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Nice Properties of Theories

ω-stable

A theory is **\(\omega\)-stable** if \(S^M_n(A)\) is countable for all models \(M\) and countable subsets \(A\subseteq M\).

See , , .

See , , .

superstable

Given an infinite cardinal \(\kappa\), a theory is \(\kappa\)-stable if \[|S_n^M(A)|\leq\kappa\] for all models \(M\) and subsets \(A\subseteq M\) with \(|A|\leq\kappa\).

A theory \(T\) is**superstable** if there is some cardinal \(\lambda\) such that \(T\) is \(\kappa\)-stable for all \(\kappa\geq\lambda\).

See , , .

*Remarks*

In the definition, \(\lambda\) can be made equal to \(2^{|T|}\). Some sources define superstable as "stable and supersimple".

A theory \(T\) is

See , , .

In the definition, \(\lambda\) can be made equal to \(2^{|T|}\). Some sources define superstable as "stable and supersimple".

stable (NOP)

Given an infinite cardinal \(\kappa\), a theory is \(\kappa\)-stable if \[|S_n^M(A)|\leq\kappa\] for all models \(M\) and subsets \(A\subseteq M\) with \(|A|\leq\kappa\).

A theory is**stable** if it is \(\kappa\)-stable for some infinite cardinal \(\kappa\).

See , , .

*Alternate definition:*

A formula \(\varphi(x,y)\) has the order property (OP) if there are \((a_i)_{i\lt\omega}\) and \((b_i)_{i\lt\omega}\) such that \[\models\varphi(a_i,b_j)~\Leftrightarrow~ i\lt j.\] A theory is**stable (NOP)** if no formula has the order property.

See .

A theory is

See , , .

A formula \(\varphi(x,y)\) has the order property (OP) if there are \((a_i)_{i\lt\omega}\) and \((b_i)_{i\lt\omega}\) such that \[\models\varphi(a_i,b_j)~\Leftrightarrow~ i\lt j.\] A theory is

See .

o-minimal

Assume the language \(\mathcal{L}\) contains a binary relation <, and that *T* is an \(\mathcal{L}\)-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 , , .

characterization of forking

See , , .

dp-minimal

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

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

See , .

A theory is

See , .

NIP (dependent)

A formula \(\varphi(x,y)\) as the independence property (IP) if there are \((a_i)_{i\lt\omega}\) and \((b_I)_{I\subseteq\omega}\) such that \[\models\varphi(a_i,b_I)~\Leftrightarrow~i\in I.\]
A theory is **NIP (dependent)** if no formula has the independence property.

See , .

See , .

supersimple

A theory is **supersimple** if for all sets \(B\) and complete types \(p\in S_n(B)\) there is a finite subset \(A\subseteq B\) such that \(p\) does not fork over \(A\).

See , .

See , .

simple (NTP)

A theory \(T\) is **simple** if for all sets \(B\) and complete types \(p\in S_n(B)\) there is a subset \(A\subseteq B\), with \(|A|\leq|T|\), such that \(p\) does not fork over \(A\).

*Alternate Definition:*

A formula \(\varphi(x,y)\) has the tree property (TP) if there are \((a_\eta:\eta~\in\omega^{\lt\omega})\) and some \(k\geq 2\) such that**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 , ):

A formula \(\varphi(x,y)\) has the tree property (TP) if there are \((a_\eta:\eta~\in\omega^{\lt\omega})\) and some \(k\geq 2\) such that

- \(\forall~\sigma\in\omega^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
- \(\forall~\eta\in\omega^{\lt\omega}\), \(\{\varphi(x,a_{\sigma\hat{~}n}:n\lt\omega\}\) is \(k\)-inconsistent.

See , .

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 ).

NSOP1

A formula \(\varphi(x,y)\) has SOP_{1} if there are \((a_\eta:\eta~\in 2^{\lt\omega})\) such that
**NSOP**_{1} if no formula has SOP_{1}.

See ,, .

- \(\forall~\sigma\in 2^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
- \(\forall~\eta,\mu\in\omega^{\lt\omega}\), if \(\mu\hat{~}0\prec\eta\), then \(\{\varphi(x,a_{\mu\hat{~}1}),~\varphi(x,a_{\eta})\}\) is inconsistent.

See ,, .

NTP1 (NSOP2)

no proper examples known

A formula \(\varphi(x,y)\) has the tree property 1 (TP_{1}) if there are \((a_\eta:\eta~\in \omega^{\lt\omega})\) such that
**NTP**_{1} if no formula has TP_{1}.

See , .

*Remarks*

As a property of the theory, TP_{1} is equivalent to SOP_{2} (see ), which was recently shown to be maximal in Keisler's order (see ).

- \(\forall~\sigma\in \omega^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
- \(\forall\) incomparable \(\eta,\mu\in\omega^{\lt\omega}\), \(\{\varphi(x,a_\mu),~\varphi(x,a_{\eta})\}\) is inconsistent.

See , .

As a property of the theory, TP

NSOP3

no proper examples known

For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOP_{n}) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i \lt j \); but
\[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

A theory is**NSOP**_{n} if no formula has SOP_{n}.

See , .

A theory is

See , .

NSOP4

For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOP_{n}) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but
\[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

A theory is**NSOP**_{n} if no formula has SOP_{n}.

See , , .

*Remarks:*

There are results about non-existence of universal models of cardinality λ (for certain λ) when*T* has SOP_{4} (see , ).

For example:

If*T* has SOP_{4} and λ is regular such that κ^{+} < λ < 2^{κ} for some κ, then *T* has no universal model of cardinality λ.

A theory is

See , , .

There are results about non-existence of universal models of cardinality λ (for certain λ) when

For example:

If

NSOPn+1

For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOP_{n}) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but
\[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

A theory is**NSOP**_{n} if no formula has SOP_{n}.

See , .

A theory is

See , .

NSOP∞

A formula \(\varphi(x,y)\) has the *fully finitary strong order property* (SOP_{∞}) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but, for all \(n\geq 3\),
\[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is unsatisfiable.

A theory is**NSOP**_{∞} if no formula has SOP_{∞}.

See , .

*Remark*: SOP_{∞} is a not currently the standard abbreviation.

Two other dividing lines with a close relationship to the fully finitary strong order property are the*finitary strong order property* and the *strong order property*.
The definitions of these properties are obtained by replacing the formula \(\varphi(x,y)\) in the definition above with, respectively, a type in finitely many variables and a type in arbitrarily many variables.
Note that compactness does not imply these are the same, due to the requirement that the type omit *n*-cycles for all *n*.

A theory is

See , .

Two other dividing lines with a close relationship to the fully finitary strong order property are the

NSOP

A formula \(\varphi(x,y)\) has the strict order property (SOP) if there are \((a_i)_{i\lt\omega}\) 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 , .

See , .

NTP2

A formula \(\varphi(x,y)\) has the tree property 2 (TP_{2}) if there are \((a_{i,j})_{i,j\lt\omega}\) such that
**NTP**_{2} if no formula has the tree property 2.

See , .

- \(\forall~\sigma\in\omega^\omega\), \(\{\varphi(x,a_{n,\sigma(n)}):n\lt\omega\}\) is consistent.
- \(\forall~n\lt\omega,~\forall~ i\lt j\lt\omega\), \(\{\varphi(x,a_{n,i}),~\varphi(x,a_{n,j})\}\) is inconsistent.

See , .

forking = dividing

nonforking exists

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, a strongly minimal theory is \(\omega\)-stable, as indicated by the map.

See , .

For simplicity, this region of the map is for strongly minimal theories in a

Click a property above to highlight region and display details. Or click the map for specific region information.

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List of Examples

Implications Between Properties

Click the link to see the implications between all the above properties:

Diagram of Implications

Some other important equivalences:

Some of the proofs have been collected here.

Diagram of Implications

Some other important equivalences:

- stable ⇔ NIP and NSOP
- simple ⇔ NTP
_{1}and NTP_{2}

Some of the proofs have been collected here.

Open Regions

Each small region outlined in red currently contains no known examples, due to the following open questions:

- Are the implications, SOP
_{3}⇒ TP_{1}⇒ SOP_{1}, proper? See , , . - Is the SOP hierarchy strict inside of NTP2? See , Exercise III.7.12.

Open Examples

This list is intended for popular and/or important examples of theories with unknown positions on the map. Suggestions for such theories are welcomed.