Editing Elementary equivalence (section)

==Tarski–Vaught test==

The '''Tarski–Vaught test''' (or '''Tarski–Vaught criterion''') is a necessary and sufficient condition for a substructure ''N'' of a structure ''M'' to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure.

Let ''M'' be a structure of signature ''σ'' and ''N'' a substructure of ''M''. Then ''N'' is an elementary substructure of ''M'' if and only if for every first-order formula ''φ''(''x'',&nbsp;''y''<sub>1</sub>,&nbsp;…,&nbsp;''y''<sub>''n''</sub>) over ''σ'' and all elements ''b''<sub>1</sub>,&nbsp;…,&nbsp;''b''<sub>''n''</sub> from ''N'', if ''M'' <math>\models</math> {{exist}}''x''&nbsp;''φ''(''x'',&nbsp;''b''<sub>1</sub>,&nbsp;…,&nbsp;''b''<sub>''n''</sub>), then there is an element ''a'' in ''N'' such that ''M'' <math>\models</math> ''φ''(''a'',&nbsp;''b''<sub>1</sub>,&nbsp;…,&nbsp;''b''<sub>''n''</sub>).