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Opposite Group
In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids, groups, rings, and algebras can be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Definition Let G be a group under the operation *. The opposite group of G, denoted G^, has the same underlying set as G, and its group operation \mathbin is defined by g_1 \mathbin g_2 = g_2 * g_1. If G is abelian, then it is equal to its opposite group. Also, every group G (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism \varphi: G \to G^ is given by \varphi(x) = x^. More generally, any antiautomorphism \psi: G \to G gives rise to a corresponding isomorphism \psi': G \to G^ via \psi'(g)=\psi(g), since : \psi'(g * h) = \psi(g * h) = \psi(h) * \psi(g) = \psi(g) \mathbin \psi(h)=\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Group Nature
Opposite or Opposites may refer to: * Opposite (semantics), a word that means the reverse of a word * Opposite (leaf), an arrangement of leaves on a stem * Opposite (mathematics), the negative of a number; numbers that, when added, yield zero *"The Opposite", a 1994 episode of ''Seinfeld'' Music * The Opposites, Dutch rap group * Opposites (album), ''Opposites'' (album), 2013 album by Scottish alternative rock band Biffy Clyro ** Opposite (song), "Opposite" (song), 2013 song by Biffy Clyro * Opposites (EP), ''Opposites'' (EP), 2010 album by Tracey Thorn *"The Opposite", 1964 song by Johnny Burnette See also * Opposite hitter, a position in volleyball * Antinomy, opposites in a certain form from Kant * * Anti (other) * Contrary (other) * Flipside (other) * Inverse (other) * Opposite sex (other) * Opposition (other) * Polar opposite (other) * The House Opposite (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/ up-set, ideal/filter etc. This order theoretic duality is in turn a special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/ up-set, ideal/filter etc. This order theoretic duality is in turn a special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Ring
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in ''R''. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see '). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Relation to automorphisms and antiautomorphisms In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusion with some unary operation. A ring R having isomorphic opposite ring is called a ''self-opposite'' ring, which name indicates that R^\text is essentially the same as R. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiautomorphism
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism. Definition Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures X and Y is a homomorphism \phi\colon X \to Y^, where Y^ equals Y as a set, but has its multiplication reversed to that defined on Y. Denoting the (generally non-commutative) multiplication on Y by \cdot, the multiplication on Y^, denoted by *, is defined by x*y := y \cdot x. The object Y^ is called the opposite object to Y (respectively, opposite group, opposite algebra, opposite category etc.). This definition is equivalent to that of a homomorphism \phi\colon X^ \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naturally Isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, 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 the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Ring
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in ''R''. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see '). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Relation to automorphisms and antiautomorphisms In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusion with some unary operation. A ring R having isomorphic opposite ring is called a ''self-opposite'' ring, which name indicates that R^\text is essentially the same as R. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Ring
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
