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

See also coalgebra, 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