Embedding

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an embedding (or imbedding) is one instance of some <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a> contained within another instance, such as a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> that is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a>.
When some object 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is said to be embedded in another object 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, the embedding is given by some <a href="/facts/Injective_function/5ES6tqf5">injective</a> and structure-preserving map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\rightarrow Y}
 
. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are instances. In the terminology of <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a structure-preserving map is called a <a href="/facts/Morphism/Kdpuyz3S">morphism</a>.
The fact that a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\rightarrow Y}
 
 is an embedding is often indicated by the use of a "hooked arrow" (<a href="/facts/Unicode/DDu3MfPV">U+</a>21AA ↪ RIGHTWARDS ARROW WITH HOOK); thus: 
 
 
 
 f
 :
 X
 ↪
 Y
 .
 
 
 {\displaystyle f:X\hookrightarrow Y.}
 
 (On the other hand, this notation is sometimes reserved for <a href="/facts/Inclusion_map/VUFNdaDH">inclusion maps</a>.)
Given 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, several different embeddings of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the <a href="/facts/Natural_number/u65og8JE">natural numbers</a> in the <a href="/facts/Integer/PUwwrawx">integers</a>, the integers in the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, the rational numbers in the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, and the real numbers in the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. In such cases it is common to identify the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with its <a href="/facts/Image_(mathematics)/KANefMAl">image</a> 
 
 
 
 f
 (
 X
 )
 
 
 {\displaystyle f(X)}
 
 contained in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, so that 
 
 
 
 X
 ⊆
 Y
 
 
 {\displaystyle X\subseteq Y}
 
.

Embedding open-in-new

Embedding