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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".
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 -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.
See also coalgebra, the dual notion.
- In logic:
- In measure theory:
The term "algebra" can also describe more general structures:
- Algebra Blessett, singer from the U.S, goes by the stage name Algebra.
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