Editing Embedding (section)

===Universal algebra and model theory===
{{further|Substructure (mathematics)|Elementary equivalence}}
If <math>\sigma</math> is a [[signature (logic)|signature]] and <math>A,B</math> are <math>\sigma</math>-[[structure (mathematical logic)|structures]] (also called <math>\sigma</math>-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a <math>\sigma</math>-embedding exactly if all of the following hold:
* <math>h</math> is injective,
* for every <math>n</math>-ary function symbol <math>f \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))</math>,
* for every <math>n</math>-ary relation symbol <math>R \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>A \models R(a_1,\ldots,a_n)</math> iff <math>B \models R(h(a_1),\ldots,h(a_n)).</math>

Here <math>A\models R (a_1,\ldots,a_n)</math> is a model theoretical notation equivalent to <math>(a_1,\ldots,a_n)\in R^A</math>. In model theory there is also a stronger notion of [[elementary embedding]].