{{Short description|Concept in model theory}} In model theory, a first-order theory is called '''model complete''' if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.

==Model companion and model completion==

A '''companion''' of a theory ''T'' is a theory ''T''* such that every model of ''T'' can be embedded in a model of ''T''* and vice versa.

A '''model companion''' of a theory ''T'' is a companion of ''T'' that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if ''T'' is an <math>\aleph_0</math>-categorical theory, then it always has a model companion.{{sfn|Saracino|1973}}{{sfn|Simmons|1976}}

A '''model completion''' for a theory ''T'' is a model companion ''T''* such that for any model ''M'' of ''T'', the theory of ''T''* together with the diagram of ''M'' is complete. Roughly speaking, this means every model of ''T'' is embeddable in a model of ''T''* in a unique way.

If ''T''* is a model companion of ''T'' then the following conditions are equivalent:{{sfn|Chang|Keisler|2012}} * ''T''* is a model completion of ''T'' * ''T'' has the amalgamation property.

If ''T'' also has universal axiomatization, both of the above are also equivalent to: * ''T''* has elimination of quantifiers

==Examples==

*Any theory with elimination of quantifiers is model complete. *The theory of algebraically closed fields is the model completion of the theory of fields. It is model complete but not complete. *The model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements. * The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains). *The theory of real closed fields, in the language of rings, is the model companion for the theory of formally real fields, but is not a model completion.

==Non-examples==

*The theory of dense linear orders with a first and last element is complete but not model complete. *The theory of groups (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

==Sufficient condition for completeness of model-complete theories==

If ''T'' is a model complete theory and there is a model of ''T'' that embeds into any model of ''T'', then ''T'' is complete.{{sfn| Marker|2002}}

==Notes==

{{reflist}}

==References==

* {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=1990|orig-year=1973|publisher=Elsevier|edition=3rd|series=Studies in Logic and the Foundations of Mathematics|isbn=978-0-444-88054-3}}

* {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=2012|orig-year=1990|publisher=Dover Publications|edition=3rd|series=Dover Books on Mathematics|pages=672|isbn=978-0-486-48821-9}}

* {{cite book|last=Hirschfeld|first=Joram|last2=Wheeler|first2=William H.|chapter=Model-completions and model-companions|title=Forcing, Arithmetic, Division Rings|series=Lecture Notes in Mathematics|publisher=Springer|volume=454|pages=44–54|year=1975|isbn=978-3-540-07157-0|mr=0389581|doi=10.1007/BFb0064085}}

* {{cite book | last=Marker | first=David | title= Model Theory: An Introduction | publisher=Springer-Verlag|location= New York| year=2002 | isbn=0-387-98760-6| series=Graduate Texts in Mathematics 217}}

* {{cite journal |last=Saracino |first=D. |title=Model Companions for ℵ<sub>0</sub>-Categorical Theories |journal=Proceedings of the American Mathematical Society |volume=39 |issue=3 |date=August 1973 |pages=591–598 }}

* {{cite journal |last=Simmons |first=H. |title=Large and Small Existentially Closed Structures |journal=Journal of Symbolic Logic |volume=41 |issue=2 |year=1976 |pages=379–390 }}

{{Mathematical logic}} {{Authority control}}

Category:Mathematical logic Category:Model theory