Algebra

Look up algebra in Wiktionary, the free dictionary.

The word 'Algebra' describes one of the main branches of mathematics. It can also be used as a name for several different types of mathematical structures. The word itself comes from the Arabic version of the name of a procedure used by Persian mathematician al Khwārizmī.

As a branch of mathematics

The term algebra may also refer to a more specialized branch of mathematics within the general field of Algebra:

• 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".

As a mathematical structure

Several different classes of algebraic structures are known as "algebras".

One such class consists of objects consisting of a vector space (or more generally a module over a ring) equipped with a notion of "vector multiplication." Examples include:

• In ring theory and linear algebra:
  • Algebra over a commutative ring: a module equipped with a bilinear product
  • Algebra over a field: a vector space equipped with a bilinear vector product
  • Associative algebra: a module equipped with an associative bilinear vector product
  • Superalgebra: a -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.

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

• In logic:
  • 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:
  • -algebra
  • -coalgebra

Other

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

See also

• Algebraic (disambiguation)

This disambiguation page lists mathematics articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.