Algebra (disambiguation)

Last updated September 29, 2025

The word algebra is used for various branches and structures of mathematics. For their overview, see Algebra. The name comes from the famous 10th century book Al-Jabr by Al-Khwarizmi.

Branches of mathematics

Mathematical structures

Vector space with multiplication

An "algebra", or to be verbose, an algebra over a field, is a vector space equipped with a bilinear vector product. Some notable algebras in this sense are:

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 $\mathbb {Z} _{2}$ -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

Other structures

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:

In category theory and computer science:
- F-algebra and F-coalgebra
- T-algebra

Other uses

Algebra Blessett, singer from the U.S, goes by the stage name Algebra
Algebra (book)