Subalgebra
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In universal algebra, a subalgebra of an 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. Since the axioms of algebraic structures in universal algebra are described by equational laws, the only thing that is necessary to check is that S is closed under the operations.
For example, 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-1 (so that S is closed under the inverse operation);
- whenever x and y belong to S, so does x * y (so that S is closed under the group's multiplication operation).
In the case of groups, it's actually enough to check that S is not the empty set and with x and y always also contains x-1 * y. However, in more general situations, it's not safe to make analogous assumptions, and every operation must be checked.
The term subalgebra is also used in the context of specific types of algebras such as associative algebras and Lie algebras. In those contexts, you should think specifically of the algebraic structures relevant to them.
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