Inmodel theory, afirst-order theory is calledmodel complete if every embedding of its models is anelementary embedding. Equivalently, everyfirst-order formula is equivalent to a universal formula.This notion was introduced byAbraham Robinson.

Acompanion of a theoryT is a theoryT* such that every model ofT can be embedded in a model ofT* and vice versa.

Amodel companion of a theoryT is a companion ofT 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 ifT is an${\displaystyle {\mathrm{\aleph }}_0}$-categorical theory, then it always has a model companion.^[1][2]

Amodel completion for a theoryT is a model companionT* such that for any modelM ofT, the theory ofT* together with thediagram ofM iscomplete. Roughly speaking, this means every model ofT is embeddable in a model ofT* in a unique way.

IfT* is a model companion ofT then the following conditions are equivalent:^[3]

• T* is a model completion ofT
• T has theamalgamation property.

IfT also has universal axiomatization, both of the above are also equivalent to:

• T* haselimination of quantifiers

• Any theory withelimination of quantifiers is model complete.
• The theory ofalgebraically closed fields is the model completion of the theory of fields. It is model complete but not complete.
• The model completion of the theory ofequivalence relations is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.
• The theory ofreal closed fields, in the language ofordered rings, is a model completion of the theory ofordered fields (or even ordereddomains).
• The theory of real closed fields, in the language ofrings, is the model companion for the theory offormally real fields, but is not a model completion.

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

IfT is a model complete theory and there is a model ofT that embeds into any model ofT, thenT is complete.^[4]

• Chang, Chen Chung;Keisler, H. Jerome (1990) [1973].Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier.ISBN 978-0-444-88054-3.

• Chang, Chen Chung;Keisler, H. Jerome (2012) [1990].Model Theory. Dover Books on Mathematics (3rd ed.).Dover Publications. p. 672.ISBN 978-0-486-48821-9.

• Hirschfeld, Joram; Wheeler, William H. (1975). "Model-completions and model-companions".Forcing, Arithmetic, Division Rings. Lecture Notes in Mathematics. Vol. 454. Springer. pp. 44–54.doi:10.1007/BFb0064085.ISBN 978-3-540-07157-0.MR 0389581.

• Marker, David (2002).Model Theory: An Introduction.Graduate Texts in Mathematics 217. New York: Springer-Verlag.ISBN 0-387-98760-6.

• Saracino, D. (August 1973). "Model Companions for ℵ₀-Categorical Theories".Proceedings of the American Mathematical Society.39 (3):591–598.

• Simmons, H. (1976). "Large and Small Existentially Closed Structures".Journal of Symbolic Logic.41 (2):379–390.

• Mathematical logic
• Model theory

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