Isomorphism

In mathematics , an isomorphism (from ancient Greek ἴσος (ísos) - "equal" and μορφή (morphḗ) - "form", "shape") is a mapping between two mathematical structures through which parts of one structure can be reversed to parts of the same meaning in another structure clearly (bijectively) mapped.

definition

Universal algebra

In universal algebra , a function between two algebraic structures (for example groups , rings , bodies or vector spaces ) is called an isomorphism if: ${\ displaystyle \ varphi}$

${\ displaystyle \ varphi}$ is bijective ,
${\ displaystyle \ varphi}$ is a homomorphism .

If there is an isomorphism between two algebraic structures, then the two structures are called isomorphic to one another . Isomorphic structures are in a certain way "the same", namely if one disregards the representation of the elements of the underlying sets and the names of the relations and connections.

The statement “ and are isomorphic” is usually written through or through . ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle \ simeq}$ ${\ displaystyle X \ cong Y}$

If there is a bijective homomorphism between two algebraic structures, then there is always also a bijective homomorphism. ${\ displaystyle \ varphi}$ ${\ displaystyle \ varphi ^ {- 1}}$

Relational structures

Let and two relational structures of the same type be denoted so that for each the arity of the relations and . A bijection is called isomorphism if it has the following compatibility property for each and every one of them : ${\ displaystyle {\ varvec {A}} = (A, (R_ {i}))}$ ${\ displaystyle {\ boldsymbol {B}} = (B, (S_ {i}))}$ ${\ displaystyle (n_ {i}),}$ ${\ displaystyle n_ {i} \ in \ mathbb {N}}$ ${\ displaystyle i}$ ${\ displaystyle R_ {i}}$ ${\ displaystyle S_ {i}}$ ${\ displaystyle \ varphi \ colon A \ to B}$ ${\ displaystyle i}$ ${\ displaystyle a_ {1}, \ ldots, a_ {n_ {i}} \ in A}$

{\ displaystyle (a_ {1}, \ ldots, a_ {n_ {i}}) \ in R_ {i} \ Leftrightarrow (\ varphi (a_ {1}), \ ldots, \ varphi (a_ {n_ {i}) })) \ in S_ {i}.}

In contrast to algebraic structures, not every bijective homomorphism between relational structures is an isomorphism. An example of isomorphisms between relational structures are isomorphisms between graphs .

Category theory

In category theory, an isomorphism is generally defined as a morphism that has a two-sided inverse : ${\ displaystyle f \ colon X \ to Y,}$ ${\ displaystyle f ^ {- 1} \ colon \, Y \ to X}$

{\ displaystyle f \ circ f ^ {- 1} = \ operatorname {id} _ {Y}}

and

{\ displaystyle f ^ {- 1} \ circ f = \ operatorname {id} _ {X}.}

The isomorphisms between algebraic structures and between relational structures defined above are special cases of this definition. Further special cases of this concept of isomorphism are, for example, homeomorphisms as isomorphisms in the category of topological spaces and continuous mappings or homotopy equivalences as isomorphisms in the category of topological spaces with the homotopy classes of mappings as morphisms.

meaning

In category theory it is of crucial importance that the property isomorphism is preserved under each functor ; H. is an isomorphism in a category and a functor, then is ${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle C}$ ${\ displaystyle F \ colon C \ to D}$

{\ displaystyle F (f) \ colon F (X) \ to F (Y)}

also an isomorphism, in the category . In algebraic topology , this property is often determined in order to be able to bring spaces into relation: if, for example, the fundamental groups of two spaces are isomorphic, the spaces are homeomorphic . ${\ displaystyle D}$

Examples

If and are sets with a binary connection , then there is a bijection with ${\ displaystyle (X, \ cdot)}$ ${\ displaystyle \ left (Y, + \ right)}$ ${\ displaystyle f \ colon X \ to Y}$

{\ displaystyle f (u) + f (v) = f (u \ cdot v)}

for all

{\ displaystyle u, v \ in X}

an isomorphism from to . For example, the logarithm is an isomorphism from to , there . ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle (\ mathbb {R} ^ {+}, /)}$ ${\ displaystyle (\ mathbb {R}, -)}$ ${\ displaystyle \ log (x) - \ log (y) = \ log \ left ({\ tfrac {x} {y}} \ right)}$

A binary link is a three-digit relation. But homo- and isomorphisms can also be defined for two-digit relations (see below #Order isomorphism ).

For some isomorphisms, the homomorphism of the function also implies that of the inverse function; with the others you have to prove it separately.

Group isomorphism

If the structures are groups , then such an isomorphism is called a group isomorphism . With isomorphisms, one usually means those between algebraic structures such as groups, rings , bodies or vector spaces .

Isometric isomorphism

Are and metric spaces and is a bijection from to with the property ${\ displaystyle \ left (X, d \ right)}$ ${\ displaystyle \ left (Y, D \ right)}$ ${\ displaystyle f}$ ${\ displaystyle X}$ ${\ displaystyle Y}$

{\ displaystyle D \ left (f (u), f (v) \ right) = d (u, v)}

for all ,

{\ displaystyle u, v \ in X}

then one calls an isometric isomorphism . ${\ displaystyle f}$

In the previous examples, isomorphisms are exactly the homomorphic bijections - the inverse mapping is automatically homomorphic. In the following examples it must also be required that the inverse mapping is also homomorphic.

In functional analysis , a mapping between normalized spaces is called an isomorphism if it has the following properties: ${\ displaystyle T \ colon X \ to Y}$ ${\ displaystyle (X, \ | \ cdot \ | _ {X}), (Y, \ | \ cdot \ | _ {Y})}$

{\ displaystyle T}

is linear

{\ displaystyle T}

is steady

The inverse function is also continuous ${\ displaystyle T ^ {- 1}}$

Additionally, if all is true , it is called an isometric isomorphism . ${\ displaystyle x \ in X}$ ${\ displaystyle \ | T (x) \ | _ {Y} = \ | x \ | _ {X}}$ ${\ displaystyle T}$

Order isomorphism

If and are ordered sets, then an (order) isomorphism from to is an order-preserving bijection whose inverse function is also order-preserving. Order-preserving bijections between totally ordered sets are automatically isomorphisms; This does not apply to partial orders: it is obviously an order-preserving bijection from with the partial relation to with the normal order, but not in the opposite direction. Order isomorphisms play an important role in the theory of ordinal numbers . It also says, and are ordnungsisomorph or the same type of order . The order type of natural numbers is denoted by and that of rational numbers by . The order type of the rational numbers in the open interval is also. Both are dense in their respective completion . The order types of the real numbers and the interval are also the same, but different from since there is no bijection between and . ${\ displaystyle (X, \ leq _ {X})}$ ${\ displaystyle (Y, \ leq _ {Y})}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle n \ mapsto n}$ ${\ displaystyle \ mathbb {N} ^ {+}}$ ${\ displaystyle \ mathbb {N} ^ {+}}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ eta}$ ${\ displaystyle \ {q \ in \ mathbb {Q} \ mid 0 <q <1 \}}$ ${\ displaystyle \ eta.}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle (0,1)}$ ${\ displaystyle \ eta,}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {Q}}$