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Algebraic structure

In abstract algebra, an algebraic structure consists of a set together with one or more operations on the set which satisfy certain axioms. In case there are no ambiguities, we usually identify the set with the algebraic structure. For example, a group (G,*) is usually referred simply as a group G.

Depending on the operations and axioms, the algebraic structures get their names. The following is a partial list of algebraic structures:

Those statements that apply to all algebraic structures collectively are investigated in the branch of mathematics known as universal algebra.

Algebraic structures can also be defined on sets with additional non-algebraic structures, such as topological spaces. For example, a topological group is a topological space with a group structure such that the operations of multiplication and taking inverses are continuous; a topological group has both a topological and an algebraic structure. Other common examples are topological vector spaces and Lie groups.

Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s). In this way, every algebraic structure defines a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure.\n