|
Locally Finite Category
In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the notions of group algebras and incidence algebras, just as categories generalize the notions of groups and partially ordered sets. Definition If the given category is finite (has finitely many objects and morphisms), then the following two definitions of the category algebra agree. Group algebra-style definition Given a group ''G'' and a commutative ring ''R'', one can construct ''RG'', known as the group algebra; it is an ''R''-module equipped with a multiplication. A group is the same as a category with a single object in which all morphisms are isomorphisms (where the elements of the group correspond to the morphisms of the category), so the following construction generalizes the definition of the group algebra from groups to arbitrary categories. Let ''C'' be a category and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Ring
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Definition Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' over ''R'', denoted ''R'' 'G''or ''RG'', is the set of formal sums \sum_ r_g g, where r_g \in R for each g \in G and ''r''''g'' = 0 for all but finitely many ''g'', equipped with coefficient-wise addition, and the multiplication in which the elements of ''R'' commute with the elements of ''G''. More formally, ''R'' 'G''is the set of functions such that is finite, equipped with addition of functions, and with multiplication defined by : (\phi \psi)(g) = \sum_ \phi(k) \psi(\ell). If ''G'' is a group, then ''R'' 'G''is also called the group ring of ''G'' over ''R''. Universal property Given ''R'' and ''G'', there is a ring homomorphism sending each ''r'' to ''r''1 (where 1 is the identity element of ''G''), and a monoid homomorphism (w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires ''Invisible Empires'' is the seventh studio album and tenth album overall from Christian singer and songwriter Sara Groves, and it released on October 18, 2011 by Fair Trade and Columbia Records. The producers on the album were Steve Hindalong an ...'' See also * * Nonfinite (other) {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-correlation of and , or and . For complex-valued fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its compleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
