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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 X is embedded in another space Y when the properties of Y restricted to X are the same as the properties of X. For example, the rationals are embedded in the reals, and the integers are embedded in the rationals. In geometry, the sphere is embedded in R^3 as the unit sphere.

Let A=(A,(c_(c in C)^A,(P^A)_(P in P),(f^A)_(f in F)) and B=(B,(c_(c in C)^B,(P^B)_(P in P),(f^B)_(f in F)) be structures for the same first-order language L, and let h:A->B be a homomorphism from A to B. Then h is an embedding provided that it is injective (Enderton 1972, Grätzer 1979, Burris and Sankappanavar 1981).

For example, if (X,<=) and (Y,<=) are partially ordered sets, then an injective monotone mapping h:X->Y may not be an embedding from (X,<=) into (Y,<=). To be an embedding, such a mapping must preserve order "both ways":

 h(x)<=h(y)<==>x<=y.

See also

Campbell's Theorem, Embeddable Knot, Embedded Surface, Extrinsic Curvature, Field, Graph Embedding, Hyperboloid Embedding, Injection, Manifold, Nash's Embedding Theorem, Sphere Embedding, 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.

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Embedding

Cite this as:

Insall, Matt; Rowland, Todd; and Weisstein, Eric W. "Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Embedding.html

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