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"
[edit]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
[edit]- 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
[edit]Vector space with multiplication
[edit]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
- C*-algebra, a Banach algebra equipped with a unary involution operation
- Von Neumann algebra (or W*-algebra)
See also coalgebra, the dual notion.
Other structures
[edit]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
"Algebra" can also describe more general structures:
- In category theory and computer science:
Other uses
[edit]- Algebra Blessett, singer from the U.S, goes by the stage name Algebra
See also
[edit]- Algebraic (disambiguation)
- List of all articles whose title begins with "algebra"