Isomorphism

In abstract algebra, an isomorphism (Greek: ἴσος isos "equal", and μορφή morphe "shape") is a bijective map f such that both f and its inverse f⁻¹ are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f⁻¹: Y → X, with the property that both f⁻¹f = id_X and f f⁻¹ = id_Y.

Informally, an isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, isomorphic structures are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined.

Purpose

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property which is preserved by an isomorphism and which is true of one of the objects, is also true of the other. If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to "solid ground" where the problem is easier to understand and work with.

Practical example

The following are examples of isomorphisms from ordinary algebra.

Consider the logarithm function: For any fixed base b, the logarithm function log_b maps from the positive real numbers $\mathbb {R} ^{+}$ onto the real numbers $\mathbb {R}$ ; formally:
$\log _{b}:\mathbb {R} ^{+}\to \mathbb {R} .\!$
This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group $(\mathbb {R} ^{+},\times )$ of positive real numbers under ordinary multiplication. The logarithm function obeys the following identity:
$\log _{b}(xy)=\log _{b}(x)+\log _{b}(y).\!$
But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group $(\mathbb {R} ^{+},\times )$ to the group $(\mathbb {R} ,+)$ .
Logarithms can therefore be used to simplify multiplication of real numbers. By working with logarithms, multiplication of positive real numbers is replaced by addition of logs. This way it is possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.
Consider the group Z₆, the numbers from 0 to 5 with addition modulo 6. Also consider the group Z₂ × Z₃, 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, if you identify them using the following scheme:
(0,0) → 0

(1,1) → 1

(0,2) → 2

(1,0) → 3

(0,1) → 4

(1,2) → 5
or in general (a,b) → ( 3a + 4 b ) mod 6. For example note that (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 Z_n and Z_m is cyclic if and only if n and m are coprime.

Abstract examples

A 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 such that

f(u) S f(v) if and only if u R v.

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, {{#invoke:Binary_relation|Relations_over_a_set|trichotomous}}, a partial order, total 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 $\sqsubseteq$ , then an isomorphism from X to Y is a bijective function f : X → Y such that

f(u)\sqsubseteq f(v)

if and only if u ≤ v.

Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.

If X = Y we have a relation-preserving automorphism.

An operation-preserving isomorphism

Suppose that on these sets X and Y, there are two binary operations $\star$ and $\Diamond$ which happen to constitute the groups (X, $\star$ ) and (Y, $\Diamond$ ). Note that the operators operate on elements from the domain and range, respectively, of the "one-to-one" and "onto" function f. There is an isomorphism from X to Y if the bijective function f : X → Y happens to produce results, that sets up a correspondence between the operator $\star$ and the operator $\Diamond$ .

f(u)\Diamond f(v)=f(u\star v)

for all u, v in X.

Applications

In abstract algebra, two basic isomorphisms are defined:

Group isomorphism, an isomorphism between groups
Ring isomorphism, an isomorphism between rings. (Note that isomorphisms between fields are actually ring isomorphisms)

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 category theory, Iet the category C consist of two classes, one of objects and the other of morphisms. Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism f : a → b that has an inverse, i.e. there exists a morphism g : b → a with 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.

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 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 regulator part and the processing part of the system.

Relation with equality

In certain areas of mathematics, notably category theory, it is very valuable to distinguish between equality on the one hand and isomorphism on the other.^[1] Equality is when two objects are "literally the same", while isomorphism is when two objects "can be made to correspond via an isomorphism". For example, the sets $A=\{x\in \mathbf {Z} \mid x^{2}<2\}$ and $B=\{-1,0,1\}$ are equal – they are two different presentations (one in set builder notation, one by an enumeration) of the same subset of the integers. By contrast, the sets $\{{\text{A}},{\text{B}},{\text{C}}\}$ and $\{1,2,3\}$ are not equal – the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their cardinality (number of elements) and these both have three elements, but there are many choices of isomorphism – 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 better than any other.^{[note 1]}^{[note 2]} Thus one cannot identify these two sets: one can choose an isomorphism between them, but any statement that identifies these two sets depends on the choice of isomorphism.

A motivating example of this distinction is the distinction between a finite-dimensional vector space V and its dual space $V^{*}.$ These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism $V{\overset {\sim }{\to }}V^{*}.$ If one chooses a basis for V, then this yields an isomorphism; this corresponds to transforming a column vector (element of V) to a row vector (element of $V^{*}$ ) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis". More subtly, there is a map from a vector space to its double dual $V\to V^{**}$ that does not depend on the choice of basis. This leads to a third notion, that of a natural isomorphism: while $V$ and $V^{**}$ are different sets (the first consists of vectors v ( $V=\{v\}$ ), while the latter consists of functions from the dual space $V^{*}$ to the field K of scalars ( $V^{**}=\{\phi \colon V^{*}\to K\}$ )), there is a "natural" choice of isomorphism between them. This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a natural transformation: briefly, that one may consistently identify, or more generally map from, a vector space to its double dual, across all vector spaces: one may map $V{\overset {\sim }{\to }}V^{**}$ and $W{\overset {\sim }{\to }}W^{**}$ and so forth for all vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory.

If one wishes to draw a distinction between an arbitrary isomorphism (one that depends on a choice) and a natural isomorphism (one that can be done consistently), one may write $\approx$ for an unnatural isomorphism and $\cong$ for a natural isomorphism, as in $V\approx V^{*}$ and $V\cong V^{**};$ this convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.

Generally, saying that two objects are equal is reserved for when there is a notion of a larger (ambient) space which these objects live within. Most often, one speaks of equality of two subsets of a given set (as in the integer set example above), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensional space $S^{2}:=\{(x,y,z)\in \mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}$ and the Riemann sphere ${\widehat {\mathbf {C} }}$ – which can be presented as the one-point compactification of the complex plane $\mathbf {C} \cup \{\infty \}$ or as the complex projective line (a quotient space) $\mathbf {P} _{\mathbf {C} }^{1}:=(\mathbf {C} ^{2}\setminus \{(0,0)\})/(\mathbf {C} ^{*})$ – are three different descriptions for a mathematical object, all of which are isomorphic, but which are not equal because they are not all subsets of a single space: the first is a subset of $\mathbf {R} ^{3},$ the second is $\mathbf {C} \cong \mathbf {R} ^{2}$ ^{[note 3]} plus an additional point, and the third is a subquotient of $\mathbf {C} ^{2}.$

Notes

^ The careful reader may note that A, B, C have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus in some sense there is a "natural" isomorphism, namely ${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$ This intuition can be formalized by saying that any two finite ordered sets of the same cardinality have a natural isomorphism, by sending the least element of the first to the least element of the second, the least element of what remains in the first to the least element of what remains in the second, and so forth, but conversely two finite unordered sets are not naturally isomorphic because there is more than one choice of map – except if the cardinality is 0 or 1, where there is a unique choice.
^ In fact, there are precisely $3!=6$ different isomorphism between two sets with three elements. This is equal to the number of automorphisms of a given three element set (the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted $\operatorname {Iso} (A,B),$ is a torsor for the automorphism group of A, $\operatorname {Aut} (A)$ and also a torsor for the automorphism group of B. In fact, automorphisms of an object are a key reason to be concerned with the distinction between isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or with its double dual, as elaborated in the sequel.
^ Being precise, the identification of the complex numbers with the real plane, $\mathbf {C} \cong \mathbf {R} \cdot 1\oplus \mathbf {R} \cdot i=\mathbf {R} ^{2}$ depends on a choice of $i;$ one can just as easily choose $(-i),$ which yields a different identification – formally, complex conjugation is an automorphism – but in practice one often assumes that one has made such an identification.