# Forking

(in logic)

A notion introduced by S. Shelah [a8]. The general theory of forking is also known as stability theory, but more commonly, non-forking (the negation of forking) is defined as a certain well-behaved relation between a type and its extension (cf. Types, theory of).

Let $M$ be a sufficiently saturated model of a theory $T$ in a countable first-order language (cf. also Formal language; Model (in logic); Model theory). Given an $n$- tuple of variables ${\overline{x}\; }$ and $A \subset M$, a collection of formulas $\phi ( {\overline{x}\; } , {\overline{a}\; } )$ with parameters ${\overline{a}\; }$ in $A$ is called an $n$- type over $A$. For simplicity, only $1$- types will be considered; these are simply called types over $A$. A complete type is one which is maximal consistent. Let $S ( A )$ be the set of complete types over $A$.

Given a type $t = t ( x )$ and a formula $\phi = \phi ( x, {\overline{y}\; } )$, one defines the Morley $\phi$- rank of $t$, $\phi roman \AAh { \mathop{\rm rk} } ( t )$, inductively as follows: $\phi roman \AAh { \mathop{\rm rk} } ( t ) \geq 0$ if $t$ is consistent, for each natural number $n$, $\phi roman \AAh { \mathop{\rm rk} } ( t ) \geq n + 1$ if for every finite $s \subset t$ and natural number $m$ there are collections $p _ {1} \dots p _ {m}$ of $\phi$- formulas (with parameters from $M$) such that:

i) for $i \neq j$, $p _ {i}$ and $p _ {j}$ are contradictory, i.e. for some ${\overline{a}\; }$, $\phi ( x, {\overline{a}\; } )$ belongs to one of $p _ {i}$ and $p _ {j}$, and $\neg \phi ( x, {\overline{a}\; } )$ belongs to the other;

ii) $\phi roman \AAh { \mathop{\rm rk} } ( s \cup p _ {i} ) \geq n$.

Assume that $T$ is stable, i.e. for some infinite $\kappa$, whenever $| A | \leq \kappa$, then also $| {S ( A ) } | \leq \kappa$. (Equivalently, $\phi roman \AAh { \mathop{\rm rk} } ( t ) < \infty$ for every type $t$ and formula $\phi$.) Let $A \subset B$, $t \in S ( A )$, $u \in S ( B )$ be such that $u \supset t$. Then $u$ is called a non-forking extension of $t$, or it is said that $u$ does not fork over $A$, if for every formula $\phi$ with $\phi ( x, {\overline{b}\; } ) \in u$,

$$\phi roman \AAh { \mathop{\rm rk} } ( p ) = \phi roman \AAh { \mathop{\rm rk} } ( p \cap \phi ( x, {\overline{b}\; } ) ) ,$$

where $p \cap \phi ( x, {\overline{b}\; } )$ denotes the set $\{ {\theta \wedge \phi ( x, {\overline{b}\; } ) } : {\theta \in p } \}$.

Let $t \Sbs u$ mean that $u$ is a non-forking extension of $t$. Then $\Sbs$ is the unique relation on complete types satisfying the following Lascar axioms:

1) $\Sbs$ is preserved under automorphisms of $M$;

2) if $t \subset u \subset v$, then $t \Sbs v$ if and only if $t \Sbs u$ and $u \Sbs v$;

3) for any $t \in S ( A )$ and $B \supset A$ there exists a $u \in S ( B )$ such that $t \Sbs u$;

4) for any $t \in S ( A )$ there exist countable $A _ {0} \subset A$ and $t _ {0} \Sbs t$, where $t _ {0}$ is the restriction of $t$ to formulas with parameters from $A _ {0}$;

5) for any $t \in S ( A )$ and $A \subset B$,

$$\left | {\left \{ {u \in S ( B ) } : {t \Sbs u } \right \} } \right | \leq 2 ^ {\aleph _ {0} } .$$

The ultrapower construction (cf. also Ultrafilter) gives a systematic way of building non-forking extensions [a4].

For $c \in M$ one writes ${ \mathop{\rm tp} } ( c/A )$ for the type in $S ( A )$ realized by $c$. Given a set $A$ and $b,c \in M$, the following important symmetry property holds: ${ \mathop{\rm tp} } ( b/A \cup \{ c \} )$ does not fork over $A$ if and only if ${ \mathop{\rm tp} } ( c/A \cup \{ b \} )$ does not fork over $A$. If either holds, one says that $b$, $c$ are independent over $A$, and this notion is viewed as a generalization of algebraic independence.

Given $t \in S ( A )$, $B \supset A$, $u \in S ( B )$, and $u \supset t$, one says that $u$ is an heir of $t$ if for every $\phi ( x, {\overline{y}\; } )$( with parameters in $A$), $\phi ( x, {\overline{b}\; } ) \in u$ for some ${\overline{b}\; }$ in $B$ if and only if $\phi ( x, {\overline{a}\; } ) \in t$ for some ${\overline{a}\; }$ in $A$. One says that $u$ is definable over $A$ if for every $\phi ( x, {\overline{y}\; } )$ there is a formula $\theta ( {\overline{y}\; } )$ with parameters from $A$ such that for any ${\overline{b}\; }$ in $B$, $\phi ( x, {\overline{b}\; } ) \in u$ if and only if $M \vDash \theta ( {\overline{b}\; } )$.

$u$ is said to be a coheir of $t$ if $u$ is finitely satisfiable in $A$. So, for $b,c \in M$, ${ \mathop{\rm tp} } ( c/ A \cup \{ b \} )$ is an heir of ${ \mathop{\rm tp} } ( c/A )$ if and only if ${ \mathop{\rm tp} } ( b/A \cup \{ c \} )$ is a coheir of ${ \mathop{\rm tp} } ( b/A )$.

If $A$ is an elementary submodel of $M$, then $u \Sps t$ if and only if $u$ is an heir of $t$ if and only if $u$ is definable over $A$. In particular, in that case $t$ has a unique non-forking extension over any $B \supset A$. Then it follows from the forking symmetry that when $A$ is an elementary submodel, ${ \mathop{\rm tp} } ( b/A \cup \{ c \} )$ being a coheir of ${ \mathop{\rm tp} } ( b/A )$ is equivalent to being an heir.

For a comprehensive introduction of forking see [a1], [a2], [a4], [a5], and [a9]. For applications in algebra, see [a7] and [a6].

The techniques of forking have been extended to unstable theories. In [a2], this is done by considering only types that satisfy stable conditions. In [a3], types are viewed as probability measures and forking is treated as a special kind of measure extension. The stability assumption is then weakened to theories that do not have the independence property.

#### References

 [a1] J.T. Baldwin, "Fundamentals of stability theory" , Springer (1987) [a2] V. Harnik, L. Harrington, "Fundamentals of forking" Ann. Pure and Applied Logic , 26 (1984) pp. 245–286 [a3] H.J. Keisler, "Measures and forking" Ann. Pure and Applied Logic , 34 (1987) pp. 119–169 [a4] D. Lascar, B. Poizat, "An introduction to forking" J. Symb. Logic , 44 (1979) pp. 330–350 [a5] A. Pillay, "Introduction to stability theory" , Oxford Univ. Press (1983) [a6] A. Pillay, "The geometry of forking and groups of finite Morley rank" J. Symb. Logic , 60 (1995) pp. 1251–1259 [a7] M. Prest, "Model theory and modules" , Cambridge Univ. Press (1988) [a8] S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1990) (Edition: Revised) [a9] M. Makkai, "A survey of basic stability theory" Israel J. Math. , 49 (1984) pp. 181–238
How to Cite This Entry:
Forking. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Forking&oldid=46953
This article was adapted from an original article by Siu-Ah Ng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article