Algebra (disambiguation)

Branches of mathematics

Algebra over a field, or algebra, a vector space equipped with a bilinear vector product.

In ring theory and linear algebra:
- Algebra over a commutative ring, a module equipped with a bilinear product. Generalization of algebras over a field
- Associative algebra, a module equipped with an associative bilinear vector product
- Superalgebra, a $ℤ_{2}$ -graded algebra
- Lie algebras, Poisson algebras, and Jordan algebras, important examples of (potentially) nonassociative algebras
In functional analysis:
- Banach algebra, an associative algebra A over the real or complex numbers which at the same time is also a Banach space
- Operator algebra, continuous linear operators on a topological vector space with multiplication given by the composition
- *-algebra, An algebra with a notion of adjoints
  - C*-algebra, a Banach algebra equipped with a unary involution operation
  - Von Neumann algebra (or W*-algebra)
Coalgebra is the dual notion

A different class of "algebras" consists of objects which generalize logical connectives, sets, and lattices.

In logic:
- Relational algebra, in which a set of finitary relations that is closed under certain operators
- Boolean algebra and Boolean algebra (structure)
- Heyting algebra
In measure theory:
- Algebra over a set, a collection of sets closed under finite unions and complementation
- Sigma algebra, a collection of sets closed under countable unions and complementation

"Algebra" can also describe more general structures: