Isomorphism

The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "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 are the same up to an isomorphism.^{[citation needed]}

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 $p$ , all fields with $p$ elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.

The term isomorphism is mainly used for algebraic structures. In this case, mappings are called homomorphisms, 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 spaces.
A homeomorphism is an isomorphism of topological spaces.
A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
A symplectomorphism is an isomorphism of symplectic manifolds.
A permutation is an automorphism of a set.
In geometry, isomorphisms and automorphisms are often called transformations, for example rigid 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 the multiplicative group of positive real numbers, and let $\mathbb {R}$ be the additive group of real numbers.

The logarithm function $\log :\mathbb {R} ^{+}\to \mathbb {R}$ satisfies $\log(xy)=\log x+\log y$ for all $x,y\in \mathbb {R} ^{+},$ so it is a group homomorphism. The exponential 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 $\exp$ are inverses of 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 a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.

Integers modulo 6

Consider the group $(\mathbb {Z} _{6},+),$ the integers from 0 to 5 with addition modulo 6. Also consider the group $\left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),$ 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: ${\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}$ 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 indeed isomorphic: their structures are exactly the same. More generally, the direct product of two cyclic groups $\mathbb {Z} _{m}$ and $\mathbb {Z} _{n}$ is isomorphic to $(\mathbb {Z} _{mn},+)$ 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 $f:X\to Y$ such that:^[1] $\operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)$

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, well-order, strict weak order, 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 ordering ≤ and S an ordering $\scriptstyle \sqsubseteq ,$ then an isomorphism from X to Y 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 an order isomorphism or (less commonly) an isotone isomorphism.

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

Applications

In algebra, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example:

Linear isomorphisms between vector spaces; they are specified by invertible matrices.
Group isomorphisms between groups; the classification of isomorphism classes of finite groups is an open problem.
Ring isomorphism between rings.
Field isomorphisms are the same as ring isomorphism between fields; their study, and more specifically the study of field automorphisms is an important part of Galois theory.

Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing a common structure form a 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 algebraic 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 u to vertex v in G if and only if there is an edge from $f(u)$ to $f(v)$ in H. See graph isomorphism.

In mathematical analysis, an isomorphism between two Hilbert spaces 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 or 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

In category theory, given a category C, 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 bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

Two categories $C$ and $D$ are isomorphic if there exist functors $F:C\to D$ 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 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 sets. In algebraic categories (specifically, categories of 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).

Relation with equality

Although, there are cases where isomorphic objects can be considered as equal, one must distinguish equality and isomorphism.^[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 $A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}$ are equal; they are merely different representations—the first an intensional one (in set builder notation), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets $\{A,B,C\}$ and $\{1,2,3\}$ are not equal since they have not 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 them 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, integers and even numbers are isomorphic as ordered sets and abelian group (for addition), cannot be considered as equal sets, since one is a proper subset of the other.

On the other hand, when sets (or other mathematical objects) are definied only by their properties, without considering the nature of their elements, one consider often them as equal. This is generally the case with solutions of universal properties.

For example, the rational numbers are usually defined as equivalence classes of pairs of integers, although nobody think of a rational number as a set (equivalence class). The universal property of the rational numbers is essentially that they form a field that 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 transfered to the other through the isomorphism. For example the real numbers that 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 (as 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 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 ${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$

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

[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]