Isomorphism

Thegroupof fifthroots of unityunder multiplication is isomorphic to the group of rotations of the regular pentagon under composition.

Inmathematics,anisomorphismis a structure-preservingmappingbetween twostructuresof the same type that can be reversed by aninverse mapping.Two mathematical structures areisomorphicif an isomorphism exists between them. The word isomorphism is derived from theAncient Greek:ἴσοςisos"equal", andμορφήmorphe"form" or "shape".

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects arethe sameup toan isomorphism.^{[citation needed]}

Anautomorphismis an isomorphism from a structure to itself. An isomorphism between two structures is acanonical isomorphism(acanonical mapthat is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of auniversal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for everyprime number $p$ ,allfieldswith $p$ elements are canonically isomorphic, with a unique isomorphism. Theisomorphism theoremsprovide canonical isomorphisms that are not unique.

The termisomorphismis mainly used foralgebraic structures.In this case, mappings are calledhomomorphisms,and a homomorphism is an isomorphismif and only ifit isbijective.

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:

Anisometryis an isomorphism ofmetric spaces.
Ahomeomorphismis an isomorphism oftopological spaces.
Adiffeomorphismis an isomorphism of spaces equipped with adifferential structure,typicallydifferentiable manifolds.
Asymplectomorphismis an isomorphism ofsymplectic manifolds.
Apermutationis an automorphism of aset.
Ingeometry,isomorphisms and automorphisms are often calledtransformations,for examplerigid transformations,affine transformations,projective transformations.

Category theory,which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.

Examples

Logarithm and exponential

Let $\mathbb {R} ^{+}$ be themultiplicative groupofpositive real numbers,and let $\mathbb {R}$ be the additive group of real numbers.

Thelogarithm function $\log:\mathbb {R} ^{+}\to \mathbb {R}$ satisfies $\log(xy)=\log x+\log y$ for all $x,y\in \mathbb {R} ^{+},$ so it is agroup homomorphism.Theexponential function $\exp:\mathbb {R} \to \mathbb {R} ^{+}$ satisfies $\exp(x+y)=(\exp x)(\exp y)$ for all $x,y\in \mathbb {R},$ so it too is a homomorphism.

The identities $\log \exp x=x$ and $\exp \log y=y$ show that $\log$ and $\exp$ areinversesof each other. Since $\log$ is a homomorphism that has an inverse that is also a homomorphism, $\log$ is an isomorphism of groups.

The $\log$ function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using arulerand atable of logarithms,or using aslide rulewith a logarithmic scale.

Integers modulo 6

Consider the group $(\mathbb {Z} _{6},+),$ the integers from 0 to 5 with additionmodulo6. Also consider the group $\left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),$ the ordered pairs where thexcoordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in thex-coordinate is modulo 2 and addition in they-coordinate is modulo 3.

These structures are isomorphic under addition, under the following scheme: ${\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}$ or in general $(a,b)\mapsto (3a+4b)\mod 6.$

For example, $(1,1)+(1,0)=(0,1),$ which translates in the other system as $1+3=4.$

Even though these two groups "look" different in that the sets contain different elements, they are indeedisomorphic:their structures are exactly the same. More generally, thedirect productof twocyclic groups $\mathbb {Z} _{m}$ and $\mathbb {Z} _{n}$ is isomorphic to $(\mathbb {Z} _{mn},+)$ if and only ifmandnarecoprime,per theChinese remainder theorem.

Relation-preserving isomorphism

If one object consists of a setXwith abinary relationR and the other object consists of a setYwith a binary relation S then an isomorphism fromXtoYis a bijective function $f:X\to Y$ such that:^[1] $\operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)$

S isreflexive,irreflexive,symmetric,antisymmetric,asymmetric,transitive,total,trichotomous,apartial order,total order,well-order,strict weak order,total preorder(weak order), anequivalence relation,or a relation with any other special properties, if and only if R is.

For example, R is anordering≤ and S an ordering $\scriptstyle \sqsubseteq,$ then an isomorphism fromXtoYis a bijective function $f:X\to Y$ such that $f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.$ Such an isomorphism is called anorder isomorphismor (less commonly) anisotone isomorphism.

If $X=Y,$ then this is a relation-preservingautomorphism.

Applications

Inalgebra,isomorphisms are defined for allalgebraic structures.Some are more specifically studied; for example:

Linear isomorphismsbetweenvector spaces;they are specified byinvertible matrices.
Group isomorphismsbetweengroups;the classification ofisomorphism classesoffinite groupsis an open problem.
Ring isomorphismbetweenrings.
Field isomorphisms are the same as ring isomorphism betweenfields;their study, and more specifically the study offield automorphismsis an important part ofGalois theory.

Just as theautomorphismsof analgebraic structureform agroup,the isomorphisms between two algebras sharing a common structure form aheap.Letting a particular isomorphism identify the two structures turns this heap into a group.

Inmathematical analysis,theLaplace transformis an isomorphism mapping harddifferential equationsinto easieralgebraicequations.

Ingraph theory,an isomorphism between two graphsGandHis abijectivemapffrom the vertices ofGto the vertices ofHthat preserves the "edge structure" in the sense that there is an edge fromvertexuto vertexvinGif and only if there is an edge from $f(u)$ to $f(v)$ inH.Seegraph isomorphism.

In mathematical analysis, an isomorphism between twoHilbert spacesis a bijection preserving addition, scalar multiplication, and inner product.

In early theories oflogical atomism,the formal relationship between facts and true propositions was theorized byBertrand RussellandLudwig Wittgensteinto be isomorphic. An example of this line of thinking can be found in Russell'sIntroduction to Mathematical Philosophy.

Incybernetics,thegood regulatoror Conant–Ashby theorem is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system.

Category theoretic view

Incategory theory,given acategoryC,an isomorphism is a morphism $f:a\to b$ that has an inverse morphism $g:b\to a,$ that is, $fg=1_{b}$ and $gf=1_{a}.$ For example, a bijectivelinear mapis an isomorphism betweenvector spaces,and a bijectivecontinuous functionwhose inverse is also continuous is an isomorphism betweentopological spaces,called ahomeomorphism.

Two categories $C$ and $D$ areisomorphicif there existfunctors $F:C\to D$ and $G:D\to C$ which are mutually inverse to each other, that is, $FG=1_{D}$ (the identity functor on $D$ ) and $GF=1_{C}$ (the identity functor on $C$ ).

Isomorphism vs. bijective morphism

In aconcrete category(roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as thecategory of topological spacesor categories of algebraic objects (like thecategory of groups,thecategory of rings,and thecategory of modules), an isomorphism must be bijective on theunderlying sets.In algebraic categories (specifically, categories ofvarieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).

Relation to equality

Although there are cases where isomorphic objects can be considered equal, one must distinguishequalityandisomorphism.^[2]Equality is when two objects are the same, and therefore everything that is true about one object is true about the other. On the other hand, isomorphisms are related to some structure, and two isomorphic objects share only the properties that are related to this structure.

For example, the sets $A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}$ areequal;they are merely different representations—the first anintensionalone (inset builder notation), and the secondextensional(by explicit enumeration)—of the same subset of the integers. By contrast, the sets $\{A,B,C\}$ and $\{1,2,3\}$ are notequalsince they do not have the same elements. They are isomorphic as sets, but there are many choices (in fact 6) of an isomorphism between them: one isomorphism is

{\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3,

while another is

{\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,

and no one isomorphism is intrinsically better than any other.^{[note 1]}On this view and in this sense, these two sets are not equal because one cannot consider themidentical:one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism.

Also,integersandeven numbersare isomorphic asordered setsandabelian groups(for addition), but cannot be considered equal sets, since one is aproper subsetof the other.

On the other hand, when sets (or othermathematical objects) are defined only by their properties, without considering the nature of their elements, one often considers them to be equal. This is generally the case with solutions ofuniversal properties.

For example, therational numbersare usually defined asequivalence classesof pairs of integers, although nobody thinks of a rational number as a set (equivalence class). The universal property of the rational numbers is essentially that they form afieldthat contains the integers and does not contain any proper subfield. It results that given two fields with these properties, there is a unique field isomorphism between them. This allows identifying these two fields, since every property of one of them can be transferred to the other through the isomorphism. For example thereal numbersthat are obtained by dividing two integers (inside the real numbers) form the smallest subfield of the real numbers. There is thus a unique isomorphism from the rational numbers (defined as equivalence classes of pairs) to the quotients of two real numbers that are integers. This allows identifying these two sorts of rational numbers.

Notes

^ $A,B,C$ have a conventional order, namely the Alpha betical order, and similarly 1, 2, 3 have the usual order of the integers. Viewed as ordered sets, there is only one isomorphism between them, namely ${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$

References

^Vinberg, Ėrnest Borisovich (2003).A Course in Algebra.American Mathematical Society. p. 3.ISBN 9780821834138.
^Mazur 2007

External links

"Isomorphism",Encyclopedia of Mathematics,EMS Press,2001 [1994]
Weisstein, Eric W."Isomorphism".MathWorld.

[3] $A,B,C$ have a conventional order, namely the Alpha betical order, and similarly 1, 2, 3 have the usual order of the integers. Viewed as ordered sets, there is only one isomorphism between them, namely ${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$

[1] Vinberg, Ėrnest Borisovich (2003).A Course in Algebra.American Mathematical Society. p. 3.ISBN 9780821834138.

[2] Mazur 2007

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[note 1]