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Isomorphism

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Inversible mapping (mathematics)

This article is about mathematics. For other uses, seeIsomorphism (disambiguation).

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Thegroup of fifthroots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.

Inmathematics, anisomorphism is a structure-preservingmapping (amorphism) between twostructures of the same type that can be reversed by aninverse mapping. Two mathematical structures areisomorphic if an isomorphism exists between them. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, 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 to an isomorphism.^{[citation needed]}

Anautomorphism is an isomorphism from a structure to itself. An isomorphism between two structures is acanonical isomorphism (acanonical map that 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 numberp, allfields withp elements are canonically isomorphic, with a unique isomorphism. Theisomorphism theorems provide canonical isomorphisms that are not unique.

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

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

Anisometry is an isomorphism ofmetric spaces.
Ahomeomorphism is an isomorphism oftopological spaces.
Adiffeomorphism is an isomorphism of spaces equipped with adifferential structure, typicallydifferentiable manifolds.
Asymplectomorphism is an isomorphism ofsymplectic manifolds.
Apermutation is 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

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Logarithm and exponential

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Let $\mathbb {R} ^{+}$ be themultiplicative group ofpositive 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 $\exp$ areinverses of each other. Since $\log$ is a homomorphism that has an inverse that is also a homomorphism, $\log$ is anisomorphism of groups, i.e., $\mathbb {R} ^{+}\cong \mathbb {R}$ via the isomorphism $\log x$ .

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 aruler and atable of logarithms, or using aslide rule with a logarithmic scale.

Integers modulo 6

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Consider the group $(\mathbb {Z} _{6},+),$ the integers from 0 to 5 with additionmodulo 6. Also consider the group $\left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),$ the ordered pairs where thex coordinates 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 product of twocyclic groups $\mathbb {Z} _{m}$ and $\mathbb {Z} _{n}$ is isomorphic to $(\mathbb {Z} _{mn},+)$ if and only ifm andn arecoprime, per theChinese remainder theorem.

Relation-preserving isomorphism

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If one object consists of a setX with abinary relation R and the other object consists of a setY with a binary relation S then an isomorphism fromX toY is 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 fromX toY is 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 isomorphism or (less commonly) anisotone isomorphism.

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

Applications

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

Linear isomorphisms betweenvector spaces; they are specified byinvertible matrices.
Group isomorphisms betweengroups; the classification ofisomorphism classes offinite groups is an open problem.
Ring isomorphism betweenrings.
Field isomorphisms are the same as ring isomorphism betweenfields; their study, and more specifically the study offield automorphisms is an important part ofGalois theory.

Just as theautomorphisms of analgebraic structure form 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 transform is an isomorphism mapping harddifferential equations into easieralgebraic equations.

Ingraph theory, an isomorphism between two graphsG andH is abijective mapf from the vertices ofG to the vertices ofH that preserves the "edge structure" in the sense that there is an edge fromvertexu to vertexv inG if and only if there is an edge from $f(u)$ to $f(v)$ inH. Seegraph isomorphism.

Inorder theory, an isomorphism between two partially ordered setsP andQ is abijective map $f {\displaystyle f}$ fromP toQ that preserves the order structure in the sense that for any elements $x {\displaystyle x}$ and $y {\displaystyle y}$ ofP we have $x {\displaystyle x}$ less than $y {\displaystyle y}$ inP if and only if $f(x)$ is less than $f(y)$ inQ. As an example, the set {1,2,3,6} of whole numbers ordered by theis-a-factor-of relation is isomorphic to the set {O,A,B,AB} ofblood types ordered by thecan-donate-to relation. Seeorder isomorphism.

In mathematical analysis, an isomorphism between twoHilbert spaces is 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 Russell andLudwig Wittgenstein to be isomorphic. An example of this line of thinking can be found in Russell'sIntroduction to Mathematical Philosophy.

Incybernetics, thegood regulator theorem or Conant–Ashby theorem is stated as "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

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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 $gf=1_{a}.$

Two categoriesC andD areisomorphic if 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 onD) and $GF=1_{C}$ (the identity functor onC).

Isomorphism vs. bijective morphism

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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 spaces or 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).

Isomorphism class

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Since a composition of isomorphisms is an isomorphism, since the identity is an isomorphism and since the inverse of an isomorphism is an isomorphism, the relation that two mathematical objects are isomorphic is anequivalence relation. Anequivalence class given by isomorphisms is commonly called anisomorphism class.^[2]

Examples

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Examples of isomorphism classes are plentiful in mathematics.

Two sets are isomorphic if there is abijection between them. The isomorphism class of a finite set can be identified with the non-negative integer representing the number of elements it contains.
The isomorphism class of afinite-dimensional vector space can be identified with the non-negative integer representing its dimension.
Theclassification of finite simple groups enumerates the isomorphism classes of allfinite simple groups.
Theclassification of closed surfaces enumerates the isomorphism classes of all connectedclosed surfaces.
Ordinals are essentially defined as isomorphism classes of well-ordered sets (though there are technical issues involved).
There are three isomorphism classes of the planarsubalgebras of M(2,R), the 2 x 2 real matrices.

However, there are circumstances in which the isomorphism class of an object conceals vital information about it.

Given amathematical structure, it is common that twosubstructures belong to the same isomorphism class. However, the way they are included in the whole structure can not be studied if they are identified. For example, in a finite-dimensional vector space, allsubspaces of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc.
Inhomotopy theory, thefundamental group of aspace $X {\displaystyle X}$ at a point $p {\displaystyle p}$ , though technically denoted $\pi _{1}(X,p)$ to emphasize the dependence on the base point, is often written lazily as simply $\pi _{1}(X)$ if $X {\displaystyle X}$ ispath connected. The reason for this is that the existence of a path between two points allows one to identifyloops at one with loops at the other; however, unless $\pi _{1}(X,p)$ isabelian this isomorphism is non-unique. Furthermore, the classification ofcovering spaces makes strict reference to particularsubgroups of $\pi _{1}(X,p)$ , specifically distinguishing between isomorphic butconjugate subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.

Relation to equality

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Although there are cases where isomorphic objects can be considered equal, one must distinguishequality andisomorphism.^[3] 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 anintensional one (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 notequal since 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,integers andeven numbers are isomorphic asordered sets andabelian groups (for addition), but cannot be considered equal sets, since one is aproper subset of 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 numbers are formally defined asequivalence classes of 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 afield that contains the integers and does not contain any proper subfield. 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. Thereal numbers that can be expressed as a quotient of integers form the smallest subfield of the reals. There is thus a unique isomorphism from this subfield of the reals to the rational numbers defined by equivalence classes.

Notes

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^ $A, B, C {\displaystyle A,B,C}$ have a conventional order, namely the alphabetical 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 ${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$