Algebra (disambiguation)

The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra.

The bare word "algebra"

The bare word "algebra" can mean:

• Elementary algebra
• Abstract algebra
• Algebra over a field

In universal algebra, algebra has an axiomatic definition, roughly as an instance of any of a number of algebraic structures, such as groups, rings, etc.

Branches of mathematics

• Elementary algebra, i.e. "high-school algebra."
• Abstract algebra
• Linear algebra
• Relational algebra
• Universal algebra

The term is also traditionally used for the field of:

• Computer algebra, dealing with software systems for symbolic mathematical computation, which often offer capabilities beyond what is normally understood to be "algebra".

Mathematical structures

Vector space with multiplication

• "Algebra", or to be verbose, algebra over a field: 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 are 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

The term "algebra" can also describe more general structures:

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

Other

• Algebra Blessett, singer from the U.S, goes by the stage name Algebra.

See also

• Algebraic (disambiguation)
• List of all articles whose title begins with "algebra"