and
$\pi _{\beta }(x)\cup \{\varphi _{\eta _{\beta }(\alpha )}(x;b^{\beta ,\alpha }_0)\;|\;\alpha <\lambda \}$
is consistent). Consider
$d\models \pi (x)\,\cup \,\{\psi (x;c^{\alpha }_0)\;|\;\alpha <\lambda \}$
, then for each
$\alpha <\lambda $
there is
$i_{\alpha }<r$
such that
$\models \varphi _{i_{\alpha }}(d,c^{\alpha }_0)$
. Let
$\eta \in n^{\lambda }$
be given by
$\eta :\alpha \mapsto i_{\alpha }$
.
As we want to use Lemma 3.12 to show that
$D_Q(\pi )\geqslant \lambda $
, and we have already defined
$\eta $
,
$N\preceq \mathfrak {C}$
and
$(c^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
, we need to verify whether all the five conditions from Lemma 3.12 hold. Obviously,
$\operatorname {dom}(\pi )\subseteq N$
and q is N-invariant, so we have the first and the second condition. The fifth condition is naturally satisfied by the choice of
$(c^{\alpha }_i)_{\alpha <\lambda ,i<\omega }$
. The third condition says that
$\pi (x)\,\cup \,\{\varphi _{\eta (\alpha }(x,c^{\alpha }_0)\;|\;\alpha <\lambda \}$
is consistent, which is witnessed by element d. For the fourth condition, we need to note that
$\{\varphi _{\eta (\alpha )}(x,c^{\alpha }_i)\;|\;i<\omega \}$
is inconsistent for every
$\alpha <\lambda $
. Because
$\eta (\alpha )< r$
, there exist
$\beta $
and
$\alpha '<\lambda $
such that
$\eta (\alpha )=\eta _{\beta }(\alpha ')$
. We know that
$\{\varphi _{\eta _{\beta }(\alpha ')}(x,b^{\beta ,\alpha '}_i)\;|\;i<\omega \}$
is inconsistent, that
$(b^{\beta ,\alpha '}_i)_{i<\omega }\models q^{\otimes \omega }|_{M^{\beta }}$
and that
$(c^{\alpha }_i)_{i<\omega }\models q^{\otimes \omega }|_{M^{\beta }}$
. Thus also
$\{\varphi _{\eta (\alpha )}(x,c^{\alpha }_i)\;|\;i<\omega \}$
is inconsistent.
Corollary 3.18. Let
$q_0(y_0)=\dots =q_{n-1}(y_{n-1})=q(y)$
(in Q) and let
$\pi (x)$
be a partial type over A. Then there exists
$p(x)\in S(A)$
extending
$\pi (x)$
such that
$D_Q(p)=D_Q(\pi )$
.
Proof The proof is completely standard, but let us sketch it anyway. Consider

By Lemmas 3.15 and 3.16, the set
$\tilde {\pi }$
is a partial type over A. Let
$p(x)\in S(A)$
be any extension of
$\tilde {\pi }$
.
If
$D_Q(p)<D_Q(\pi )$
then by Lemma 3.17 there exists a finite subset
$p_0(x)\subseteq p(x)$
such that
$D_Q(p_0)<D_Q(\pi )$
. Let
$\psi (x):=\bigwedge p_0(x)\in L(A)$
, we have that

Thus
$\neg \psi (x)\in \tilde {\pi }(x)\subseteq p(x)$
and we got a contradiction with
$\psi (x)\in p(x)$
.
Let
$\Delta =\{\varphi _1(x;y),\ldots ,\varphi _n(x;y)\}$
and
$1<k<\omega $
. Recall that there is a local rank used in simple theories, denoted
$D(\,\cdot \,,\Delta ,k)$
(cf. Chapter 3 in [Reference Casanovas4]). This local rank may be used to characterize simplicity as: T is simple if and only if
$D(\{x=x\},\{\varphi \},k)<\omega $
for all
$\varphi $
and k (e.g., Proposition 3.13 in [Reference Casanovas4]). As our rank is also local, we can treat our rank as an analogon of the local rank
$D(\,\cdot \,,\Delta ,k)$
. Let us compare now the both local ranks.
Remark 3.19. For each
$Q=((\varphi _0,q_0),\ldots ,(\varphi _{n-1},q_{n-1}))$
and any partial type
$\pi $
, there exists
$K<\omega $
such that for any
$k\geqslant K$
we have

Proof As T is NSOP
$_1$
, there is some
$\lambda <\omega $
such that
$D_Q(\pi )=\lambda $
. Let
$\eta $
and
$(b^\alpha ,M^\alpha )_{\alpha <\lambda }$
be as in Definition 3.1. For each
$\alpha <\lambda $
there exists
$k_{\alpha }<\omega $
such that
$\varphi _{\eta (\alpha )}(x;b^{\alpha })$
$k_{\alpha }$
-divides over
$M^{\alpha }$
. Set
$K:=\max \{k_0,\ldots ,k_{\lambda -1}\}$
. Then, by definition,
$D(\pi ,\{\varphi _0,\ldots ,\varphi _{n-1}\},k)\geqslant \lambda $
, provided
$k\geqslant K$
.
On the other hand, if T is NSOP
$_1$
but not simple, then there must be a formula
$\varphi $
and some
$k_0<\omega $
such that
$D(\{x=x\},\{\varphi \},k_0)\geqslant \omega $
. Thus for every
$K<\omega $
there exists
$k\geqslant K$
(e.g.,
$k=K+k_0$
) such that
$D(\{x=x\},\{\varphi \},k)\geqslant \omega $
but,
$D_{((\varphi ,q))}(\{x=x\})<\omega $
for any choice of
$q\in S(\mathfrak {C})$
. Therefore sharp inequality in Remark 3.19 happens outside of the class of simple theories. One could ask about equality under the assumption of 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
$\omega $
-stable of Morley rank
$2$
. Let
$\varphi _0(x,y)=E(x,y)$
and
$\varphi _1(x,y)=(x=y)$
. Then it is easy to see that for any
$k>1$
we have that
$D(\{x=x\},\{\varphi _0(x,y),\varphi _1(x,y)\},k)=2$
. Now, fix two arbitrary invariant global types
$q_0(x),q_1(x)$
and put
$Q=\{(\varphi _0,q_0),(\varphi _1,q_1)\}$
. We claim that
$D_Q(\{x=x\})\leq 1$
. Suppose for a contradiction that
$D_Q(\{x=x\})\geqslant 2$
witnessed by
$M^0,M^1,b^0,b^1$
and
$\eta :2 \to 2$
. The case where
$\eta (0)=1$
can be excluded immediately, so assume
$\eta (0)=0$
. Observe that
$\models E(b^0,b^1)$
, as otherwise
$\varphi _{\eta (0)}(x,b^0)\wedge \varphi _{\eta (1)}(x,b^1)$
would be inconsistent (note
$\varphi _i(x,y)\vdash E(x,y)$
for
$i=0,1$
). On the other hand, as
$b^1\models q_{\eta (1)}|_{M^0b^0}$
, we have in particular that
, so
$E(b^0,M^0)\neq \emptyset $
, which contradicts that
$\varphi _0(x,b^0)$
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 [Reference Dobrowolski14]), which is NSOP
$_1$
. Since in the case of simple theories, a notion of finite local rank, which is compatible with forking and invariant under shifts by elements of a definable group (so-called stratified local rank, see, e.g., Fact 3.7 in [Reference Tent and Ziegler26]), would lead to the existence of forking generics (see Lemma 3.8 in [Reference Tent and Ziegler26]), 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 not 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 [Reference Chernikov, Byungham and Ramsey10].
Lemma 3.21. Let
$M\preceq N\preceq \mathfrak {C}$
and
$a\in \mathfrak {C}$
. If
$D_Q(\operatorname {tp}(a/M))=D_Q(\operatorname {tp}(a/N))$
for each M-invariant Q such that
$|Q|=1$
, then
.
Proof Assume that , which means that
$\operatorname {tp}(a/N)$
Kim-divides over M. Let
$\varphi (x,b)\in \operatorname {tp}(a/N)$
Kim-divide over M. There exists an M-invariant
$q(y)\in S(\mathfrak {C})$
extending
$\operatorname {tp}(b/M)$
such that
$\varphi (x,b)\ q$
-divides over M. We set
$Q=\big ((\varphi (x;y),q(y))\big )$
. By Lemma 3.15 and Corollary 3.11, it follows that

Therefore there exists a sequence
$(b^{\alpha },M^{\alpha })_{\alpha <\lambda }$
such that:
-
(1)
$Mb\subseteq M^0$ ,
-
(2) q is
$M^0$ -invariant,
-
(3)
$M^{\alpha }\preceq \mathfrak {C}$ and
$M^{\alpha +1}$ is
$|M^{\alpha }|^+$ -saturated for each
$\alpha <\lambda $ ,
-
(4)
$b^{\alpha }M^{\alpha }\subseteq M^{\alpha +1}$ for each
$\alpha +1<\lambda $ ,
-
(5)
$b^{\alpha }\models q|_{M^{\alpha }}$ for each
$\alpha <\lambda $ ,
-
(6)
$\operatorname {tp}(a/M)\;\cup \;\{\varphi (x;b)\}\;\cup \;\{\varphi (x;b^{\alpha })\;|\;\alpha <\lambda \}$ is consistent,
-
(7)
$\varphi (x;b^{\alpha })$ Kim-divides over
$M^\alpha $ for each
$\alpha <\lambda $ .
By Lemma 3.12, we can modify
$M^0$
and so assume that
$M^0$
is
$|M|^+$
-saturated, which we do. We set
$M^{-1}:=M$
and
$b^{-1}:=b$
. Checking that
$(b^{\alpha },M^{\alpha })_{-1\leqslant \alpha <\lambda }$
witnesses that
$D_Q(\operatorname {tp}(a/M))\geqslant \lambda +1$
is routine (note that, as
$\lambda <\omega $
,
$(b^{\alpha },M^{\alpha })_{-1\leqslant \alpha <\lambda }$
can be naturally indexed by the elements of
$\lambda +1$
).
Lemma 3.22. Let T be NSOP
$_1$
with existence. Assume that
$a\in \mathfrak {C}$
,
$M\preceq N\preceq \mathfrak {C}$
, N is
$|M|^+$
-saturated,
$q(y)\in S(\mathfrak {C})$
is M-invariant, and that
$q^{\otimes \omega \cdot n}|_M$
is stationary for any
$n<\omega $
. Let
$Q=\big ((\varphi (x;y),q(y))\big )$
. If
then
$D_Q(a/M)=D_Q(a/N)$
.
Proof Let
$D_Q(a/M)=:\lambda <\omega $
, it means by Lemma 3.9 that there exists
$\big ( (b^{\alpha }_i)_{i<\omega }, M^{\alpha }\big )_{\alpha <\lambda }$
such that:
-
(1)
$M\subseteq M^0$ ,
-
(2) q is
$M^0$ -invariant (which comes for free as q is M-invariant and
$M\subseteq M^0$ ),
-
(3)
$M^{\alpha }\preceq \mathfrak {C}$ for all
$\alpha <\lambda $ ,
$(M^{\alpha })_{\alpha <\lambda }$ is continuous, and
$M^{\alpha +1}$ is
$|M^{\alpha }|^+$ -saturated for all
$\alpha +1<\lambda $ ,
-
(4)
$b^{\alpha }_{<\omega }M^{\alpha }\subseteq M^{\alpha +1}$ for all
$\alpha +1<\lambda $ ,
-
(5)
$b^{\alpha }_{<\omega }\models q^{\otimes \omega }|_{M^{\alpha }}$ for all
$\alpha <\lambda $ ,
-
(6)