What are the main types of algebra?
Algebra
Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification
where none of the first level areas (two digit entries) is called "algebra". Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.
Reductive Lie algebra
The most basic example is the Lie algebra formula_2 of formula_3 matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an "n"-dimensional vector space, formula_4 This is the Lie algebra of the general linear group GL("n"), and is reductive as it decomposes as formula_5 corresponding to traceless matrices and scalar matrices.
Quadratic algebra
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
Mu Alpha Theta
Competition is divided into six levels or divisions, Calculus, Pre-calculus, Algebra II, Geometry, Algebra I, and Statistics. At state and national competitions, only three levels are used: Theta (Geometry and Algebra II), Alpha (Pre-Calculus), and Calculus. There is only a Mu division at the nationals level. Additionally, there are usually Open tests, which can be taken by students from any division, including Statistics, Number Theory, and History of Math. Most students start at the level of math that they are currently enrolled in or have last taken and progress to higher levels. A student can begin at another level, but it must be higher. The only exception to this is that students enrolled in either Algebra II or Geometry can take whichever of the two they want because not all schools offer these courses in the same sequence. If a student competes in a higher level, such as Pre-Calculus, he/she cannot then go back and compete at the Algebra II level. This encourages students to compete with other students who are taking classes of similar mathematical difficulty.
History of algebra
Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.
where p and q are positive.
This trichotomy comes about because quadratic equations of the form formula_4, with p and q positive, have no positive roots.
Algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as "a", "n", "x", "y" or "z"). This is useful because:
Geometry
The study of low-dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.
Structure (mathematical logic)
Both universal algebra and model theory study classes of (structures or) algebras that are defined by a signature and a set of axioms. In the case of model theory these axioms have the form of first-order sentences. The formalism of universal algebra is much more restrictive; essentially it only allows first-order sentences that have the form of universally quantified equations between terms, e.g.  "x" "y" ("x" + "y" = "y" + "x"). One consequence is that the choice of a signature is more significant in universal algebra than it is in model theory. For example, the class of groups, in the signature consisting of the binary function symbol × and the constant symbol 1, is an elementary class, but it is not a variety. Universal algebra solves this problem by adding a unary function symbol .
Geometric algebra
Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.
Associative algebra
The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
Severi–Brauer variety
In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra ("a","b") corresponds to the conic "C"("a","b") with equation 
and the algebra ("a","b") "splits", that is, ("a","b") is isomorphic to a matrix algebra over "K", if and only if "C"("a","b") has a point defined over "K": this is in turn equivalent to "C"("a","b") being isomorphic to the projective line over "K".
Variety (universal algebra)
A theory is a signature, a set of variables and a set of equational laws. Any theory gives a variety of algebras as follows. Given a theory formula_8, an algebra of formula_8 consists of a set formula_10 together with, for each operation formula_3 of formula_8 with arity formula_13, a function formula_14 such that for each axiom formula_7 and each assignment of elements of formula_10 to the variables in that axiom, the equation holds that is given by applying the operations to the elements of formula_10 as indicated by the trees defining formula_5 and formula_6. We call the class of algebras of a given theory formula_8 a variety of algebras.
Classification of Clifford algebras
It may be seen that of all matrix ring types mentioned, there is only one type shared between both complex and real algebras: the type M(2,C). For example, Cℓ(C) and Cℓ(R) are both determined to be M(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cℓ(C) is algebra isomorphic via a C-linear map (which is necessarily R-linear), and Cℓ(R) is algebra isomorphic via an R-linear map, Cℓ(C) and Cℓ(R) are R-algebra isomorphic.
Algebraic curve
There is a triple equivalence of categories between the category of smooth irreducible projective algebraic curves over C (with non-constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function fields in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing. It allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis, and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry.
Tensor algebra
The tensor algebra is important because many other algebras arise as quotient algebras of "T"("V"). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
Algebra
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in formula_1 the letter formula_2 is unknown, but the law of inverses can be used to discover its value: formula_3. In , the letters formula_4 and formula_5 are variables, and the letter formula_6 is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
Quadratic-linear algebra
In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by . An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form "XY" – "YX" – ["X", "Y"] = 0.
E8 (mathematics)
As well as the complex Lie group of type E, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.
Space (mathematics)
Von Neumann then proposed that non-commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called "factors". Von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. Types II and III exhibited new phenomena. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any non-negative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors.
Kunihiko Kodaira
In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds. This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically; the other two being non-algebraic. He provided also detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over algebraic function fields, a theory whose arithmetic analogue proved important soon afterwards. This work also included a characterisation of K3 surfaces as deformations of quartic surfaces in "P", and the theorem that they form a single diffeomorphism class. Again, this work has proved foundational. (The K3 surfaces were named after Ernst Kummer, Erich Kähler, and Kodaira).
Algebra tile
Algebra tiles are made up of small squares, rectangles, and large squares. The small square, the unit tile, represents the number one; the rectangle represents the variable formula_1; and the large square represents formula_2. The side of the formula_2 tile is equal to the length of the formula_1 tile. The width of the formula_1 tile is the same as the side of the unit tile. Additionally, the length of the formula_1 tile is often not an integer multiple of the side of the unit tile.
Algebra
Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.
Real algebraic geometry
Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields) and their applications to the study of positive polynomials and sums-of-squares of polynomials. (See Hilbert's 17th problem and Krivine's Positivestellensatz.) The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra to complex algebraic geometry. Related fields are the theory of moment problems, convex optimization, the theory of quadratic forms, valuation theory and model theory.
Outline of algebra
Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields.An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree.
Space (mathematics)
Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined from the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa.
Algebra
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are listed fundamental concepts in abstract algebra.
F4 (mathematics)
There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.
Comparison of vector algebra and geometric algebra
Vector algebra is specific to Euclidean 3-space, while geometric algebra uses multilinear algebra and applies in all dimensions and signatures, notably 3+1 spacetime as well as 2 dimensions. They are mathematically equivalent in 3 dimensions, where this article is focused, though the approaches differ. Vector algebra is more widely used in elementary multivariable calculus, while geometric algebra is used in more advanced treatments, and is proposed for elementary use as well.
Elementary algebra
has solutions
Subalgebra
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of "all" algebraic structures. Subalgebra can be a subset of both cases.
Cuntz algebra
Cuntz algebras have been generalised in many ways. Notable amongst which are the Cuntz–Krieger algebras, graph C*-algebras and k-graph C*-algebras.
Magma (algebra)
Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:
Abstract algebra
These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's "Moderne algebra", the two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word "algebra" from "the theory of equations" to the "theory of algebraic structures".
History of algebra
A conic section is a curve that results from the intersection of a cone with a plane. There are three primary types of conic sections: ellipses (including circles), parabolas, and hyperbolas. The conic sections are reputed to have been discovered by Menaechmus (c. 380 BC – c. 320 BC) and since dealing with conic sections is equivalent to dealing with their respective equations, they played geometric roles equivalent to cubic equations and other higher order equations.
Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give "A" the structure of a ring; the addition and scalar multiplication operations together give "A" the structure of a vector space over "K". In this article we will also use the term "K"-algebra"' to mean an associative algebra over the field "K". A standard first example of a "K"-algebra is a ring of square matrices over a field "K", with the usual matrix multiplication.
List of commutative algebra topics
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers formula_1, and p-adic integers.
Structure (mathematical logic)
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.
Algebra
The rational numbers, the real numbers and the complex numbers are all examples of fields.
Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra.
Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:
Universal algebra
Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.
Monad (category theory)
The example about free groups given above can be generalized to any type of algebra in the sense of a variety of algebras in universal algebra. Thus, every such type of algebra gives rise to a monad on the category of sets. Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg–Moore algebras), so monads can also be seen as generalizing varieties of universal algebras. Even more generally, any adjunction is said to be monadic (or tripleable) if it shares this property of being (equivalent to) the Eilenberg–Moore category of its associated monad. Consequently, Beck's monadicity theorem, which gives a criterion for monadicity, can be used to show that an arbitrary adjunction can be treated as a category of algebras in this way.
George Peacock
Peacock's main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis. He founded what has been called the philological or symbolical school of mathematicians; to which Gregory, De Morgan and Boole belonged. His answer to Maseres and Frend was that the science of algebra consisted of two parts—arithmetical algebra and symbolical algebra—and that they erred in restricting the science to the arithmetical part. His view of arithmetical algebra is as follows: "In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs formula_1 and formula_2 denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as formula_3 we must suppose formula_4 and formula_5 to be quantities of the same kind; in others, like formula_6, we must suppose formula_4 greater than formula_5 and therefore homogeneous with it; in products and quotients, like formula_9 and formula_10 we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."
Albert algebra
Over any algebraically closed field, there is just one Albert algebra, and its automorphism group "G" is the simple split group of type F. (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field "F", the Albert algebras are classified by the Galois cohomology group H("F","G").
Comparison of linear algebra libraries
Matrix types (special types like bidiagonal/tridiagonal are not listed):
History of algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property).
Algebra over a field
The real numbers may be viewed as a "one"-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself. Likewise, as we saw above, the complex numbers form a "two"-dimensional vector space over the field of real numbers, and hence form a two dimensional algebra over the reals. In both these examples, every non-zero vector has an inverse, making them both division algebras. 
Although there are no division algebras in 3 dimensions, in 1843, the quaternions were defined and provided the now famous 4-dimensional example of an algebra over the real numbers, where one can not only multiply vectors, but also divide. Any quaternion may be written as ("a", "b", "c", "d") = "a" + "bi + "cj + "d"k. Unlike the complex numbers, the quaternions are an example of a non-commutative algebra: for instance, (0,1,0,0) · (0,0,1,0) = (0,0,0,1) but (0,0,1,0) · (0,1,0,0) = (0,0,0,−1).
Laws of Form
The "primary algebra" contains three kinds of proved assertions:
Equation
Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form "P"("x") = 0, where "P" is a polynomial, and linear equations have the form "ax" + "b" = 0, where "a" and "b" are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.
Non-associative algebra
Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities which simplify multiplication somewhat. These include the following identities.
Universal algebra
We assume that the type, formula_2, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values "true" and "false", usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction "and" denoted as ∧, the disjunction "or" denoted as ∨, and the negation "not" denoted as ¬. It is thus a formalism for describing logical relations in the same way that elementary algebra describes numeric relations.
Cellular algebra
Further examples include q-Schur algebras, the Brauer algebra, the Temperley–Lieb algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category formula_75 of a semisimple Lie algebra.
Commutative algebra
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers formula_1; and "p"-adic integers.
Generalizations of the derivative
Many abstract data types in mathematics and computer science can be described as the algebra generated by a transformation that maps structures based on the type back into the type. For example, the type T of binary trees containing values of type A can be represented as the algebra generated by the transformation 1+A×T→T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed either way.
Outline of algebraic structures
In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.
Severi–Brauer variety
In mathematics, a Severi–Brauer variety over a field "K" is an algebraic variety "V" which becomes isomorphic to a projective space over an algebraic closure of "K". The varieties are associated to central simple algebras in such a way that the algebra splits over "K" if and only if the variety has a point rational over "K". studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group.
Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Equation
Algebraic geometry is a branch of mathematics, classically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.
Abstract algebra
As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, manyperhaps mostof these problems were in some way related to the theory of algebraic equations. Major themes include:
Lorentz group
The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
(Strictly speaking the last corresponds to infinitely many classes, since distinct formula_29 give different classes.)
The two-dimensional subalgebras are:
The three-dimensional subalgebras are:
(Here, the Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematician Luigi Bianchi.)
The four-dimensional subalgebras are all conjugate to
Algebraic surface
Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The "general type" class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P lies in it, for example).
Common Core State Standards Initiative
High school example: As an example of a high school category, here are the domains and clusters for algebra. There are four algebra domains (in bold below), each of which is broken down into as many as four clusters (bullet points below). Each cluster contains one to five detailed standards (not listed here). Starred standards, such as the Creating Equations domain (A-CED), are also intended to be part of the modeling category.
Discrete mathematics
Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
Algebraic structure
Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra "T". Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure "E". The quotient algebra "T"/"E" is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator "m", taking two arguments, and the inverse operator "i", taking one argument, and the identity element "e", a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables "x", "y", "z", etc. the term algebra is the collection of all possible terms involving "m", "i", "e" and the variables; so for example, "m(i(x), m(x,m(y,e)))" would be an element of the term algebra. One of the axioms defining a group is the identity "m(x, i(x)) = e"; another is "m(x,e) = x". The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
However, it may happen that an algebra admits no involution at all.
Okubo algebra
In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo . Okubo algebras are composition algebras, flexible algebras ("A"("BA") = ("AB")"A"), Lie admissible algebras, and power associative, but are not associative, not alternative algebras, and do not have an identity element.
Abstract algebra
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called "variety of groups".
Boolean algebra
Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory.
Algebraic structure
Simple structures: no binary operation:Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers.
Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition.Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.Arithmetics: two binary operations, addition and multiplication. "S" is an infinite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.Module-like structures: composite systems involving two sets and employing at least two binary operations.Algebra-like structures: composite system defined over two sets, a ring "R" and a "R" module "M" equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on "R", two on "M" and one involving both "R" and "M".Four or more binary operations:Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure.Algebraic structures are defined through different configurations of axioms. Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by "identities" and structures that are not. If all axioms defining a class of algebras are identities, then the class of objects is a variety (not to be confused with algebraic variety in the sense of algebraic geometry).
Division algebra
Over the reals there are (up to isomorphism) only two unitary commutative finite-dimensional division algebras: the reals themselves, and the complex numbers. These are of course both associative. For a non-associative example, consider the complex numbers with multiplication defined by taking the complex conjugate of the usual multiplication:
This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element. There are infinitely many other non-isomorphic commutative, non-associative, finite-dimensional real divisional algebras, but they all have dimension 2.
List of abstract algebra topics
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called "elementary algebra". The distinction is rarely made in more recent writings.
Tensor algebra
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotient algebras of "T"("V"). Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
Lie-admissible algebra
In algebra, a Lie-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket ["a","b"] = "ab–ba". Examples include associative algebras, Lie algebras, and Okubo algebras.
Universal geometric algebra
In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case.
Algebra representation
Because a weight is a map to a commutative ring, the map factors through the abelianization of the algebra formula_15 – equivalently, it vanishes on the derived algebra – in terms of matrices, if formula_16 is a common eigenvector of operators formula_17 and formula_18, then formula_19 (because in both cases it is just multiplication by scalars), so common eigenvectors of an algebra must be in the set on which the algebra acts commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this particularly simple and important case of the polynomial algebra formula_20 in a set of commuting matrices, a weight vector of this algebra is a simultaneous eigenvector of the matrices, while a weight of this algebra is simply a formula_21-tuple of scalars formula_22 corresponding to the eigenvalue of each matrix, and hence geometrically to a point in formula_21-space. These weights – in particularly their geometry – are of central importance in understanding the representation theory of Lie algebras, specifically the finite-dimensional representations of semisimple Lie algebras.
Algebra
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of "sets": a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials ("ax" + "bx" + "c"), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo "n". Set theory is a branch of logic and not technically a branch of algebra.
List of algebras
This is a list of possibly nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the base ring).
Outline of algebraic structures
The following structures consist of a set with a binary operation. The most common structure is that of a "group". Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.The main types of structures with one set having two binary operations are "rings" and "lattices". The axioms defining many of the other structures are modifications of the axioms for rings and lattices. One major difference between rings and lattices is that their two operations are related to each other in different ways. In ring-like structures, the two operations are linked by the distributive law; in lattice-like structures, the operations are linked by the absorption law.The following structures have the common feature of having two sets, "A" and "B", so that there is a binary operation from "A"×"A" into "A" and another operation from "A"×"B" into "A".Many structures here are actually hybrid structures of the previously mentioned ones.There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.
Algebra representation
Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in variables over the field "K" is concretely a "K"-vector space with commuting operators, and is often denoted formula_3 meaning the representation of the abstract algebra formula_4 where formula_5
Elementary algebra
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
Geometric algebra
The scalars and vectors have their usual interpretation, and make up distinct subspaces of a GA. Bivectors provide a more natural representation of pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum, electromagnetic field and the Poynting vector. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of formula_2 and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike vector algebra, a GA naturally accommodates any number of dimensions and any quadratic form such as in relativity.
Exterior algebra
The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.
Variety (universal algebra)
The monad formula_41 is thus enough to recover the finitary algebraic category, which allows the following generalization. One says a category is an algebraic category if it is monadic over formula_42. This is a more general notion than "finitary algebraic category" because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of sigma algebras also has infinitary operations, but their arity is countable whence its signature is small (forms a set).
Structurable algebra
In abstract algebra, a structurable algebra is a certain kind of unital involutive non-associative algebra over a field. For example, all Jordan algebras are structurable algebras (with the trivial involution), as is any alternative algebra with involution, or any central simple algebra with involution. An "involution" here means a linear anti-homomorphism whose square is the identity.
ALGOL 68
The basic data types (called modes in Algol 68 parlance) are real, int, compl (complex number), bool, char, bits and bytes. For example:
Equation
The most common type of equation is an algebraic equation, in which the two sides are algebraic expressions.
Each side of an algebraic equation will contain one or more terms. For example, the equation
Geometric algebra
There are a number of different ways to define a geometric or Clifford algebra. Hestenes's original approach was axiomatic, "full of geometric significance" and equivalent to the universal Clifford algebra.
Given a finite-dimensional quadratic space formula_2 over a field formula_1 with a symmetric bilinear form (the "inner product", e.g. the Euclidean or Lorentzian metric) formula_6, the geometric algebra for this quadratic space is the geometric algebra formula_7, together with the exterior algebra formula_8. Since it is usual in GA, for the remainder of this article, only the real case, formula_9, will be considered. The notation formula_10 or formula_11 will be used to describe the geometric algebra.
Exterior algebra
The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not only "k"-blades, but sums of "k"-blades; such a sum is called a "k"-vector. The "k"-blades, because they are simple products of vectors, are called the simple elements of the algebra. The "rank" of any "k"-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that for any elements "α", "β", "γ". The "k"-vectors have degree "k", meaning that they are sums of products of "k" vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.
Algebra
The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite simple groups into roughly 30 basic types.
Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.
Universal algebra
The 1956 paper by Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially defined, typical examples for this being categories and groupoids. This leads on to the subject of higher-dimensional algebra which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.
Differential algebra
and
Algebra (disambiguation)
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:
See also coalgebra, the dual notion.
Algebra tile
Algebra tiles are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students. They have also been used to prepare prison inmates for their General Educational Development (GED) tests. Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation. The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their "Curriculum and Evaluation Standards for Mathematics". According to the NCTM 1989 standards "[r]elating models to one another builds a better understanding of each".
Universal algebra
Most algebraic structures are examples of universal algebras.Examples of relational algebras include semilattices, lattices, and Boolean algebras.
Boolean algebras canonically defined
Just as there are basic examples of groups, such as the group Z of integers and the permutation group "S" of permutations of "n" objects, there are also basic examples of Boolean algebra such as the following.
Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic, digital logic, and the set-theoretic foundations of mathematics.
F-algebra
Algebras over a field can be viewed as F-algebras of signature 1 + 1 + "A" + "A"×"A" + "A"×"A" + "k"×"A" over the category of sets, of signature 1 + "A"×"A" over the category of modules (a module with an internal multiplication), and of signature "k"×"A" over the category of rings (a ring with a scalar multiplication), when they are associative and unitary.
Computer algebra system
Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.
School for the Creative and Performing Arts
All of the general classes (Social Studies, Science, Language Arts, P.E., Health, etc.) are the same. Math is the only area that has different levels available. In 7th grade, students take a test to determine whether they are put into Pre-Algebra, or Algebra I in 8th grade. The class is usually split unevenly into three classes (one large Algebra I, one small Algebra I, and one small Pre-Algebra). Sometimes, depending on the scores, the entire grade is put into Algebra I.