Editing Subalgebra (section)

== Subalgebras in universal algebra ==
{{main article|Substructure (mathematics)}}
In [[universal algebra]], a '''subalgebra''' of an [[structure (mathematical logic)|algebra]] ''A'' is a [[subset]] ''S'' of ''A'' that also has the structure of an algebra of the same type when the algebraic operations are restricted to ''S''. If the axioms of a kind of [[algebraic structure]] is described by [[variety (universal algebra)|equational laws]], as is typically the case in universal algebra, then the only thing that needs to be checked is that ''S'' is [[closed set|''closed'']] under the operations.

Some authors consider algebras with [[partial functions]]. There are various ways of defining subalgebras for these. Another generalization of algebras is to allow relations. These more general algebras are usually called [[structure (mathematical logic)|structures]], and they are studied in [[model theory]] and in [[theoretical computer science]]. For structures with relations there are notions of weak and of induced [[Substructure (mathematics)|substructure]]s.

=== Example ===
For example, the standard signature for [[group (mathematics)|groups]] in universal algebra is {{nowrap|(•,  <sup>−1</sup>, 1)}}. (Inversion and unit are needed to get the right notions of homomorphism and so that the group laws can be expressed as equations.) Therefore, a [[subgroup]] of a group ''G'' is a subset ''S'' of ''G'' such that:
* the identity ''e'' of ''G'' belongs to ''S'' (so that ''S'' is closed under the identity constant operation);
* whenever ''x'' belongs to ''S'', so does ''x''<sup>−1</sup> (so that ''S'' is closed under the inverse operation);
* whenever ''x'' and ''y'' belong to ''S'', so does {{nowrap|''x'' • ''y''}} (so that ''S'' is closed under the group's multiplication operation).