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In model theory, a first-order theory is called model complete if every embedding of 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.

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:

• 
  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.


• 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.

References

• 
  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.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


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