Embedding

For other uses, see Embedding (disambiguation).

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f : X → Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

The fact that a map f : X → Y is an embedding is often indicated by the use of a "hooked arrow", thus: f : X ↪ Y.

Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In such cases it is common to identify the domain X with its image f(X) contained in Y, so that then X ⊆ Y. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

1 Topology and geometry
2 Algebra
- 2.1 Field theory
- 2.2 Universal algebra and model theory
3 Order theory and domain theory
4 Metric spaces
- 4.1 Normed spaces
5 Category theory
6 See also
7 References

Topology and geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures Topological equivalence redirects here see also Topological equivalence (dynamical systems. More explicitly, a map f : X → Y between topological spaces X and Y is an embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Intuitively then, the embedding f : X → Y lets us treat X as a subspace of Y. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Every embedding is injective and continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets The latter happens if the image f(X) is neither an open set nor a closed set in Y. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open.

For a given space X, the existence of an embedding X → Y is a topological invariant of X. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is This allows two spaces to be distinguished if one is able to be embedded into a space which the other is not.

Differential topology

In differential topology: Let M and N be smooth manifolds and $f:M\to N$ be a smooth map, it is called an immersion if the derivative of f is everywhere injective. In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, an immersion is a Differentiable map between Differentiable manifolds whose derivative is everywhere Injective. Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Then an embedding, or a smooth embedding, is defined to be an immersion which is an embedding in the above sense (i. e. homeomorphism onto its image). Topological equivalence redirects here see also Topological equivalence (dynamical systems.

In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion An immersion is a local embedding (i. e. for any point $x\in M$ there is a neighborhood $x\in U\subset M$ such that $f:U\to N$ is an embedding. )

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

An important case is N=Rⁿ. The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough. In Mathematics, particularly in Differential topology,there are two Whitney embedding theorems The strong Whitney embedding theorem states that any For example the real projective plane of dimension 2 requires n = 4 for an embedding. Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" An immersion of this surface is, however, possible in R³, and one example is Boy's surface—which has self-intersections. In Geometry, Boy's surface is an immersion of the Real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it The Roman surface fails to be an immersion as it contains cross-caps. The Roman surface (so called because Jakob Steiner was in Rome when he thought of it is a self-intersecting mapping of the Real projective plane into

An embedding is proper if it behaves well w.r.t. boundaries: one requires the map $f: X \rightarrow Y$ to be such that

$f(\partial X) = f(X) \cap \partial Y$ , and
$f (X)$ is transversal to $\partial Y$ in any point of $f(\partial X)$ . (Main list of acronyms and initialisms W - (s Tungsten (from German Wolfram) - Watt - West W0-9 In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below Transversality in Mathematics is a notion that describes how spaces can intersect transversality can be seen as the "opposite" of tangency, and plays a

The first condition is equivalent to having $f(\partial X) \subseteq \partial Y$ and $f(X \setminus \partial X) \subseteq Y \setminus \partial Y$ . The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.

Riemannian geometry

In Riemannian geometry: Let (M,g) and (N,h) be Riemannian manifolds. Elliptic geometry is also sometimes called Riemannian geometry. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from e. g = f*h. Explicitly, for any two tangent vectors

$v,w\in T_x(M)$

we have

$g(v,w)=h(df(v),df(w)).\,$

Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics.

Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Nash embedding theorem). The Nash embedding theorems (or imbedding theorems) named after John Forbes Nash, state that every n-dimensional Riemannian manifold can be isometrically

Algebra

In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism e:X→Y which is injective.

Field theory

In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

The kernel of σ is an ideal of E which cannot be the whole field E, because of the condition σ(1)=1. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Furthermore, it is a well-known property of fields that their only ideals are the zero ideal and the whole field itself. Therefore, the kernel is 0, so any embedding of fields is a monomorphism. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. Moreover, E is isomorphic to the subfield σ(E) of F. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This justifies the name embedding for an arbitrary homomorphism of fields.

Universal algebra and model theory

If σ is a signature and $A, B$ are σ-structures (also called σ-algebras in universal algebra or models in model theory), then a map $h:A \to B$ is a σ-embedding iff all the following holds:

$h$ is injective,
for every $n$ -ary function symbol $f \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))$ ,
for every $n$ -ary relation symbol $R \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $A \models R(a_1,\ldots,a_n)$ iff $B \models R(h(a_1),\ldots,h(a_n)).$

Here $A\models R (a_1,\ldots,a_n)$ is a model theoretical notation equivalent to $(a_1,\ldots,a_n)\in R^A$ . In Logic, especially Mathematical logic, a signature lists and describes the Non-logical symbols of a Formal language. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models ↔ In model theory there is also a stronger notion of elementary embedding. In Model theory, an elementary embedding is a special case of an embedding that preserves all first-order formulas

Order theory and domain theory

In order theory, an embedding of partial orders is a function F from X to Y such that :

$\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)$ . Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

In domain theory, an additional requirement is :

$\forall y\in Y:\{x: F(x)\leq y\}$ is directed. Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains.

Based on an article from FOLDOC, used by permission.

Metric spaces

A mapping $\phi: X \to Y$ of metric spaces is called an embedding (with distortion $C > 0$ ) if

$L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y)$

for some constant $L > 0$ . In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

Normed spaces

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to

One of the basic questions that can be asked about a finite-dimensional normed space $(X, \| \cdot \|)$ is, what is the maximal dimension $k$ such that the Hilbert space $\ell_2^k$ can be linearly embedded into $X$ with constant distortion?

The answer is given by Dvoretzky's theorem. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, in the theory of Banach spaces Dvoretzky's theorem is an important structural theorem proved by Aryeh Dvoretzky in the early 1960s

Category theory

In category theory, it is not possible to define an embedding without additional structures on the base category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets However, in all generality, it is possible to define what properties should satisfy a class of embeddings in a given category.

In all cases, the class of embeddings should contain all isomorphisms. Most of the time, embeddings are required to be stable under composition and be monic. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a

A common property of embeddings is that the class of all embedded subobjects of a given object, thought equivalent up to an isomorphism, is small, and thus an ordered set. In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously Ordered set is used with distinct meanings in Order theory. A set with a Binary relation R on its elements that is reflexive (for In this case, the category is said to be well powered with respect to the class of embeddings. This allows to define new local structures on the category (such as a closure operator). A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S

The kind of structures on a category allowing to define embeddings are:

a concrete category structure, embeddings are then defined as the morphisms with injective underlying function satisfying an initiality condition
a factorization system $(E, M)$ , embeddings are then defined as the morphisms in $M$ (in this case, the category is often required to be well powered with respect to $M$ ). In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving In Mathematics, it can be shown that every function can be written as the composite of a Surjective function followed by an Injective function

In most cases, concrete categories have a factorization structure $(E, M)$ where $M$ is the class of embeddings defined by the concrete structure. This is the case of the majority of the examples given in this article.

As usual in category theory, there is a dual concept, known as quotient. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the All the preceding properties can be dualized.

References

Adámek, Jiří; Horst Herrlich, George Strecker [1990] (2006). In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the Abstract and Concrete Categories (The Joy of Cats).

Dictionary

-noun

(mathematics) A map which maps a subspace (smaller structure) to the whole space (larger structure).
(computing) To inject, insert a code (malicious) in other code or into the operating system.

-verb

Present participle of embed.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents