Isomorphicism
in mathematics is an isomorphicism an illustration between two mathematical structures, by which parts of a structure are illustrated on „meaning-meaning “parts of another structure reversibly clearly (bijektiv).
For further meanings see also isomorphism.
Definition
a function f between two structures is called isomorphicism, if:
- f is bijektiv,
- f a Homomorphismus is,
- the inverse function f -1 a Homomorphismus is.
There is an isomorphicism between two structures, then the two structures are called to each other isomorphic. Isomorphic structures are in certain way “the same”, i.e. if one refrains from the representation of the elements of the underlying quantities and the name of the relations and linkages.
The statement “<math> X< /math> and <math> Y< /math> " become isomorphic usually as math <X> \ cong Y /math< are >written; in addition, there is the indications <math> \ simeq< /math> or <math> \ approx< /math> usually.
One does not consider that at groups, rings, bodies, vector spaces and unite other structures the third condition from the other two follows, one generally however upit to do without can, which is shown in the article Homöomorphismus by an example.
Often one can determine certain structures only up to isomorphism clearly, like e.g.
- the quotient body of an integrity ring,
- the only finite body of the order p n,
- that algebraic conclusion of a body,
- the completion of a metric area.
Isomorphicisms are gladly used in mathematics, in order to walk on a lighter calculation method. By the definitions (bijektiv), specified above, this is possible.
e.g.: Laplace transform; S-transformation
Examples
are <math> (X, *)< /math> and <math> (Y, +)< /math> Mix with a binary linkage, then is an isomorphicism of <math> X< /math> after <math> Y< /math> a Bijektion <math> f: X \ rightarrow Y< /math> with
- < math> f (u) + f (v) = f (u * v)< /math>
for all <math> u, v \ inX< /math>.
e.g.: <math> \ log (5) - \ log (2) = \ logs (5/2)< /math>
If the structures are groups, then such isomorphicism group isomorphicism is called. Usually one means such with isomorphicisms between algebraic structures such as groups, rings, bodies or vector spaces.
Are <math> (X, \ leq_X)< /math> and <math> (Y, \ leq_Y)< /math>totally arranged quantities, then is an isomorphicism from X to Y calls one order-receiving Bijektionen. They play an important role in the theory of the ordinal numbers.
Are (X, D) and (Y, D) metric areas, then is an isomorphicism from X to Y a Bijektion f with the characteristic
- D (f (u), f (v)) = D (u, v)
for all u, v in X. One calls such isomorphicisms isometries.
In universal algebra one can indicate a general definition of an isomorphicism, which covers these and other situations. The definition of an isomorphicism in the category theory is still more general.
One leaves into thataway, one receives Homomorphismen in each case to given examples the demand of the Bijektivität.
see also
| Wiktionary: Isomorphicism - word origin, synonyms and translations |
