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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 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, Sphere Embedding, Submanifold Portions of this entry contributed by Todd Rowland
Portions of this entry contributed by Matt Insall (author's link)
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: Embedding
CITE THIS AS:Insall, Matt; Rowland, Todd; and Weisstein, Eric W. "Embedding." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Embedding.html
site: http://mathworld.wolfram.com/Embedding.html
Dennis
Dennis Mazur
Dear Dennis do you mean by a monotone injective map, something like
total partial order that did not clearly define the equality relation such as a total probability order, partial order without a non strict part that came out partially re-presentable
x>y then F(x)>F(y) ,
x F(y)
contradicting F(x)=F(y)
or xy or x=y iff x>= y. cant say which one is which
(3) a surjective order imbedding , is well defined appears to be a strict weak ordwith a total pre-order, where
x=y iff x=x;er, that
gets F(x)=F(y) iff x=y part right and x>=y iff F(x)>=F(y)
but may not only be a strict trichotomout total order, as the symbols > or < have not been introced, with a strict weak order, and so is is totally ordered', under = and = 'where x=y iff x=x; so it may be a completely comparability relation of a kind between totality and trichotomotomy which may induce a strict part, over the equivalence classes.
(4) an order automorhism is presumably x=y iff F(x)=F(y)
F(x)>F(y) iff F(x)>F(y)
F(x)
hat is not into, something that does not satisfy the function requirement, in the usual sense,
'in the usual sense' x=y then F(x)=F(y)
by not ' from into' or 'onto'; or something like a galois connection or a strict partial order,
fails in both injective monotone map, as in an order embedding
in the context of a partial order,
which if x=y then x=y (anit symmetry(
anby monotonicity and which gives F(x)=F(y), F(x)=F(y)
As even an order embedding
xy or x=y
where its (1)total x=y but not trichotmous,
(2)and no specific meaning for x=y is given, x=y in not completely defintely terms of x=y but only in one direction, by anti symmetry;
but which comes about as strongly re presentable over the total quasi comparability relation >= =F(y) that is also injective , appears to be like strong ;partial representation.
an attempt at a strong strong representation that failed, fine grained representation than an order embedding ie a trichotomy, x=y or x< y or x> y that failed due to certain axioms failing where either only obtained x>y then PR(x)>PR(y) with the additional PR(X)=PR(Y) then x=y due to the total order
(2) where x=y , and its a total strict partial order, but something when wrong so its just a totally ordered, partially representable order,
may be defined clearly. and was only partially represented, partial reprsentationin qualitative probability
here as x=y as there are no incomparables then if F(x)=F(y), then its trichotomous and partially representable then if x not equal to y, then it must be that xF(y) contradicting F(x)=F(y), and x>y F(x)>F(y)
appear to be strict partial order,
as anti symmetry is not a bi-conditional - unlike in a surjective order imbedding,
then such that hat an order embedding does not preserve x=y then F(x)=F(y) or F(x)=F(y) then x= y,
may not be sur-jective, unless its a surjective order imbedding, or order isomophism
IF this is just a function from orders, to different order than to a real valued range then it makes sense (or
although generally injective monotone functions in the context of functions are equivalent to which gives F(x)=F(y) to strictly monotone functions ,
x< y then F(x) y then F(x)>F(y)and its a function
x=y implies F(x)=F(y)
x=y then F9
and being a function are and in the context of a total order, to a biconditional strictly monotone functions (which are numerical functions generally)
and when one has an injective monotone function
one has (A)x
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