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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" may refer to:
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
An "algebra", or to be verbose, an algebra over a field, is 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, 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:
"Algebra" can also describe more general structures:
- Algebra Blessett, singer from the U.S, goes by the stage name Algebra
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