Algebra (disambiguation)

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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:

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)

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.