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Universal Properties
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field (mathematics), field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of category (mathematics), categories and functors by mean of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as Initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Morphism Definition
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Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind–MacNeille Completion
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers. It is also called the completion by cuts or normal completion. Order embeddings and lattice completions A partially ordered set (poset) consists of a set of elements together with a binary relation on pairs of elements that is reflexive ( for every ''x''), transitive (if and then ), and antisymmetric (if both and hold, then ). The usual numeric orderings on the integers or real numbers satisfy these properties; however, unlike the orderings on the numbers, a partial order may have two elements that are ''incomparable'': neither nor holds. Another familiar example of a par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion Of A Ring
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \, of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion Of A Metric Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: :#Every |
Grothendieck Group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic image of will also contain a homomorphic image of the Grothendieck group of . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation. Grothendieck group of a commutative monoid Motivation Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing inverse elements to all elements of . Such an abelian group always exists; it is called the Grothendieck group of . It is character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lattice
In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property. Formal definition Any set ''X'' may be used to generate the free semilattice ''FX''. The free semilattice is defined to consist of all of the finite subsets of ''X'', with the semilattice operation given by ordinary set union. The free semilattice has the universal property. The universal morphism is , where η is the unit map η: ''X'' → ''FX'' which takes ''x'' ∈ ''X'' to the singleton set . The universal property is then as follows: given any map ''f'': ''X'' → ''L'' from ''X'' to some arbitrary semilattice ''L'', there exists a unique semilattice homomorphism \tilde:FX \to L such that f = \tilde\circ\eta. The map \tilde may be explicitly written down; it is given by :S \in FX \mapsto \bigvee\left\ where \bigvee denotes the semilattice operation in ''L''. This construction may be promoted from semilattices to lattice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms. Definition Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function between vector spaces is entirely determined by its values on a basis of the vector space The following definition translates this to any category. A concrete category is a category that is equipped with a faithf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization (commutative Algebra)
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
