In
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
, a forking extension of a type is an extension of that type that is not whereas a non-forking extension is an extension that is as free as possible. This can be used to extend the notions of
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
or
algebraic independence to
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 In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model theory, model of a theory (logic), theory, generalizing the notion of dimension in algebraic geometry.
Definition
Fix a theory ''T'' with a ...
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''
0 of ''A'' of cardinality less than ''κ'' so that (''p'', ''A''
0) ≤ ''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''.
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Model theory