Algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, named for the Persian mathematician al Khwārizmī

Muhammad ibn Musa al-Khwarizmi

'There is some confusion in the literature on whether al-Khwārizmī's full name is ' or '. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." . 'There is some confusion in the literature on...

, describes one of the main branches of mathematics. It can also be used as a name for several different types of mathematical structures.

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
  Elementary algebra
  Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
  
  , i.e. "high-school algebra."
• Abstract algebra
  Abstract algebra
  Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
  
  
• Linear algebra
  Linear algebra
  Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
  
  
• Relational algebra
  Relational algebra
  Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations that is closed under certain operators. These operators operate on one or more relations to yield a relation...
  
  
• Universal algebra
  Universal algebra
  Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
  
  

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

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 (or more generally a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over a ring) equipped with a notion of "vector multiplication." Examples include:

• In ring theory
  Ring theory
  In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
  
   and linear algebra
  Linear algebra
  Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
  
  :
  • Algebra
    Algebra (ring theory)
    In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
    
     over a commutative ring: a module equipped with a bilinear product
  • Algebra over a field
    Algebra over a field
    In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
    
    : a vector space
    Vector space
    A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
    
     equipped with a bilinear vector product
  • Associative algebra
    Associative algebra
    In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
    
    : a module
    Module (mathematics)
    In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
    
     equipped with an associative bilinear vector product
  • Superalgebra
    Superalgebra
    In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
    
    : a -graded algebra
    Graded algebra
    In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
    
    
  • Lie algebra
    Lie algebra
    In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
    
    s, Poisson algebra
    Poisson algebra
    In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups...
    
    s, and Jordan algebra
    Jordan algebra
    In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
    
    s are important examples of (potentially) nonassociative algebras.
• In functional analysis
  Functional analysis
  Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
  
  :
  • Banach algebra
    Banach algebra
    In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
    
    : an associative algebra
    Associative algebra
    In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
    
     A over the real
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
    
     or complex
    Complex number
    A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
    
     numbers which at the same time is also a Banach space
    Banach space
    In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
    
    .
  • Operator algebra
    Operator algebra
    In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
    
    : continuous linear operators on a topological vector space
    Topological vector space
    In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
    
     with multiplication given by the composition.
  • *-algebra: An algebra with a notion of adjoints.
    • C* algebra: a Banach algebra
      Banach algebra
      In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
      
       equipped with a unary involution operation.
    • Von Neumann algebra
      Von Neumann algebra
      In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
      
       (or W*-algebra)

See also coalgebra

Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...

, the dual notion.

A different class of "algebras" consists of objects which generalize logical connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s, sets, and lattices

Lattice (order)

In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

.

• In logic
  Logic
  In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
  
  :
  • Boolean algebra (structure)
  • Heyting algebra
    Heyting algebra
    In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
    
    
• 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
  Category theory
  Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
  
   and computer science
  Computer science
  Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
  
  :
  • -algebra
    F-algebra
    In mathematics, specifically in category theory, an F-algebra is a structure defined according to a functor F. F-algebras can be used to represent data structures used in programming, such as lists and trees...
    
    
  • -coalgebra
    F-coalgebra
    In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F. For both algebra and coalgebra, a functor is a convenient and general way of organizing a signature...
    
    

Other

• Algebra Blessett
  Algebra (singer)
  Algebra is a contemporary R&B singer.-Early life:Blessett's mother was a gospel singer and bass player, and she grew up to the sounds of soul music, gospel and R&B. Like so many R&B singers, she sang in a gospel choir, when she was at school...
  
  , singer from the U.S, goes by the stage name Algebra.