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(ĕm-bĕd') pronunciation also im·bed (ĭm-)

v., -bed·ded, also -bed·ded, -bed·ding, -bed·ding, -beds, -beds.

v.tr.
  1. To fix firmly in a surrounding mass: embed a post in concrete; fossils embedded in shale.
  2. To enclose snugly or firmly.
  3. To cause to be an integral part of a surrounding whole: "a minor accuracy embedded in a larger untruth" (Ian Jack).
  4. To assign (a journalist) to travel with a military unit during an armed conflict.
  5. Biology. To enclose (a specimen) in a supporting material before sectioning for microscopic examination.
v.intr.
To become embedded: The harpoon struck but did not embed.

n. (ĕm'bĕd')
One that is embedded, especially a journalist who is assigned to an active military unit.

embedment em·bed'ment n.


is spelt em-, not im-.

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Inserted into. See embedded system.

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also imbed

verb

    To implant so deeply as to make change nearly impossible: entrench, fasten, fix, infix, ingrain, lodge, root1. See move/halt.


v

Definition: sink, implant
Antonyms: dig up

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pronunciation

IN BRIEF: v. - Fix or set securely or deeply.

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Fixation of tissue in a firm medium, in order to keep it intact during cutting of thin sections for pathological examination.


adj

Referring to a tooth, root tip, or foreign body that is covered in bone.

  See crossword solutions for the clue Embed.

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.

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 : XY. 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.

The fact that a map f : XY is an embedding is often indicated by the use of a "hooked arrow", thus:  f : X \hookrightarrow Y. On the other hand, this notation is sometimes reserved for inclusion maps.

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 such cases it is common to identify the domain X with its image f(X) contained in Y, so that then XY.

Contents

Topology and geometry

General topology

In general topology, an embedding is a one-to-one function (i.e., an injection) that is a homeomorphism onto its image.[1] More explicitly, an injective continuous map f : XY between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : XY lets us treat X as a subspace of Y. Every embedding is injective and continuous. 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. The latter happens if the image f(X) is neither an open set nor a closed set in Y.

For a given space X, the existence of an embedding X → Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded into a space while 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. Then f is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).[2]

In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. 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 [3] states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding. An immersion of this surface is, however, possible in R3, and one example is Boy's surface—which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

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

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. An isometric embedding is a smooth embedding f : MN which preserves the metric in the sense that g is equal to the pullback of h by f, i.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. Nash embedding theorem).[4]

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 : EF.

The kernel of σ is an ideal of E which cannot be the whole field E, because of the condition σ(1)=1. 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. Hence, E is isomorphic to the subfield σ(E) of F. 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 model theory there is also a stronger notion of elementary embedding.

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).

In domain theory, an additional requirement is:

 \forall y\in Y:\{x: F(x)\leq y\} is directed.

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.

Normed spaces

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.

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.

Category theory

In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.

Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set. 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).

In a concrete category, an embedding is a morphism ƒA → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒgC → B, then g itself is a morphism.

A factorization system for a category also gives rise to a notion of embedding. If (EM) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense. 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. All the preceding properties can be dualized.

An embedding can also refer to an embedding functor.

See also

Notes

  1. ^ Sharpe, R.W. (1997) , page 16.
  2. ^ Warner, F.W. (1983) , page 22.
  3. ^ Whitney H., Differentiable manifolds, Ann. of Math. (2), 37 (1936), 645-680.
  4. ^ Nash J., The embedding problem for Riemannian manifolds, Ann. of Math. (2), 63 (1956), 20-63.

References

  • Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9 .
  • Warner, F.W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, ISBN 0-387-90894-3 .

External links


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Dansk (Danish)
v. tr. - omslutte, indstøbe, omgive, fastholde, indlejre, indlægge
v. intr. - putte i seng, anbringe i et leje

idioms:

  • embedded command    indlejret kode
  • embedded in concrete    indstøbt i cement

Nederlands (Dutch)
omsluiten, inbedden, vastzetten

Français (French)
v. tr. - enfoncer (dans le bois), couler (dans le ciment), sceller (dans la pierre), enchâsser (un bijou), incruster, (Ling) enchâsser, (fig) être gravé dans
v. intr. - être enfoncé dans, être scellé (dans la pierre), être enchâssé (un bijou), être coulé (dans le ciment)

idioms:

  • embedded command    (Comput) commande intégrée
  • embedded in concrete    noyé dans le béton

Deutsch (German)
v. - einlassen

idioms:

  • embedded command    eingebettete Befehle
  • embedded in concrete    einzementieren

Ελληνική (Greek)
v. - χώνω, θάβω, εντοιχίζω, εμφυτεύω

idioms:

  • embedded command    (Η/Υ) ένθετη εντολή
  • embedded in concrete    τσιμενταρισμένος

Italiano (Italian)
incastrare, immergere

Português (Portuguese)
v. - embutir, encaixar, implantar

Русский (Russian)
вставлять, врезать, внедрять

Español (Spanish)
v. tr. - empotrar, clavar, hincar, meter, fijar
v. intr. - clavarse, hincarse

idioms:

  • embedded command    orden sobreentendida, orden incluida en otra
  • embedded in concrete    definitivo, reforzado con hormigón

Svenska (Swedish)
v. - bädda in, förankra, innesluta

中文(简体)(Chinese (Simplified))
使插入, 深留, 使嵌入, 嵌入

idioms:

  • embedded command    嵌入的命令
  • embedded in concrete    在水泥中砌进

中文(繁體)(Chinese (Traditional))
v. tr. - 使插入, 深留, 使嵌入
v. intr. - 嵌入

idioms:

  • embedded command    嵌入的命令
  • embedded in concrete    在水泥中砌進

한국어 (Korean)
v. tr. - 끼워 넣다 , 깊이 간직하다
v. intr. - 파묻히다

日本語 (Japanese)
v. - ぴったりとはめ込む, 植え込む, 深くとどめる, はまり込む, はめ込む, 埋める

العربيه (Arabic)
‏(فعل) يطمر, يطوق باحكام‏

עברית (Hebrew)
v. tr. - ‮שיבץ, קבע, נעץ, שיכן‬
v. intr. - ‮השתבץ, נקבע במקומו‬


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