Embedding
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":
REFERENCES:
Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. https://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.
