In abstract algebra, an algebraic structure consists of a set together with one or more operations on the set which satisfy certain axioms. Depending on the operations and axioms, the algebraic structures get their names. The following is a partial list of algebraic structures:
- Magma: a set with a single binary operation
- Semigroup: an associative magma
- Monoid: a semigroup with an identity element
- Group: a monoid in which every element has an inverse
- Abelian group: a commutative group
- Ring: a set with an abelian group operation as addition, together with a monoid operation as multiplication, satisfying distributivity
- Field: a ring in which the the non-zero elements form an abelian group under multiplication
- Module over a given ring R: a set with an abelian group operation as addition, together with an additive unary operation of scalar multiplication for every element of R, with an associativity condition linking scalar multiplication to multiplication in R
- Vector space: a module over a field
- Algebra: a module or vector space together with a bilinear operation as multiplication
- Associative algebra: an algebra whose multiplication is associative
- Commutative algebra: an associative algebra whose multiplication is commutative
- Lattice: a set with two commutative, associative, idempotent operations satisfying the absorption law
- Boolean algebra: a bounded, distributive, complemented lattice
- Set: although some mathematicians would not count it, a set can itself be thought of as a degenerate algebraic structure, one that has zero operations defined on it
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.
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, which may be regarded as a category of sets with extra structure in the category-theoretic sense. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms.