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''(in logic)''
 
''(in logic)''
  
 
A notion introduced by S. Shelah [[#References|[a8]]]. The general theory of forking is also known as [[Stability theory|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|Types, theory of]]).
 
A notion introduced by S. Shelah [[#References|[a8]]]. The general theory of forking is also known as [[Stability theory|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|Types, theory of]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101501.png" /> be a sufficiently saturated model of a theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101502.png" /> in a countable first-order language (cf. also [[Formal language|Formal language]]; [[Model (in logic)|Model (in logic)]]; [[Model theory|Model theory]]). Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101503.png" />-tuple of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101505.png" />, a collection of formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101506.png" /> with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101507.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f1101508.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015010.png" />-type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015011.png" />. For simplicity, only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015012.png" />-types will be considered; these are simply called types over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015013.png" />. A complete type is one which is maximal consistent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015014.png" /> be the set of complete types over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015015.png" />.
+
Let $  M $
 +
be a sufficiently saturated model of a theory $  T $
 +
in a countable first-order language (cf. also [[Formal language|Formal language]]; [[Model (in logic)|Model (in logic)]]; [[Model theory|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015016.png" /> and a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015017.png" />, one defines the Morley <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015019.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015021.png" />, inductively as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015023.png" /> is consistent, for each natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015025.png" /> if for every finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015026.png" /> and natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015027.png" /> there are collections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015029.png" />-formulas (with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015030.png" />) such that:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015033.png" /> are contradictory, i.e. for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015035.png" /> belongs to one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015037.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015038.png" /> belongs to the other;
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015039.png" />.
+
ii) $  \phi roman \AAh { \mathop{\rm rk} } ( s \cup p _ {i} ) \geq  n $.
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015040.png" /> is stable, i.e. for some infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015041.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015042.png" />, then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015043.png" />. (Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015044.png" /> for every type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015045.png" /> and formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015046.png" />.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015049.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015050.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015051.png" /> is called a non-forking extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015052.png" />, or it is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015053.png" /> does not fork over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015054.png" />, if for every formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015056.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015057.png" /></td> </tr></table>
+
$$
 +
\phi roman \AAh { \mathop{\rm rk} } ( p ) = \phi roman \AAh { \mathop{\rm rk} } ( p \cap \phi ( x, {\overline{b}\; } ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015058.png" /> denotes the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015059.png" />.
+
where $  p \cap \phi ( x, {\overline{b}\; } ) $
 +
denotes the set $  \{ {\theta \wedge \phi ( x, {\overline{b}\; } ) } : {\theta \in p } \} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015060.png" /> mean that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015061.png" /> is a non-forking extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015062.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015063.png" /> is the unique relation on complete types satisfying the following Lascar axioms:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015064.png" /> is preserved under automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015065.png" />;
+
1) $  \Sbs $
 +
is preserved under automorphisms of $  M $;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015067.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015069.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015071.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015073.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015074.png" /> there exist countable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015077.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015078.png" /> to formulas with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015079.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015081.png" />,
+
5) for any $  t \in S ( A ) $
 +
and $  A \subset  B $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015082.png" /></td> </tr></table>
+
$$
 +
\left | {\left \{ {u \in S ( B ) } : {t \Sbs u } \right \} } \right | \leq  2 ^ {\aleph _ {0} } .
 +
$$
  
 
The ultrapower construction (cf. also [[Ultrafilter|Ultrafilter]]) gives a systematic way of building non-forking extensions [[#References|[a4]]].
 
The ultrapower construction (cf. also [[Ultrafilter|Ultrafilter]]) gives a systematic way of building non-forking extensions [[#References|[a4]]].
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015083.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015084.png" /> for the type in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015085.png" /> realized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015086.png" />. Given a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015088.png" />, the following important symmetry property holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015089.png" /> does not fork over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015090.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015091.png" /> does not fork over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015092.png" />. If either holds, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015094.png" /> are independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015095.png" />, and this notion is viewed as a generalization of [[Algebraic independence|algebraic independence]].
+
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|algebraic independence]].
  
Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015098.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f11015099.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150100.png" /> is an heir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150101.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150102.png" /> (with parameters in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150103.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150104.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150105.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150106.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150107.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150108.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150109.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150110.png" /> is definable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150111.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150112.png" /> there is a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150113.png" /> with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150114.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150115.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150117.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150118.png" />.
+
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}\; } ) $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150119.png" /> is said to be a coheir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150120.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150121.png" /> is finitely satisfiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150122.png" />. So, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150124.png" /> is an heir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150125.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150126.png" /> is a coheir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150127.png" />.
+
$  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150128.png" /> is an elementary submodel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150129.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150130.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150131.png" /> is an heir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150132.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150133.png" /> is definable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150134.png" />. In particular, in that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150135.png" /> has a unique non-forking extension over any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150136.png" />. Then it follows from the forking symmetry that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150137.png" /> is an elementary submodel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150138.png" /> being a coheir of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110150/f110150139.png" /> is equivalent to being an heir.
+
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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], and [[#References|[a9]]]. For applications in algebra, see [[#References|[a7]]] and [[#References|[a6]]].
 
For a comprehensive introduction of forking see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], and [[#References|[a9]]]. For applications in algebra, see [[#References|[a7]]] and [[#References|[a6]]].

Latest revision as of 19:39, 5 June 2020


(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