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55 Algebraic structure topics matching:
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| x Module |
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring. Modules also generalize the notion of...
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| x Ring |
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In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set...
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| x Field |
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In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the...
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| x Set |
A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of...
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| x Pointed set |
In mathematics, a pointed set is a set X with a distinguished basepoint . Maps of pointed sets (based maps) are functions preserving basepoints, i.e. a map such that f(x0) = y0. This is usually denoted
Pointed sets may be regarded as a rather...
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| x Unary system | ||
| x Pointed unary system | ||
| x Magma |
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In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M. A binary operation...
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| x Steiner magma | ||
| x Squag | ||
| x Sloop | ||
| x Semigroup |
In mathematics, a semigroup is an algebraic structure consisting of a nonempty set S together with an associative binary operation. In other words, a semigroup is an associative magma. The terminology is derived from the anterior notion of a group....
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| x Monoid |
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids...
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| x Group |
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called...
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| x Abelian group |
An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition...
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| x Band |
In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). The lattice of varieties of bands was described independently by Birjukov, Fennemore and Gerhard....
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| x Semilattice |
In mathematics, a join-semilattice is a partially ordered set which has a join (a least upper bound) for any finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet (or greatest lower bound) for any finite subset. Every...
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| x Boundary algebra | ||
| x Quasigroup |
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative. A quasigroup with...
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| x Moufang loop |
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.
A Moufang loop is a loop Q that satisfies any one of the...
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| x Lattice |
In mathematics, a lattice is a partially ordered set (also called a poset) in which any two elements have a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet). Lattices can...
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| x Bounded lattice | ||
| x Complemented lattice |
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0. A relatively complemented lattice is a lattice such that...
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| x Modular lattice |
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:
Modular lattices arise naturally in algebra and in many other areas of mathematics. For example, the subspaces of a...
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| x Distributive lattice |
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In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union...
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| x Kleene algebra |
In mathematics, a Kleene algebra (named after Stephen Cole Kleene, pronounced /ˈkleɪni/ KLAY-nee) is either of two different things:
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. See [1...
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| x Boolean algebra |
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a...
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| x 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...
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| x Relation algebra |
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra equipped with an involution called "converse". The motivating example of a relation algebra is the algebra 2 of all binary relations on a set X, with R•S...
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| x Heyting algebra |
In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras, named after Arend Heyting. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded...
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| x Semiring |
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs are...
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| x Commutative semiring | ||
| x Rng |
In abstract algebra, a rng (also called a pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" (pronounced rung) is meant to...
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| x Commutative ring |
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In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.
Some specific kinds of commutative rings are given with...
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| x Boolean ring |
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In mathematics, a Boolean ring R is a ring (with identity) for which x = x for all x in R; that is, R consists only of idempotent elements.
Boolean rings are automatically commutative and of characteristic 2 (see below for proof). A Boolean ring is...
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| x Integral domain |
In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity. Integral domains are generalizations of the integers and provide a natural setting for...
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| x Euclidean domain |
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization...
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| x Division ring |
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. More formally, a ring with 0 ≠ 1 is a division ring if every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1...
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| x Ordered field |
In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations. This concept was introduced by Emil Artin in 1927.
There are two equivalent definitions, depending...
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| x Formally real field |
In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. A formally real field F may be characterized in any of the following equivalent ways:
The equivalence of the first...
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| x Simple ring |
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra.
According to the Artin–Wedderburn theorem, every simple ring that is left or right...
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| x Weyl algebra |
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable),
More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then...
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| x Artinian ring |
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals...
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| x Vector space |
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A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also...
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| x Jordan algebra |
In abstract algebra, a Jordan algebra is an (not necessarily associative) algebra over a field whose multiplication satisfies the following axioms:
The product of two elements x and y in a Jordan algebra is also denoted x ∘ y, particularly to avoid...
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| x Lie algebra |
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In mathematics, a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) 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...
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| x Associative algebra |
In mathematics, an associative algebra is a module which also allows the multiplication of vectors in a distributive and associative manner. They are thus a special case of algebras over commutative rings.
Let R be a fixed commutative ring. An...
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| x Linear algebra |
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Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern...
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| x Commutative algebra |
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...
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| x Symmetric algebra |
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative K-algebra containing V.
It corresponds to polynomials with indeterminates in V, without choosing...
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| x Graded algebra |
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading).
A graded ring A is a ring that has a direct sum decomposition into...
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| x Exterior algebra |
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In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of...
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| x Clifford algebra |
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In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of...
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| x Geometric algebra |
In mathematical physics, a geometric algebra is a multilinear algebra described technically as a Clifford algebra over a real vector space equipped with a non-degenerate quadratic form. Informally, a geometric algebra is a Clifford algebra that...
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| x Grassmann-Cayley algebra |
Grassmann–Cayley algebra, also known as double algebra, is a form of modeling algebra for use in projective geometry. The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British...
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