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In [[model theory]], a '''forking extension''' of a type is an extension of that type that is not {{clarify-span|free|date=August 2013}} whereas a '''non-forking extension''' is an extension that is as free as possible. This can be used to extend the notions of [[Linear independence|linear]] or [[algebraic independence]] to [[Stable theory|stable theories]]. These concepts were introduced by [[S. Shelah]].
==Definitions==
Suppose that ''A'' and ''B'' are models of some complete ω-stable theory ''T''.
If ''p'' is a type of ''A'' and ''q'' is a type of ''B'' containing ''p'', then ''q'' is called a '''forking extension''' of ''p'' if its [[Morley rank]] is smaller, and a '''nonforking extension''' if it has the same Morley rank.
==Axioms==
Let ''T'' be a stable complete theory. The non-forking relation ≤ for types over ''T'' is the unique relation that satisfies the following axioms:
#If ''p''≤ ''q'' then  ''p''⊂''q''. If ''f'' is an elementary map then ''p''≤''q'' if and only if ''fp''≤''fq''
#If ''p''⊂''q''⊂''r'' then ''p''≤''r'' if and only if ''p''≤''q'' and ''q''≤ ''r''
#If ''p'' is a type of ''A'' and ''A''⊂''B'' then there is some type ''q'' of ''B'' with ''p''≤''q''.
#There is a cardinal κ such that if ''p'' is a type of ''A'' then there is a subset ''A''<sub>0</sub> of ''A'' of cardinality less than κ so that (''p''|''A''<sub>0</sub>) ≤ ''p'', where | stands for restriction.
#For any ''p'' there is a cardinal λ such that there are at most λ non-contradictory types ''q'' with ''p''≤''q''.
==References==
*{{citation|doi=10.1016/0168-0072(84)90005-8
|mr=0747686
|last1=Harnik|first1= Victor|last2= Harrington|first2=Leo
|title=Fundamentals of forking
|journal=Ann. Pure Appl. Logic |volume=26 |year=1984|issue= 3|pages= 245–286|doi-access=free}}
*{{citation|title=An Introduction to Forking
|first1=        Daniel |last1=Lascar|first2= Bruno|last2= Poizat
|journal=        The Journal of Symbolic Logic|volume= 44|issue= 3|year=1979|pages= 330–350|doi=10.2307/2273127|jstor=2273127|publisher=Association for Symbolic Logic}}
*{{citation|title=A survey of basic stability theory, with particular emphasis on orthogonality and regular types
|doi=10.1007/BF02760649 |doi-access=free
|journal=[[Israel Journal of Mathematics]]
|volume=49|year=1984
|first=M.|issue=1–3|last= Makkai|pages=181–238|s2cid=121533246
}}
*{{Citation | last1=Marker | first1=David | title=Model Theory: An Introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-98760-6 | year=2002}}
*{{springer|id=F/f110150|title=Forking|first=Siu-Ah|last= Ng}}
*{{Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Classification theory and the number of nonisomorphic models | orig-year=1978 | publisher=Elsevier | edition=2nd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-70260-9 | year=1990 | url-access=registration | url=https://archive.org/details/classificationth0092shel }}
[[Category:Model theory]]