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Stable theories are a class of theories generalizing strongly minimal theories, uncountably categorical theories, and totally transcendental theories. Stable theories are the primary focus of stability theory.

## Definition of stabilityEdit

In this article, “type” will mean complete $n$-type for some $n$.

Fix some complete theory $T$, and let $\mathbb{U}$ be a monster model of $T$. We say that $T$ is $\kappa$-stable if $\kappa \ge |T|$ and for every set $A \subset \mathbb{U}$ of cardinality at most $\kappa$, there are at most $\kappa$ types over $A$: $|S_n(A)| \le \kappa$ for every $n$.

$T$ is said to be stable if it is $\kappa$-stable for some $\kappa \ge |T|$.

There are many equivalent definitions of stability; we will list some below.

## ExamplesEdit

Notable examples of stable theories include

Any theory in which an infinite total order is definable (or interpretable) will not be stable. Consequently, theories such as DLO, RCF, ACVF, and the true theory of arithmetic are all unstable.

## Forking CalculusEdit

Main article: Forking in stable theories

If $T$ is stable and $A, B, C$ are subsets of the monster, there is a canonical notion of what it means for $A$ and $B$ to be independent over $C$, denoted $A \downarrow_C B$. There is a closely related notion, that of a “non-forking extension” of complete types: $\operatorname{tp}(a/BC)$ is a non-forking extension of $\operatorname{tp}(a/C)$ if $a \downarrow_C B$, and is a forking extension of $\operatorname{tp}(a/C)$ otherwise. The non-forking extensions of a type $p$ can be thought of as the “generic” or “canonical” extensions of $p$ to larger sets. The relation $A \downarrow_C B$ can be recovered from the notion of non-forking: $A \downarrow_C B$ holds if and only if $\operatorname{tp}(a/BC)$ is a non-forking extension of $\operatorname{tp}(a/C)$ for each finite tuple $a$ from $A$.

These notions satisfy a number of basic axioms, such as

• Symmetry: $A$ is independent from $B$ over $C$ if and only if $B$ is independent from $A$ over $C$.
• Transitivity: if $p \subset q \subset r$ are complete types over subsets of the monster, then $r$ is a non-forking extension of $p$ if and only if $r$ is a non-forking extension of $q$ and $q$ is a non-forking extension of $p$.
• Local character: there is a cardinal $\kappa$ such that every type $p$ is a non-forking extension of a type over a set of size less than $\kappa$.

There are six or seven other axioms besides these. All of these can be expressed in terms of the relation $\downarrow$. The formal manipulation of $\downarrow$ using these axioms is sometimes called forking calculus. The forking calculus actually characterizes stable theories, in the sense that if $T$ is a complete theory for which there is a ternary relation $\downarrow^*$ satisfying the axioms of forking independence in a stable theory, then $T$ is stable and $\downarrow^*$ is forking independence.

The ternary relation $\downarrow$ is the basis for other fundamental definitions in stability theory, such as Lascar rank, stationarity, independence, weight, domination, orthogonality, and regularity.

## Ranks in stable theoriesEdit

Several notions of “rank” or “dimension” play an important role in stable theories (particularly superstable theories). Each kind of rank assigns an ordinal number to a partial type $\Sigma(x)$. This number should be thought of as the number of degrees of freedom which the variable $x$ has, after being constrained by the statements in $\Sigma(x)$. If $\Sigma(x)$ is a finite type, i.e., a formula $\phi(x)$, then the rank of $\phi(x)$ can be thought of as the “dimension” of the definable set $\phi(\mathbb{U})$. On the other hand, the rank of a complete type $\operatorname{tp}(a/B)$ can be thought of as a generalization of $tr.deg(a/B)$.

The three most commonly used ranks in stability theory are Morley rank $RM(-)$, Lascar rank $U(-)$, and Shelah -rank $R^\infty(-)$. Each of these ranks is ordinal valued, but can take the failure value $\infty$, which is thought of as being greater than any ordinal number. The ranks are related as follows: $U(\Sigma) \le R^\infty(\Sigma) \le RM(\Sigma)$ for any partial type $\Sigma$. The totally transcendental theories are exactly the stable theories for which Morley rank is never $\infty$. The superstable theories are exactly the stable theories for which Lascar rank is never $\infty$, and are also exactly the stable theories for which $R^\infty(-) < \infty$. (Superstable theories can also be characterized as the theories which are $\kappa$-stable for all sufficiently large $\kappa$.) In practice, many of the mathematically interesting examples of stable theories are superstable; but SCF is a notable counter-example.

If $R(-)$ is one of these three ranks, it is customary to write $R(a/B)$ for $R(\operatorname{tp}(a/B))$. If $D$ is definable (or type-definable), it is customary to write $R(D)$ for the rank of the partial type cutting out $D$.

A number of fundamental properties are shared by these ranks. Letting $R(-)$ denote one of Morley rank, Lascar rank, or Shelah $\infty$-rank,

• If $D$ is type-definable over some small set $B$, then $R(D) = \sup_{a \in D} R(a/D)$.
• If $D \subset D'$, then $R(D) \le R(D')$.
• $R(D \cup D') = \max(R(D),R(D'))$
• If $B' \supseteq B$, then $R(a/B') \le R(a/B)$.
• $R(D) = 0$ if and only if $D$ is finite.
• $R(a/B) = 0$ if and only if $a$ is algebraic over $B$.
• If $a' \in \operatorname{acl}(Ba)$, then $R(a'/B) \le R(a/B)$.
• If $D$ and $D'$ are two type-definable sets in definable bijection, then $R(D) = R(D')$. More generally, if there is a definable surjection from $D$ to $D'$, then $R(D) \ge R(D')$.

Each of these ranks governs forking, when less than $\infty$. That is, if $R(a/BC) < \infty$, then $a \downarrow_C B \iff R(a/BC) = R(a/C),$ $a \not\downarrow_C B \iff R(a/BC) < R(a/C)$ In other words, the non-forking extensions of a complete type are exactly the extensions having the same rank.

Morley rank and Shelah $\infty$-rank both possess the following continuity property: $R(\Sigma(x))$ is the minimum of $R(\Sigma_0(x))$, for $\Sigma_0(x)$ a finite subtype of $\Sigma(x)$. For a complete type $\operatorname{tp}(a/B)$, this means that given $a$ and $B$, $R(a/B) = R(\phi(x))$ for some $\phi(x) \in \operatorname{tp}(a/B)$. In particular, all types in a neighborhood of $\operatorname{tp}(a/B)$ have rank no greater than $R(a/B)$. This means that the rank function on $S_n(B)$ is semi-continuous.

Lascar rank fails to have this property in general. However, it satisfies the extremely useful Lascar inequality: $U(a/Cb) + U(b/C) \le U(ab/C) \le U(a/Cb) \oplus U(b/C).$ Here $+$ denotes the usual ordinal sum, while $\oplus$ denotes the so-called “natural sum.” For finite numbers, these agree, making the Lascar inequality into an equality. There is also an adjunct statement to the Lascar inequality, namely $U(ab/C) = U(a/C) \oplus U(b/C) \Leftarrow a \downarrow_C b.$ From this, one can show that if $D_1$ and $D_2$ are non-empty definable sets, then $U(D_1 \times D_2) = U(D_1) \oplus U(D_2)$.

## Stable GroupsEdit

Definable groups in stable structures have a number of nice properties, including the following:

• Type-definable groups always sit inside of definable groups, and any type-definable subgroup of a definable group is an intersection of definable groups. (This fails in RCF, for example, because in the additive group, the group of infinitesimals fails to be an intersection of definable subgroups of the additive group.)
• In a totally transcendental setting, type-definable groups are always definable.
• If $G$ is a type-definable group, then $G^0$ (the intersection of all relatively definable subgroups of $G$ of finite index) exists, i.e., is type-definable over a small set. In the totally transcendental setting, $G^0$ is definable and has finite index in $G$.
• If $G = G^0$, then there is a canonical generic type of $G$. This type is uniquely characterized by the fact that it is stabilized by left and right translation. In the totally transcendental setting, this generic type is the unique type of maximal Morley rank in $G$.
• When $G \ne G^0$, there is still a notion of a generic type in $G$; there is one generic type in each coset of $G^0$.
• The Baldwin-Saxl condition: if $\mathcal{H}$ is a uniformly definable family of subgroups of a stable group $G$, then any intersection of members of $\mathcal{H}$ is a finite intersection of members of $\mathcal{H}$.

In the case of groups of finite Morley rank, many structural results are known, using tools such as Zilber’s indecomposability theorem. For example, it is known that if $G$ is a group of finite Morley rank, then $G$ is simple if and only if $G$ is definably simple. That is, if $G$ has a normal subgroup, then $G$ has a definable normal subgroup. Additionally, Morley rank in $G$ agrees with Lascar rank and is definable. Conjecturally, all simple groups of finite Morley rank are (as abstract groups) algebraic groups over algebraically closed fields.

Groups arise naturally in stability theory in several ways. In the analysis of uncountably categorical theories, or more generally when studying internality, binding groups appear and govern some of the structure. For example, in Poizat’s book on stable groups, there is a theorem saying that if $T$ is uncountably categorical and has no definable groups, then $T$ is almost strongly minimal.

Groups also arise from the group configuration theorem, which constructs a group from a certain configuration occurring in the pregeometry of a strongly minimal set (or more generally, a rank 1 type).

One-based groups also play an important role in the model-theoretic proofs of function field Mordell-Lang.

## Stability SpectraEdit

The set of $\kappa$ such that $T$ is $\kappa$-stable is called the stability spectrum of $T$. In Poizat’s introductory Model Theory book (in chapter 13$\pm$1?), the possible stability spectra are classified. It turns out that the stability spectrum of $T$ depends on $\lambda$, the smallest cardinal such that $T$ is $\lambda$-stable, and $\kappa$, the smallest cardinal $\kappa$ which works in the local character property of forking. (That is, $\kappa$ is the smallest cardinal with the property that if $a$ is a finite tuple and $C$ is a set, then there is some $C' \subseteq C$ with $|C'| < \kappa$ such that $a \downarrow_{C'} C$.) Somebody should add the correct statement of the theorem here.

For countable theories, it turns out that there are three possibilities:

• $T$ is $\kappa$-stable for all infinite $\kappa$. This happens exactly if $T$ is totally transcendental.
• $T$ is $\kappa$-stable for $\kappa \ge 2^{\aleph_0}$. This happens exactly if $T$ is superstable but not totally transcendental.
• $T$ is $\kappa$-stable for those $\kappa$ such that $\kappa = \kappa^{\aleph_0}$. This happens when $T$ is stable but not superstable.

## Stability and Classification TheoryEdit

Stability theory was originally motivated by things such as Morley’s Theorem, and more generally, classification theory. The goal of this subject is to classify theories $T$ according to their spectra, that is, according to how many models they have of each uncountable cardinality. A basic result in this direction is that if a countable theory $T$ is not stable, then for any uncountable $\kappa$, $T$ has $2^\kappa$ distinct models up to isomorphism, which is as large as possible. Such theories are thought of as "unclassifiable," because there is no hope of classifying their models. More generally, Shelah proved that if $T$ is not superstable, then $T$ is unclassifiable. The focus of classification theory is therefore on superstable theories. (Can someone verify the claims made in this paragraph?)

## Equivalent Definitions of StabilityEdit

There are many equivalent definitions of stability, some of which are easier to check than $\kappa$-stability in various settings.

All of the following are equivalent, for a complete theory $T$:

• $T$ is $\kappa$-stable for some $\kappa$
• No formula $\phi(x;y)$ has the order property. That is, one cannot find a formula $\phi(x;y)$ and tuples $a_1, b_1, a_2, b_2, \ldots$ from the monster $\mathbb{U}$ such that $\models \phi(a_i;b_j)$ holds if and only if $i < j$.
• Every type over a model is definable.
• Every type over any set is definable.
• Types over models have unique heirs.
• Types over models have unique coheirs.
• Every indiscernible sequence is totally indiscernible.
• $T$ is NIP and does not have the strict order property. That is, $T$ is NIP and does not interpret a partial order with an infinite chain.
• For every countable set $A$ and formula $\phi(x;y)$, the set $S_\phi(A)$ of $\phi$-types over $A$ is countable, or equivalently, contains no perfect set.

Many of these properties can be checked on the level of 1-types. Specifically, the following conditions are also equivalent to stability:

• $\kappa$-stability holds on the level of 1-types for some $\kappa$. That is, there is some $\kappa \ge |T|$ such that for every $A \subset \mathbb{U}$ with $|A| \le \kappa$, we have $|S_1(A)| \le \kappa$.
• No formula $\phi(x;y)$ with $x$ a singleton (rather than a tuple) has the order property.
• Every 1-type over a model is definable.
• Every 1-type over a model has a unique coheir.
• Every 1-type over a model has a unique heir.

These conditions are sometimes easier to check. For example, if $T$ is a strongly minimal theory, it is easy to check that no formula $\phi(x;y)$ with $x$ a singleton has the order property. (If $\phi(x;y)$ has the order property, then by compactness one can find $a_i, b_i$ for $i \in \mathbb{Z}$ such that $\models \phi(a_i;b_j)$ holds if and only if $i < j$. Then $\phi(\mathbb{U};b_0)$ contains $a_{-1}, a_{-2}, \ldots$ and doesn’t contain $a_1, a_2, \ldots$. Therefore it is neither finite nor cofinite, contradicting strong minimality.)

If $T$ is a stable theory, so is $T^{eq}$. Any definitional expansion of $T$ is stable. Any reduct of a stable theory is a stable theory. We can name any number of elements from $\mathbb{U}$ without losing stability.

## Miscellaneous Facts about Stable TheoriesEdit

Other noteworthy facts about stable theories are the following:

• An $A$-indiscernible sequence is always totally $A$-indiscernible.
• Every definable set is stably embedded. That is, if $D$ is a definable set, and $D'$ is some other definable set such that $D' \cap D^n$ makes sense, then $D' \cap D^n$ is of the form $D'' \cap D^n$ for some set $D''$ definable with parameters from $D$. In other words, definable subsets of $D$ or $D^n$ can always be defined with parameters from inside $D$. Poizat calls this the “Parameter Separation Theorem,” but the usual name is “stable embeddedness.” This turns out to be a direct consequence of the fact that complete types over $D(\mathbb{U})$ are $D(\mathbb{U})$-definable.
• Strong type, Kim-Pillay strong type, and Lascar strong type are all equivalent. One proves this as follows: if $a$ and $b$ have the same strong type over a set $C$, let $p$ be the unique non-forking extension of this strong type. Let $c_1, c_2, c_3, \ldots$ be a Morley sequence for $p$ over $abC$, so $c_i$ realizes $p | ab \operatorname{acl}^{eq}(C)$. Then the sequences $a, c_1, c_2, \ldots$ and $b, c_1, c_2, \ldots$ are both indiscernible over $C$, ensuring that $a$ and $b$ have the same Lascar strong type over $C$.
• Every model of a stable theory has an elementary extension which is saturated. Equivalently, arbitrarily large saturated models exist, without any set-theoretic assumptions like the continuum hypothesis or the existence of inaccessible cardinals. This can be seen by taking a sufficiently large cardinal $\kappa$ such that $T$ is $\kappa$-stable, taking a model of cardinality $\kappa$, realizing all types over it in some bigger model, and repeating this process $\kappa^+$ times. The result will be a $\kappa^+$-saturated model of cardinality $\kappa^+$.