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Examples

Logarithm and exponential

Let R + {\displaystyle \mathbb {R} ^{+}} be the multiplicative group of positive real numbers, and let R {\displaystyle \mathbb {R} } be the additive group of real numbers.

The logarithm function log : R + → R {\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} } satisfies log ⁡ ( x y ) = log ⁡ x + log ⁡ y {\displaystyle \log(xy)=\log x+\log y} for all x , y ∈ R + , {\displaystyle x,y\in \mathbb {R} ^{+},} so it is a group homomorphism. The exponential function exp : R → R + {\displaystyle \exp :\mathbb {R} \to \mathbb {R} ^{+}} satisfies exp ⁡ ( x + y ) = ( exp ⁡ x ) ( exp ⁡ y ) {\displaystyle \exp(x+y)=(\exp x)(\exp y)} for all x , y ∈ R , {\displaystyle x,y\in \mathbb {R} ,} so it too is a homomorphism.

The identities log ⁡ exp ⁡ x = x {\displaystyle \log \exp x=x} and exp ⁡ log ⁡ y = y {\displaystyle \exp \log y=y} show that log {\displaystyle \log } and exp {\displaystyle \exp } are inverses of each other. Since log {\displaystyle \log } is a homomorphism that has an inverse that is also a homomorphism, log {\displaystyle \log } is an isomorphism of groups, i.e., R + ≅ R {\displaystyle \mathbb {R} ^{+}\cong \mathbb {R} } via the isomorphism log ⁡ x {\displaystyle \log x} .

The log {\displaystyle \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 ( Z 6 , + ) , {\displaystyle (\mathbb {Z} _{6},+),} the integers from 0 to 5 with addition modulo 6. Also consider the group ( Z 2 × Z 3 , + ) , {\displaystyle \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: ( 0 , 0 ) ↦ 0 ( 1 , 1 ) ↦ 1 ( 0 , 2 ) ↦ 2 ( 1 , 0 ) ↦ 3 ( 0 , 1 ) ↦ 4 ( 1 , 2 ) ↦ 5 {\displaystyle {\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 ) ↦ ( 3 a + 4 b ) mod 6. {\displaystyle (a,b)\mapsto (3a+4b)\mod 6.}

For example, ( 1 , 1 ) + ( 1 , 0 ) = ( 0 , 1 ) , {\displaystyle (1,1)+(1,0)=(0,1),} which translates in the other system as 1 + 3 = 4. {\displaystyle 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 Z m {\displaystyle \mathbb {Z} _{m}} and Z n {\displaystyle \mathbb {Z} _{n}} is isomorphic to ( Z m n , + ) {\displaystyle (\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 → Y {\displaystyle f:X\to Y} such that:¹ S ⁡ ( f ( u ) , f ( v ) )  if and only if  R ⁡ ( u , v ) {\displaystyle \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 ⊑ , {\displaystyle \scriptstyle \sqsubseteq ,} then an isomorphism from X to Y is a bijective function f : X → Y {\displaystyle f:X\to Y} such that f ( u ) ⊑ f ( v )  if and only if  u ≤ v . {\displaystyle 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 , {\displaystyle 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 ) {\displaystyle f(u)} to f ( v ) {\displaystyle f(v)} in H. See graph isomorphism.

In order theory, an isomorphism between two partially ordered sets P and Q is a bijective map f {\displaystyle f} from P to Q that preserves the order structure in the sense that for any elements x {\displaystyle x} and y {\displaystyle y} of P we have x {\displaystyle x} less than y {\displaystyle y} in P if and only if f ( x ) {\displaystyle f(x)} is less than f ( y ) {\displaystyle f(y)} 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 blood types ordered by the can-donate-to relation. See order 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 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 C, an isomorphism is a morphism f : a → b {\displaystyle f:a\to b} that has an inverse morphism g : b → a , {\displaystyle g:b\to a,} that is, f g = 1 b {\displaystyle fg=1_{b}} and g f = 1 a . {\displaystyle gf=1_{a}.}

Two categories C and D are isomorphic if there exist functors F : C → D {\displaystyle F:C\to D} and G : D → C {\displaystyle G:D\to C} which are mutually inverse to each other, that is, F G = 1 D {\displaystyle FG=1_{D}} (the identity functor on D) and G F = 1 C {\displaystyle 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).

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.²

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 classification of closed surfaces enumerates the isomorphism classes of all connected closed 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 planar subalgebras 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 substructures 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 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 space X {\displaystyle X} at a point p {\displaystyle p} , though technically denoted π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} to emphasize the dependence on the base point, is often written lazily as simply π 1 ( X ) {\displaystyle \pi _{1}(X)} if X {\displaystyle X} is path connected. The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other; however, unless π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} , specifically distinguishing between isomorphic but 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) and coherent isomorphism

Although there are cases where isomorphic objects can be considered equal, one must distinguish equality and isomorphism.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 = { x ∈ Z ∣ x 2 < 2 }  and  B = { − 1 , 0 , 1 } {\displaystyle 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 } {\displaystyle \{A,B,C\}} and { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} are not 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

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

while another is

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

and no one isomorphism is intrinsically better than any other.⁴ 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 groups (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 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 of universal properties.

For example, the rational numbers are formally defined as equivalence 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 a 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 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.

See also

• Mathematics portal

• Bisimulation
• Equivalence relation
• Heap (mathematics)
• Isometry
• Isomorphism class
• Isomorphism theorem
• Universal property
• Coherent isomorphism
• Balanced category

Notes

Further reading

• Mazur, Barry (12 June 2007), When is one thing equal to some other thing? (PDF)

External links

Look up isomorphism in Wiktionary, the free dictionary.

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

References

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

2. Awodey, Steve (2006). "Isomorphisms". Category theory. Oxford University Press. p. 11. ISBN 9780198568612. 9780198568612 ↩

3. Mazur 2007 - Mazur, Barry (12 June 2007), When is one thing equal to some other thing? (PDF) 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 ↩

4. 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 A ↦ 1 , B ↦ 2 , C ↦ 3. {\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.} ↩