An embedding is a representation of a topological object, manifold, graph, field, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. For example, a field embedding preserves the algebraic structure of plus and times, an embedding of a topological space preserves open sets, and a graph embedding preserves connectivity.
One space 
is embedded in another space 
when the properties of 
restricted to 
are the same as the properties of 
. For example, the rationals are embedded in the reals, and
the integers are embedded in the rationals. In geometry, the sphere is embedded in
as the unit sphere.
Let 
and 
be structures for the same first-order language 
, and let 
be a homomorphism from 
to 
.
Then 
is an embedding provided that it is injective
(Enderton 1972, Grätzer 1979, Burris and Sankappanavar 1981).
For example, if 
and 
are partially ordered sets,
then an injective monotone mapping 
may not be an embedding from 
into 
. To be an embedding, such a mapping must preserve
order "both ways":
See also
Campbell's Theorem, Embeddable Knot, Embedded Surface, Extrinsic Curvature, Field, Graph Embedding, Hyperboloid Embedding, Injection, Manifold, Nash's Embedding Theorem, Submanifold
Portions of this entry contributed by Todd Rowland
Portions of this entry contributed by Matt Insall (author's link)
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References
Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.
Referenced on Wolfram|Alpha
Cite this as:
Insall, Matt; Rowland, Todd; and Weisstein, Eric W. "Embedding." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Embedding.html