Algebra (disambiguation)

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 $\mathbb{Z}_2$-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.

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:
  • $F$-algebra
  • $F$-coalgebra

Other

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

See also

• Algebraic (disambiguation)