HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an embedding (or imbedding) is one instance of some
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
contained within another instance, such as a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
that is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
. When some object ''X'' is said to be embedded in another object ''Y'', the embedding is given by some
injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

injective
and structure-preserving map . 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 Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
, a structure-preserving map is called a
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

morphism
. The fact that a map 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 map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s.) 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 number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s in the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s, the integers in the
rational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s, the rational numbers in the
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, and the real numbers in the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. In such cases it is common to identify the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
''X'' with its
image An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows ...
''f''(''X'') contained in ''Y'', so that .


Topology and geometry


General topology

In
general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is t ...
, an embedding is a
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
onto its image. More explicitly, an injective
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
map f : X \to Y between
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
inherited from Y). Intuitively then, the embedding f : X \to Y lets us treat X as a
subspace
subspace
of Y. Every embedding is injective and
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. Every map that is injective, continuous and either
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
nor a
closed set In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
in Y. For a given space Y, the existence of an embedding X \to Y is a
topological invariantIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
of X. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.


Differential topology

In
differential topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: Let M and N be smooth
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
s and f:M\to N be a smooth map. Then f is called an immersion if its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...
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 and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
onto its image). In other words, the domain of an embedding is
diffeomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to its image, and in particular the image of an embedding must be a
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. An immersion is precisely 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 = \mathbb^n. The interest here is in how large n must be for an embedding, in terms of the dimension m of M. The
Whitney embedding theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
states that n = 2m is enough, and is the best possible linear bound. For example, the
real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...
RP''m'' of dimension m, where m is a power of two, requires n = 2m for an embedding. However, this does not apply to immersions; for instance, RP2 can be immersed in \mathbb^3 as is explicitly shown by
Boy's surface In geometry, Boy's surface is an immersion (mathematics), immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' ...
—which has self-intersections. The
Roman surface The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion (mathematics), immersion of the projective ...
fails to be an immersion as it contains
cross-cap In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. An embedding is proper if it behaves well with respect to 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
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally *Euler force, Transverse force (or ''Euler force''), ...
to \partial Y in any point of f(\partial X). 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 and pseudo-Riemannian geometry

In
Riemannian geometry#REDIRECT Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...
and pseudo-Riemannian geometry: Let (''M'', ''g'') and (''N'', ''h'') be
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
s or more generally
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...
s. An isometric embedding is a smooth embedding ''f'' : ''M'' → ''N'' which preserves the (pseudo-)
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
in the sense that ''g'' is equal to the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
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 (pseudo)-Riemannian manifolds which preserves the (pseudo)-Riemannian metrics. Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s (cf.
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space Euclidean space is the fundamental space of classical geometry. Ori ...
).Nash J., ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), 63 (1956), 20–63.


Algebra

In general, for an algebraic category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism that is injective.


Field theory

In field theory, an embedding of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''E'' in a field ''F'' is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
. The
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
of ''σ'' is an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
of ''E'' which cannot be the whole field ''E'', because of the condition . 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 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
. Hence, ''E'' is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...
and A,B are σ-
structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ...
(also called σ-algebras in
universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...
or models in
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
), then a map h:A \to B is a σ-embedding iff all of the following hold: * 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 embeddingIn model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis ...
.


Order theory and domain theory

In
order theory Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
, an embedding of
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
s is a function ''F'' between partially ordered sets ''X'' and ''Y'' such that :\forall x_1,x_2\in X: x_1\leq x_2 \iff F(x_1)\leq F(x_2). Injectivity of ''F'' follows quickly from this definition. In
domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer s ...
, an additional requirement is that : \forall y\in Y:\ is
directed Director may refer to: Literature * Director (magazine), ''Director'' (magazine), a British magazine * The Director (novel), ''The Director'' (novel), a 1971 novel by Henry Denker * The Director (play), ''The Director'' (play), a 2000 play by Nanc ...
.


Metric spaces

A mapping \phi: X \to Y of
metric spaces Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
is called an ''embedding'' (with
distortion Distortion is the alteration of the original shape (or other characteristic) of something. In and it means the alteration of the of an information-bearing , such as an representing sound or a representing images, in an electronic device or ...
C>0) if : L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) for every x,y\in X and some constant L>0.


Normed spaces

An important special case is that of
normed spaces The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk Dunkirk (, ; french: Dunkerque ; vls, label=French Flemish, Duunkerke; nl, Duinkerke(n) ) is a Communes of France, ...
; 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(X, \, \cdot \, ) is, ''what is the maximal dimension k such that the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
\ell_2^k can be linearly embedded into X with constant distortion?'' The answer is given by
Dvoretzky's theoremIn mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimension ...
.


Category theory

In
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
, 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
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
s. Ideally the class of all embedded
subobject In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
s of a given object, up to isomorphism, should also be
small Small may refer to: Science and technology * SMALL Small may refer to: Science and technology * SMALL Small may refer to: Science and technology * SMALL Small may refer to: Science and technology * SMALL, an ALGOL-like programming language ...
, and thus an
ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a
closure operatorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
). In a
concrete category In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, an embedding is a morphism ''ƒ'': ''A'' → ''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 ''ƒg'': ''C'' → ''B'', then ''g'' itself is a morphism. A factorization system for a category also gives rise to a notion of embedding. If (''E'', ''M'') 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
concept, known as quotient. All the preceding properties can be dualized. An embedding can also refer to an embedding functor.


See also

*
Closed immersion:''For the same-name concept in differential geometry, see immersion (mathematics).'' In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algeb ...
*
Cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
*
Dimension reduction Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
* Immersion *
Johnson–Lindenstrauss lemmaIn mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson (mathematician), William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclide ...
*
Submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
* Subspace * Universal space


Notes


References

* * * * * * * * * * * * . * * .


External links

*
Embedding of manifolds
on the Manifold Atlas {{set index article Abstract algebra Category theory General topology Differential topology Functions and mappings Maps of manifolds Model theory Order theory