supported by the NSF under grant no. DMS-2204787
Loading Math Nice Properties of Theories

ω-stable
A theory is ω-stable if SM
n
(A) is countable for all models M and countable AM.

See , , .
superstable
Given an infinite cardinal κ, a theory is κ-stable if
|SM
n
(A)| ≤ κ
for all models M and AM of size at most κ.

A theory T is superstable if there is some cardinal λ such that T is κ-stable for all κ ≥ λ.

See , , .

Remarks
In the definition, λ can be made equal to 2|T|.
stable
Given an infinite cardinal κ, a theory is κ-stable if
|SM
n
(A)| ≤ κ
for all models M and AM of size at most κ.

A theory is stable if it is κ-stable for some infinite κ.

See , , .

Alternate definition:
A formula $$\large\varphi(x,y)$$ has the order property if there are $$\large(a_i)_{i\lt\omega}$$ and $$\large(b_i)_{i\lt\omega}$$ such that $\large\models\varphi(a_i,b_j)~\Leftrightarrow~ i\lt j.$ A theory is stable if no formula has the order property.

See .
o-minimal
Assume the language $$\large\mathcal{L}$$ contains a binary relation <, and that T is an $$\large\mathcal{L}$$-theory in which < is a linear order. T is o-minimal if, given a model M, every definable subset of M can be written as a finite union of points in M and intervals with endpoints in M.

characterization of forking

See , , .
dp-minimal
A theory has dp-rankn if there are formulas $$\large\varphi_1(x,y),\ldots,\varphi_n(x,y)$$ and mutually indiscernible sequences $$\large(a^1_i)_{i\lt\omega},\ldots,(a^n_i)_{i\lt\omega}$$ such that for any function $$\large\sigma:\{1,\ldots,n\}\to\omega$$, the type $\large\{\varphi_k(x,a^k_{\sigma(k)}):k\leq n\}~\cup$ $\large\{\neg\varphi_k(x,a^k_i):i\neq\sigma(k),~k\leq n\}$ is consistent.

A theory is dp-minimal if it has dp-rank 1.

See , .
NIP (dependent)
A formula $$\large\varphi(x,y)$$ has the independence property (IP) if there are $$\large(a_i)_{i\lt\omega}$$ and $$\large(b_I)_{I\subseteq\omega}$$ such that $$\large\varphi(a_i,b_I)$$ holds if and only if $$\large i\in I$$. A theory is NIP (dependent) if no formula has the independence property.

See , .
supersimple
A theory is supersimple if for all sets B and complete types p in Sn(B), there is a finite AB such that p does not fork over A.

See , .
simple
A theory T is simple if for all sets B and complete types p in Sn(B) there is AB such that |A|≤|T| and p does not fork over A.

Alternate Definition:
A formula $$\large\varphi(x,y)$$ has the tree property (TP) if there are $$y$$-tuples $$\large(a_\eta)_{\eta~\in\omega^{\lt\omega}}$$ and some $$\large k\geq 2$$ such that
• $$\large \{\varphi(x,a_{\sigma|n}):n\lt\omega\}$$ is consistent for all $$\large \sigma\in\omega^\omega$$,
• $$\large \{\varphi(x,a_{\eta\hat{~}i}:i\lt\omega\}$$ is
$$\large k$$-inconsistent for all $$\large \eta\in\omega^{\lt\omega}$$.
A theory is simple (NTP) if no formula has the tree property.

See , .

Remarks:
One often studies theories in which forking independence is well-behaved. Simple theories have been shown to strongly exemplify this. In particular, the following are equivalent (see , ):
• T is simple.
• Nonforking (or nondividing) is symmetric.
• Nonforking (or nondividing) is transitive.
• Nonforking (or nondividing) has local character.
• There is a notion of independence (which must be nonforking) satisfying certain properties (see ).
NTP1 (NSOP1/NSOP2)
Given a formula $$\large\varphi(x,y)$$ and an ordinal $$\large \alpha$$, an $$\large \alpha$$-tree for $$\large \varphi(x,y)$$ is a sequence $$\large (b_\eta)_{\eta~\in \alpha^{\lt\omega}}$$ of $$\large y$$-tuples such that for all $$\large \sigma\in \alpha^\omega$$, $$\large \{\varphi(x,b_{\sigma|n}):n\lt\omega\}$$ is consistent.

$$\large \varphi(x,y)$$ has the tree property of the first kind (TP1) if there is an $$\large \omega$$-tree $$\large (b_\eta)_{\eta~\in \omega^{\lt\omega}}$$ for $$\large \varphi(x,y)$$ such that for all incomparable $$\large \eta,\mu\in \omega^{\lt\omega}$$, $$\large \{\varphi(x,b_\eta),~\varphi(x,b_\mu)\}$$ is inconsistent.

$$\large \varphi(x,y)$$ has SOP2 if there is a 2-tree $$\large (b_\eta)_{\eta~\in 2^{\lt\omega}}$$ for $$\large \varphi(x,y)$$ such that for all incomparable $$\large \eta,\mu\in 2^{\lt\omega}$$, $$\large \{\varphi(x,b_\eta),~\varphi(x,b_\mu)\}$$ is inconsistent.

$$\large \varphi(x,y)$$ has SOP1 if there is a 2-tree $$\large (b_\eta)_{\eta~\in 2^{\lt\omega}}$$ for $$\large \varphi(x,y)$$ such that for all $$\large \eta,\mu\in 2^{\lt\omega}$$, if $$\large \mu\hat{~}0\preceq\eta$$ then $$\large \{\varphi(x,b_{\eta}),~\varphi(x,b_{\mu\hat{~}1})\}$$ is inconsistent.

A theory is NTP1 (resp., NSOP1, NSOP2) if no formula has TP1 (resp., SOP1, SOP2).

It is clear that any NSOP1 theory must be NSOP2, and easy to show that NSOP2 is equivalent to NTP1 (this is implicit in ; the proof from is given here). The equivalence of NSOP1 and NSOP2 was open for many years, until proved in .

Remarks:
In analogy to forking in simple theories, there is a canonical notion of independence in NSOP1 theories called Kim-forking. NSOP1 can be characterized by symmetry or transitivity or local character of Kim-nonforking. See , , , .

SOP2 is shown to be maximal in Keisler's order in , and also characterizes maximality in the $$\large \triangleleft^*$$-order (, , ).

removed
NSOP3
no proper examples known
Fix a formula $$\large \varphi(x,y)$$ with $$\large |x|=|y|$$. Then $$\large \varphi(x,y)$$ has the 3-strong order property (SOP3) if the set $\large \{\varphi(x_1,x_2),\varphi(x_2,x_3),\varphi(x_3,x_1)\}$ is inconsistent, and there is a sequence $$\large (a_i)_{i\lt\omega}$$ such that $$\large \varphi(a_i,a_j)$$ holds for all $$\large i\lt j$$.

A theory is NSOP3 if no formula has SOP3.

See , .
NSOP4
Fix a formula $$\large \varphi(x,y)$$ with $$\large |x|=|y|$$. Then $$\large \varphi(x,y)$$ has the 4-strong order property (SOP4) if the set $\{\varphi(x_1,x_2),\varphi(x_2,x_3),\varphi(x_3,x_4),\varphi(x_4,x_1)\}$ is inconsistent, and there is a sequence $$\large (a_i)_{i\lt\omega}$$ such that $$\large \varphi(a_i,a_j)$$ holds for all $$\large i\lt j$$.

A theory is NSOP4 if no formula has SOP4.

See , , .

Remarks:
There are results about non-existence of universal models of cardinality λ (for certain λ) when T has SOP4 (see , ). For example:
If T has SOP4 and λ is regular such that κ+<λ<2κ for some κ, then T has no universal model of cardinality λ.
NSOPn+1
Fix n ≥ 3 and a formula $$\large \varphi(x,y)$$ with $$\large |x|=|y|$$. Then $$\large \varphi(x,y)$$ has the n-strong order property (SOPn) if the set $\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}$ is inconsistent, and there is a sequence $$\large (a_i)_{i\lt\omega}$$ such that $$\large \varphi(a_i,a_j)$$ holds for all $$\large i\lt j$$.

A theory is NSOPn if no formula has SOPn.

See , .

The above definition of SOPn works equally well when n is 1 or 2, and in both cases is equivalent to well known properties (of the theory). When n is 1, the condition is equivalent to having an infinite model; and when n is 2, the condition is equivalent to having the order property (instability). However, the acronyms SOP1 and SOP2 are not defined this way in the literature, but rather are used for certain variations of the tree property. See the NTP1 (NSOP1/NSOP2) region for further details.
NFSOP
A type $$\large p(x,y)$$, with $$\large |x|=|y|$$, has the strong order property if there are $$\large (a_i)_{i\in\omega}$$ such that $$\large p(a_i,a_j)$$ holds for all $$\large i\lt j$$, but $p(x_1,x_2)\cup \ldots\cup p(x_{n-1},x_n)\cup p(x_n,x_1)$ is unsatisfiable for all $$\large n\ge 2$$.

A theory is NFSOP (no finitary strong order property) if no type $$\large p(x,y)$$ has the strong order property with $$\large |x|$$ finite.

See , .

Remarks:
Two other properties of theories with a close relationship to the finitary strong order property are the fully finitary strong order property and the strong order property. The definitions of these properties are obtained by replacing the type in the definition above with, respectively, a formula or a type in arbitrarily many variables. In particular, the strong order property for a theory is equivalent to SOPn for all n ≥ 3.
NSOP
A formula $$\large \varphi(x,y)$$ has the strict order property (SOP) if there are $$\large (a_i)_{i\lt\omega}$$ such that $\models\exists x(\varphi(x,a_j)\wedge\neg\varphi(x,a_i))~\Leftrightarrow~i\lt j.$ A theory is NSOP if no formula has the strict order property.

See , .

Remarks:
The following characterizations of the strict order property are worth noting.
• T has SOP if and only if there is a formula $$\large \psi(y_1,y_2)$$ (with $$\large |y_1|=|y_2|$$) defining a preorder with infinite chains. (In the above definition, take $$\large \psi(y_1,y_2)$$ to be$$\large \forall x(\varphi(x,y_1)\rightarrow\varphi(x,y_2))$$.)
• T has SOP if and only if there is a definable set X (in a saturated model) and an automorphism σ such that σ(X) properly contains X.
NTP2
A formula $$\large \varphi(x,y)$$ has the tree property of the second kind (TP2) if there are $$\large (a_{i,j})_{i,j\lt\omega}$$ such that
• for all $$\large \sigma\in\omega^\omega$$, $$\large \{\varphi(x,a_{n,\sigma(n)}):n\lt\omega\}$$ is consistent.
• for all $$\large n\lt\omega$$ and $$\large i\lt j\lt\omega$$, $$\large \{\varphi(x,a_{n,i}),~\varphi(x,a_{n,j})\}$$ is inconsistent.
A theory is NTP2 if no formula has the tree property 2.

See , .

distal
A theory T is distal if, for any parameter set A, any A-indiscernible sequence I, and any tuple b, if I = I1 + I2, for some sequences I1 and I2 without endpoints, and I1+b+I2 is indiscernible, then I1+b+I2 is A-indiscernible.

See , .

Remarks:
The following are two useful characterizations of distal theories.
• A theory is distal if and only if it is NIP and all generically stable Keisler measures are smooth.
• (Strong honest definitions) An NIP theory T is distal if and only if for any formula $$\large \varphi(x,y)$$ there is a formula $$\large \psi(x,y_1,\ldots,y_n)$$, with $$\large |y_i|=|y|$$, such that for any finite set B of $$\large y$$-tuples, with |B|≥ 2, and any $$\large x$$-tuple $$\large a$$, there are $$\large b_1,\ldots,b_n$$ in B such that $$\large \psi(x,b_1,\ldots,b_n)\vdash\textrm{tp}_{\varphi}(a/B)$$.
removed
strongly minimal
A theory is strongly minimal if for all models M, any definable subset of M is finite or cofinite.

See , .

Remarks
For simplicity, this region of the map is for strongly minimal theories in a countable language. With this caveat, any strongly minimal theory is ω-stable.
Click a property above to highlight region and display details. Or click the map for specific region information.
List of Examples
Implications Between Properties
Click below to see the implications between all the above properties:

Diagram of Implications

Some other important equivalences:
• stable ⇔ NIP and NSOP
• simple ⇔ NTP1 and NTP2
For proofs of these implications, see , , , , , , , . Some of the proofs have been collected here.
Open Regions
Each small region outlined in red contains no known examples, due to the following open questions:
• Is the implication:
SOP3 ⇒ TP1 (SOP1/SOP2)
proper? See , , .
• Is there a theory which is NSOP and NTP2, but not simple? See Exercise III.7.12 in , and further discussion on the fff blog.
Open Examples
The following examples are theories whose exact positions on the map are unknown. Click an example to see its definition, and to highlight the possible regions it could be in.
• $$\large (\mathbb{C}(t),+,\cdot,0,1)$$
• Farey graph
• Artin braid groups