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{{Short description|Inversible mapping (mathematics)}}
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In [[mathematics]], an '''isomorphism''' is a structure-preserving [[Map (mathematics)|mapping]] (a [[morphism]]) between two [[Mathematical structure|structures]] of the same type that can be reversed by an [[inverse function|inverse mapping]]. Two mathematical structures are '''isomorphic''' if an isomorphism exists between them. The word is derived {{ety|grc|''{{linktext|ἴσος}}'' (isos)|equal||''{{linktext|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 are {{em|the same [[up to]] an isomorphism}}.{{citation needed |date=April 2021}}

An [[automorphism]] is an isomorphism from a structure to itself. An isomorphism between two structures is a '''canonical isomorphism'''  (a [[canonical map]] that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a [[universal property]]), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every [[prime number]] {{mvar|p}}, all [[Field (mathematics)|fields]] with {{mvar|p}} elements are canonically isomorphic, with a unique isomorphism. The [[isomorphism theorems]] provide canonical isomorphisms that are not unique.

The term {{em|isomorphism}} is mainly used for [[algebraic structure]]s. In this case, mappings are called [[homomorphism]]s, and a homomorphism is an isomorphism [[if and only if]] it is [[bijective]].

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:
* An [[isometry]] is an isomorphism of [[metric space]]s.
* A [[homeomorphism]] is an isomorphism of [[topological space]]s.
* A [[diffeomorphism]] is an isomorphism of spaces equipped with a [[differential structure]], typically [[differentiable manifold]]s.
* A [[symplectomorphism]] is an isomorphism of [[symplectic manifold]]s.
* A [[permutation]] is an automorphism of a [[set (mathematics)|set]].
* In [[geometry]], isomorphisms and automorphisms are often called [[transformation (function)|transformations]], for example [[rigid transformation]]s, [[affine transformation]]s, [[projective transformation]]s.

[[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 <math> \R ^+ </math> be the [[multiplicative group]] of [[positive real numbers]], and let <math>\R</math> be the additive group of real numbers.

The [[logarithm function]] <math>\log : \R^+ \to \R</math> satisfies <math>\log(xy) = \log x + \log y</math> for all <math>x, y \in \R^+,</math> so it is a [[group homomorphism]]. The [[exponential function]] <math>\exp : \R \to \R^+</math> satisfies <math>\exp(x+y) = (\exp x)(\exp y)</math> for all <math>x, y \in \R,</math> so it too is a homomorphism.

The identities <math>\log \exp x = x</math> and <math>\exp \log y = y</math> show that <math>\log</math> and <math>\exp </math> are [[inverse function|inverses]] of each other. Since <math>\log</math> is a homomorphism that has an inverse that is also a homomorphism, <math>\log</math> is an [[Group isomorphism|isomorphism of groups]], i.e., <math>\R^+ \cong \R</math> via the isomorphism <math>\log x</math>.

The <math>\log</math> 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 a [[ruler]] and a [[table of logarithms]], or using a [[slide rule]] with a logarithmic scale.

===Integers modulo 6===
Consider the group <math>(\Z_6, +),</math> the integers from 0 to 5 with addition [[Modular arithmetic|modulo]]&nbsp;6. Also consider the group <math>\left(\Z_2 \times \Z_3, +\right),</math> the ordered pairs where the ''x'' coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the ''x''-coordinate is modulo 2 and addition in the ''y''-coordinate is modulo 3.

These structures are isomorphic under addition, under the following scheme:
<math display="block">\begin{alignat}{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{alignat}</math>
or in general <math>(a, b) \mapsto (3 a + 4 b) \mod 6.</math>

For example, <math>(1, 1) + (1, 0) = (0, 1),</math> which translates in the other system as <math>1 + 3 = 4.</math>

Even though these two groups "look" different in that the sets contain different elements, they are indeed '''isomorphic''': their structures are exactly the same. More generally, the [[direct product of groups|direct product]] of two [[cyclic group]]s <math>\Z_m</math> and <math>\Z_n</math> is isomorphic to <math>(\Z_{mn}, +)</math> if and only if ''m'' and ''n'' are [[coprime]], per the [[Chinese remainder theorem]].

===Relation-preserving isomorphism===
If one object consists of a set ''X'' with a [[binary relation]] R and the other object consists of a set ''Y'' with a binary relation S then an isomorphism from ''X'' to ''Y'' is a bijective function <math>f : X \to Y</math> such that:<ref>{{Cite book|author=Vinberg, Ėrnest Borisovich|title=A Course in Algebra|publisher=American Mathematical Society|year=2003|isbn=9780821834138|page=3|url=https://books.google.com/books?id=kd24d3mwaecC&pg=PA3}}</ref>
<math display="block">\operatorname{S}(f(u),f(v)) \quad \text{ if and only if } \quad \operatorname{R}(u,v) </math>

S is [[Reflexive relation|reflexive]], [[Irreflexive relation|irreflexive]], [[Symmetric relation|symmetric]], [[Antisymmetric relation|antisymmetric]], [[Asymmetric relation|asymmetric]], [[Transitive relation|transitive]], [[Connected relation|total]], [[Homogeneous relation#Properties|trichotomous]], a [[partial order]], [[total order]], [[well-order]], [[strict weak order]], [[Strict weak order#Total preorders|total preorder]] (weak order), an [[equivalence relation]], or a relation with any other special properties, if and only if R is.

For example, R is an [[Order theory|ordering]] ≤ and S an ordering <math>\scriptstyle \sqsubseteq,</math> then an isomorphism from ''X'' to ''Y'' is a bijective function <math>f : X \to Y</math> such that
<math display="block">f(u) \sqsubseteq f(v) \quad \text{ if and only if } \quad  u \leq v.</math>
Such an isomorphism is called an {{em|[[order isomorphism]]}} or (less commonly) an {{em|isotone isomorphism}}.

If <math>X = Y,</math> then this is a relation-preserving [[automorphism]].

==Applications==
In [[algebra]], isomorphisms are defined for all [[algebraic structure]]s. Some are more specifically studied; for example:
* [[Linear isomorphism]]s between [[vector space]]s; they are specified by [[invertible matrices]].
* [[Group isomorphism]]s between [[group (mathematics)|groups]]; the classification of [[isomorphism class]]es of [[finite group]]s is an open problem.
* [[Ring isomorphism]] between [[ring (mathematics)|rings]]. 
* Field isomorphisms are the same as ring isomorphism between [[field (mathematics)|fields]]; their study, and more specifically the study of [[field automorphism]]s is an important part of [[Galois theory]].

Just as the [[automorphism]]s of an [[algebraic structure]] form a [[group (mathematics)|group]], the isomorphisms between two algebras sharing a common structure form a [[heap (mathematics)|heap]]. Letting a particular isomorphism identify the two structures turns this heap into a group.

In [[mathematical analysis]], the [[Laplace transform]] is an isomorphism mapping hard [[differential equations]] into easier [[algebra]]ic equations.

In [[graph theory]], an isomorphism between two graphs ''G'' and ''H'' is a [[bijective]] map ''f'' from the vertices of ''G'' to the vertices of ''H'' that preserves the "edge structure" in the sense that there is an edge from [[Vertex (graph theory)|vertex]] ''u'' to vertex ''v'' in ''G'' if and only if there is an edge from <math>f(u)</math> to <math>f(v)</math> in ''H''. See [[graph isomorphism]].

In [[order theory]], an isomorphism between two partially ordered sets ''P'' and ''Q'' is a [[bijective]] map <math>f</math> from ''P'' to ''Q'' that preserves the order structure in the sense that for any elements <math>x</math> and <math>y</math> of ''P'' we have <math>x</math> less than <math>y</math> in ''P'' if and only if <math>f(x)</math> is less than <math>f(y)</math> in ''Q''. As an example, the set {1,2,3,6} of whole numbers ordered by the ''is-a-factor-of'' relation is isomorphic to the set {''O'', ''A'', ''B'', ''AB''} of [[ABO blood group system|blood types]] ordered by the ''can-donate-to'' relation. See [[order isomorphism]].

In mathematical analysis, an isomorphism between two [[Hilbert space]]s is a bijection preserving addition, scalar multiplication, and inner product.

In early theories of [[logical atomism]], the formal relationship between facts and true propositions was theorized by [[Bertrand Russell]] and [[Ludwig Wittgenstein]] to be isomorphic. An example of this line of thinking can be found in Russell's ''[[Introduction to Mathematical Philosophy]]''.

In [[cybernetics]], the [[good 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==
In [[category theory]], given a [[category (mathematics)|category]] ''C'', an isomorphism is a morphism <math>f : a \to b</math> that has an inverse morphism <math>g : b \to a,</math> that is, <math>f g = 1_b</math> and <math>g f = 1_a.</math> <!-- This is discussed below. Consider the [[equivalence relation]] that regards two objects as related if there is an isomorphism between them. The [[equivalence class]]es of this equivalence relation are called isomorphism classes. -->

Two categories {{mvar|C}} and {{mvar|D}} are [[Isomorphism of categories|isomorphic]] if there exist [[functor]]s <math>F : C \to D</math> and <math>G : D \to C</math> which are mutually inverse to each other, that is, <math>FG = 1_D</math> (the identity functor on {{mvar|D}}) and <math>GF = 1_C</math> (the identity functor on {{mvar|C}}).

===Isomorphism vs. bijective morphism===
In a [[concrete category]] (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the [[category of topological spaces]] or categories of algebraic objects (like the [[category of groups]], the [[category of rings]], and the [[category of modules]]), an isomorphism must be bijective on the [[underlying set]]s. In algebraic categories (specifically, categories of [[variety (universal algebra)|varieties 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==
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 an [[equivalence relation]]. An [[equivalence class]] given by isomorphisms is commonly called an '''isomorphism class'''.<ref>{{cite book|author=Awodey, Steve|author-link = Steve Awodey|chapter=Isomorphisms|title=Category theory|publisher=Oxford University Press|year=2006|isbn=9780198568612|page=11|chapter-url=https://books.google.com/books?id=IK_sIDI2TCwC&pg=PA11}}</ref>

=== Examples ===
Examples of isomorphism classes are plentiful in mathematics.
* Two sets are isomorphic if there is a [[bijection]] 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 a [[finite-dimensional vector space]] can be identified with the non-negative integer representing its dimension.
* The [[classification of finite simple groups]] enumerates the isomorphism classes of all [[finite simple groups]].
* The [[Surface (topology)#Classification of closed surfaces|classification of closed surfaces]] enumerates the isomorphism classes of all connected [[closed surface]]s.
* [[ordinal number|Ordinals]] are essentially defined as isomorphism classes of well-ordered sets (though there are technical issues involved).
* There are three isomorphism classes of the planar [[subalgebra]]s 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 a [[mathematical structure]], it is common that two [[substructure (mathematics)|substructure]]s 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, all [[Linear subspace|subspaces]] of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc.
* In [[homotopy theory]], the [[fundamental group]] of a [[topological space|space]] <math>X</math> at a point <math>p</math>, though technically denoted <math>\pi_1(X,p)</math> to emphasize the dependence on the base point, is often written lazily as simply <math>\pi_1(X)</math> if <math>X</math> is [[connected space#Path connectedness|path connected]]. The reason for this is that the existence of a path between two points allows one to identify [[loop (topology)|loops]] at one with loops at the other; however, unless <math>\pi_1(X,p)</math> is [[abelian group|abelian]] this isomorphism is non-unique. Furthermore, the classification of [[covering space]]s makes strict reference to particular [[subgroup]]s of <math>\pi_1(X,p)</math>, specifically distinguishing between isomorphic but [[conjugacy class|conjugate]] 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==
{{See also|Equality (mathematics)|coherent isomorphism}}

Although there are cases where isomorphic objects can be considered equal, one must distinguish  {{em|[[Equality (mathematics)|equality]]}} and {{em|isomorphism}}.<ref>{{Harvnb|Mazur|2007}}</ref> 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
<math display="block">A = \left\{ x \in \Z \mid x^2 < 2\right\} \quad \text{ and } \quad B = \{-1, 0, 1\}</math>
are {{em|equal}}; they are merely different representations—the first an [[intensional definition|intensional]] one (in [[set builder notation]]), and the second [[extensional definition|extensional]] (by explicit enumeration)—of the same subset of the integers. By contrast, the sets <math>\{A, B, C\}</math> and <math>\{1, 2, 3\}</math> are not {{em|equal}} 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
:<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3,</math> 
while another is 
:<math>\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,</math>
and no one isomorphism is intrinsically better than any other.<ref group="note"><math>A, B, C</math> 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 
<math display="block">\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.</math></ref> On this view and in this sense, these two sets are not equal because one cannot consider them {{em|identical}}: 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, [[integer]]s and [[even number]]s are isomorphic as [[ordered set]]s and [[abelian group]]s (for addition), but cannot be considered equal sets, since one is a [[proper subset]] of the other.

On the other hand, when sets (or other [[mathematical object]]s) 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 of [[universal properties]].

For example, the [[rational number]]s are formally defined as [[equivalence class]]es 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 a [[field (mathematics)|field]] 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. The [[real number]]s 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.

==See also==
{{Portal|Mathematics}}
{{Div col|colwidth=20em}}
*[[Bisimulation]]
*[[Equivalence relation]]
*[[Heap (mathematics)]]
*[[Isometry]]
*[[Isomorphism class]]
*[[Isomorphism theorem]]
*[[Universal property]]
*[[Coherent isomorphism]]
*[[Balanced category]]
{{Div col end}}

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

==Further reading==
* {{Citation|first = Barry|last = Mazur|author-link = Barry Mazur|title = When is one thing equal to some other thing?|date = 12 June 2007|url = https://bpb-us-e1.wpmucdn.com/sites.harvard.edu/dist/a/189/files/2023/01/When-is-one-thing-equal-to-some-other-thing.pdf }}

==External links==
{{Wiktionary|isomorphism}}
*{{Springer|title=Isomorphism|id=p/i052840}}
*{{MathWorld|urlname=Isomorphism|title = Isomorphism}}
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[[Category:Morphisms]]
[[Category:Equivalence (mathematics)]]