# Category: Algebra

Series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (additi
Akivis algebra
In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator and a ternary operator, the associator that satisf
Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently,
Median algebra
In mathematics, a median algebra is a set with a ternary operation satisfying a set of axioms which generalise the notions of medians of triples of real numbers and of the Boolean majority function. T
Smooth algebra
In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map , there exist
Symbolic method
In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invar
One-relator group
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group
Sklyanin algebra
In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification o
Polynomial mapping
In algebra, a polynomial map or polynomial mapping between vector spaces over an infinite field k is a polynomial in linear functionals with coefficients in k; i.e., it can be written as where the are
Graph algebra (social sciences)
Graph algebra is modeling tool for the social sciences. It was first developed by Sprague, Pzeworski, and Cortes as a hybridized version of engineering plots to describe social phenomena.
Filtration (mathematics)
In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that if in , then
Minimal algebra
Minimal algebra is an important concept in tame congruence theory, a theory that has been developed by Ralph McKenzie and David Hobby.
Operator (mathematics)
In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly the same space, sometimes required to be the same space).
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called
Reduct
In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion."
Temperley–Lieb algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable mode
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a sup
Distribution (number theory)
In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in th
Equivalence class
In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence cla
Groupoid algebra
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.
Partial fraction decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operati
A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on ad
Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nat
Amitsur complex
In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism. It was introduced by Shimshon Amitsur. When the homomorphism is faithfully flat, the Amitsur complex is exact (t
Monomial ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the i
Polynomial Diophantine equation
In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate. A Diophantine equation, in ge
Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is
Constant (mathematics)
In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: *
Distributive homomorphism
A congruence θ of a join-semilattice S is monomial, if the θ-equivalence class of any element of S has a largest element. We say that θ is distributive, if it is a join, in the congruence lattice Con
Hundred Fowls Problem
The Hundred Fowls Problem is a problem first discussed in the fifth century CE Chinese mathematics text Zhang Qiujian suanjing (The Mathematical Classic of Zhang Qiujian), a book of mathematical probl
Invertible module
In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the de
Antiisomorphism
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite o
Operand
In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Brahmagupta–Fibonacci identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under
Unitary method
The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has be
Generalized arithmetic progression
In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithm
Calabi–Yau algebra
In algebra, a Calabi–Yau algebra was introduced by Victor Ginzburg to transport the geometry of a Calabi–Yau manifold to noncommutative algebraic geometry.
Order of operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate
Height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica
Pseudoalgebra
In algebra, given a 2-monad T in a 2-category, a pseudoalgebra for T is a 2-category-version of algebra for T, that satisfies the laws up to coherent isomorphisms.
Algebra
Algebra (from Arabic ‏الجبر‎ (al-jabr) 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules fo
Series multisection
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its
Conjugate (square roots)
In mathematics, the conjugate of an expression of the form is provided that does not appear in a and b. One says also that the two expressions are conjugate. In particular, the two solutions of a quad
System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over
Homotopy associative algebra
In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily assoc
Distribution algebra
In algebra, the distribution algebra of a p-adic Lie group G is the K-algebra of K-valued distributions on G. (See the reference for a more precise definition.)
Matrix factorization of a polynomial
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p withou
De Morgan algebra
In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: * (A, ∨, ∧, 0, 1) is a bounded distributive
Casus irreducibilis
In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed t
Howson property
In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two
Board puzzles with algebra of binary variables
Board puzzles with algebra of binary variables ask players to locate the hidden objects based on a set of clue cells and their neighbors marked as variables (unknowns). A variable with value of 1 corr
Topological module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Dialgebra
In abstract algebra, a dialgebra is the generalization of both algebra and coalgebra. The notion was originally introduced by Lambek as “subequalizers”. Many algebraic notions have previously been gen
Straight-line program
In mathematics, more specifically in computational algebra, a straight-line program (SLP) for a finite group G = ⟨S⟩ is a finite sequence L of elements of G such that every element of L either belongs
Complex Lie algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra , its conjugate is a complex Lie algebra with the same underlying real vector space but wit
Digital root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result fro
Factorization of polynomials over finite fields
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for p
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include which arises in discus
Brahmagupta's identity
In algebra, Brahmagupta's identity says that, for given , the product of two numbers of the form is itself a number of that form. In other words, the set of such numbers is closed under multiplication
Cyclotomic polynomial
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. It
Omar Khayyam
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam (Persian: عمر خیّام), was a Khorasani polymath, known for his contributions to mat
Elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without f
Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one
Subfield of an algebra
In algebra, a subfield of an algebra A over a field F is an F-subalgebra that is also a field. A maximal subfield is a subfield that is not contained in a strictly larger subfield of A. If A is a fini
Algebraic signal processing
Algebraic signal processing (ASP) is an emerging area of theoretical signal processing (SP). In the algebraic theory of signal processing, a set of filters is treated as an (abstract) algebra, a set o
Monomial basis
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis
Map algebra
Map algebra is an algebra for manipulating geographic data, primarily fields. Developed by Dr. Dana Tomlin and others in the late 1970s, it is a set of primitive operations in a geographic information
Boundedly generated group
In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence sub
Tertiary ideal
In mathematics, a tertiary ideal is an (two-sided) ideal in a (perhaps noncommutative) ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertia
Multiplicative digital root
In number theory, the multiplicative digital root of a natural number in a given number base is found by multiplying the digits of together, then repeating this operation until only a single-digit rem
Wang algebra
In algebra and network theory, a Wang algebra is a commutative algebra , over a field or (more generally) a commutative unital ring, in which has two additional properties:(Rule i) For all elements x
Consistent and inconsistent equations
In mathematics and particularly in algebra, a linear or nonlinear system of equations is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the s
Regular semi-algebraic system
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
Field arithmetic
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group.It is an interdisciplinary subject as it uses tools
Resolvent cubic
In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: In each case: * The coefficients of the resolvent cubic c
Free product of associative algebras
In algebra, the free product (coproduct) of a family of associative algebras over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations o
Berlekamp–Rabin algorithm
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over a field . The method was discovered by
Infrastructure (number theory)
In mathematics, an infrastructure is a group-like structure appearing in global fields.
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and pos
Bose–Mesner algebra
In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (formin
Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in
Rational difference equation
A rational difference equation is a nonlinear difference equation of the form where the initial conditions are such that the denominator never vanishes for any n.
Symbolic power of an ideal
In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the n-th symbolic power of is the ideal where is the localization of at , we set is the canonical map from a
Mode-k flattening
In multi-linear algebra, mode-k flattening (also matricisation, matricizing, or unfolding) is an operation on tensor (a multi-dimensional array) denoted by turning it into a matrix (a two-dimensional
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In
Schur algebra
In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used t
Linearly disjoint
In mathematics, algebras A, B over a field k inside some field extension of k are said to be linearly disjoint over k if the following equivalent conditions are met: * (i) The map induced by is injec
Moderne Algebra
Moderne Algebra is a two-volume German textbook on graduate abstract algebra by Bartel Leendert van der Waerden , originally based on lectures given by Emil Artin in 1926 and by Emmy Noether from 1924
Graph dynamical system
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis
Cyclic algebra
In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field, and plays a key role in the theory of central simple algebras.
Variable (mathematics)
In mathematics, a variable (from Latin variabilis, "changeable") is a symbol and placeholder for any mathematical object. In particular, a variable may represent a number, a vector, a matrix, a functi
Idempotent matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily
Indeterminate equation
In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation is a simple indeterminate equation, as is . Indet
List of algebraic constructions
An algebraic construction is a method by which an algebraic entity is defined or derived from another. Instances include: * Cayley–Dickson construction * Proj construction * Grothendieck group * G
Substitution (algebra)
In algebra, the operation of substitution can be applied in various contexts involving formal objects containing symbols (often called variables or indeterminates); the operation consists of systemati
Classical Lie algebras
The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types , , and , where for the general linear Lie algebra and the identity matrix: * , the
In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 2
Recurrence relation
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the seque
Boolean differential calculus
Boolean differential calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables and Boolean functions. Boolean different
Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − ×
Timeline of geometry
A timeline of algebra and geometry
Conservation form
Conservation form or Eulerian form refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i
Antivector
An antivector is an element of grade n − 1 in an n-dimensional exterior algebra. An antivector is always a blade, and it gets its name from the fact that its components each involve a combination of a
Prime avoidance lemma
In algebra, the prime avoidance lemma says that if an ideal I in a commutative ring R is contained in a union of finitely many prime ideals Pi's, then it is contained in Pi for some i. There are many
K-theory (physics)
In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field stre
Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of sparse polynomial after the monomials.
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Shuffle algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of inter
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, a
In algebra, a differential graded module, or dg-module, is a -graded module together with a differential; i.e., a square-zero graded endomorphism of the module of degree 1 or −1, depending on the conv
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 Pol
Canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which pro
Inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers. Given an operation denoted here ∗, and an identity element denoted e, if x
Polynomial transformation
In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transform
Free presentation
In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules: Note the image under g of the standard basis generates M. In particular, if J is finite, then
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Bot
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as a, b a
Cramer's theorem (algebraic curves)
In algebraic geometry, Cramer's theorem on algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-d
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: 1. * A monomial, also called power product, is a product of po
Quasi-unmixed ring
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring such that for each prime ideal p, the comple
Co-Hopfian group
In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.
Vector algebra
In mathematics, vector algebra may mean: * Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. * The algebraic operations in v
In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by in their formulation of Koszul duality.
Coherent algebra
A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix and the all-ones ma
Spherical design
A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit d-sphere Sd such that the average value of any polynomial f of degree t or
Module homomorphism
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homom
Spherically complete field
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is
Cylindrical algebraic decomposition
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set S of polynomia
Gelfand–Kirillov dimension
In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a right module M over a k-algebra A is: where the supremum is taken over all finite-dimensional subspaces and . An algebra is said to ha
Suspension of a ring
In algebra, more specifically in algebraic K-theory, the suspension of a ring R is given by where is the ring of all infinite matrices with coefficients in R having only finitely many nonzero elements
First and second fundamental theorems of invariant theory
In algebra, the first and second fundamental theorems of invariant theory concern the generators and the relations of the ring of invariants in the ring of polynomial functions for classical groups (r
Laws of Form
Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. LoF describes three distinct logical systems: * The "p
Integrable module
In algebra, an integrable module (or integrable representation) of a Kac–Moody algebra (a certain infinite-dimensional Lie algebra) is a representation of such that (1) it is a sum of weight spaces an
Immanant
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let be a partition of an in
Ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six
Regular chain
In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set.
Primitive part and content
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primi
Primordial element (algebra)
In algebra, a primordial element is a particular kind of a vector in a vector space.