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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
: 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 . 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 it 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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, all fields with elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. The term is mainly used for algebraic structures. In this case, mappings are called
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s, and a homomorphism is an isomorphism
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
is an isomorphism of
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s. * A
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
is an isomorphism of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s. * 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
. * In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 \R^+ be the multiplicative group of
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
, and let \R be the additive group of real numbers. The logarithm function \log : \R^+ \to \R satisfies \log(xy) = \log x + \log y for all x, y \in \R^+, so it is a group homomorphism. The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
\exp : \R \to \R^+ satisfies \exp(x+y) = (\exp x)(\exp y) for all x, y \in \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 (\Z_6, +), the integers from 0 to 5 with addition
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
 6. Also consider the group \left(\Z_2 \times \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 (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 or in general (a, b) \mapsto (3 a + 4 b) \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 In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of two cyclic groups \Z_m and \Z_n is isomorphic to (\Z_, +) 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: \operatorname(f(u),f(v)) \quad \text \quad \operatorname(u,v) S is reflexive,
irreflexive In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
,
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
, 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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
≤ 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 \quad u \leq v. Such an isomorphism is called an or (less commonly) an . If X = Y, then this is a relation-preserving automorphism.


Applications

In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example: * Linear isomorphisms between
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s; they are specified by invertible matrices. * Group isomorphisms between
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
; 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 Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
is an isomorphism mapping hard differential equations into easier
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic equations. In
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, 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 Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
''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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
and
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian- British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is consi ...
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, f g = 1_b and g f = 1_a. For example, a bijective
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
is an isomorphism between
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, and a bijective
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
whose inverse is also continuous is an isomorphism between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, called a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
. Two categories and 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 ) and GF = 1_C (the identity functor on ).


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 In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s. 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

In certain areas of mathematics, notably category theory, it is valuable to distinguish between on the one hand and on the other. Equality is when two objects are exactly the same, and everything that is true about one object is true about the other, while an isomorphism implies everything that is true about a designated part of one object's structure is true about the other's. For example, the sets A = \left\ \quad \text \quad B = \ are ; they are merely different representations—the first an intensional one (in
set builder notation In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Def ...
), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets \ and \ are not —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 \mapsto 1, \text \mapsto 2, \text \mapsto 3, while another is \text \mapsto 3, \text \mapsto 2, \text \mapsto 1, and no one isomorphism is intrinsically better than any other.A, B, C have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus one particular isomorphism is "natural", namely \text \mapsto 1, \text \mapsto 2, \text \mapsto 3. More formally, as these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as they are naturally isomorphic (there is a unique isomorphism, given above), since
finite total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
s are uniquely determined up to unique isomorphism by cardinality. This intuition can be formalized by saying that any two finite totally ordered sets of the same cardinality have a natural isomorphism, the one that sends 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 in general, pairs of sets of a given finite cardinality 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 isomorphisms between two sets with three elements. This is equal to the number of automorphisms of a given three-element set (which in turn is equal to the order of the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted \operatorname(A,B), is a torsor for the automorphism group of ''A,'' \operatorname(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. On this view and in this sense, these two sets are not equal because one cannot consider them : 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. Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a finite-dimensional vector space ''V'' and its dual space V^* = \left\ of linear maps from ''V'' to its field of scalars \mathbf. 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 \mathrel V^*. If one chooses a basis for ''V'', then this yields an isomorphism: For all u, v \in V, v \mathrel \phi_v \in V^* \quad \text \quad \phi_v(u) = v^\mathrm u. 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 a map from a vector space ''V'' to its
double dual In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by co ...
V^ = \left\ that does not depend on the choice of basis: For all v \in V \text \varphi \in V^*, v \mathrel x_v \in V^ \quad \text \quad x_v(\phi) = \phi(v). This leads to a third notion, that of a natural isomorphism: while V and V^ are different sets, 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 identify, or more generally map from, a finite-dimensional vector space to its double dual, V \mathrel V^, for vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory. However, there is a case where the distinction between natural isomorphism and equality is usually not made. That is for the objects that may be characterized by a universal property. In fact, there is a unique isomorphism, necessarily natural, between two objects sharing the same universal property. A typical example is the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, which may be defined through infinite decimal expansion, infinite binary expansion, Cauchy sequences, Dedekind cuts and many other ways. Formally, these constructions define different objects which are all solutions with the same universal property. As these objects have exactly the same properties, one may forget the method of construction and consider them as equal. This is what everybody does when referring to " set of the real numbers". The same occurs with quotient spaces: they are commonly constructed as sets of equivalence classes. However, referring to a set of sets may be counterintuitive, and so quotient spaces are commonly considered as a pair of a set of undetermined objects, often called "points", and a surjective map onto this set. If one wishes to distinguish 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 In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
and 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 is reserved for when there is a notion of a larger (ambient) space that these objects live in. 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 := \left\ and the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
\widehat which can be presented as the one-point compactification of the complex plane \Complex \cup \ as the complex projective line (a quotient space) \mathbf_^1 := \left(\Complex^2\setminus \\right) / \left(\Complex^*\right) are three different descriptions for a mathematical object, all of which are isomorphic, but not because they are not all subsets of a single space: the first is a subset of \R^3, the second is \Complex \cong \R^2Being precise, the identification of the complex numbers with the real plane, \Complex \cong \R \cdot 1 \oplus \R \cdot i = \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. plus an additional point, and the third is a
subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
of \Complex^2. In the context of category theory, objects are usually at most isomorphic—indeed, a motivation for the development of category theory was showing that different constructions in homology theory yielded equivalent (isomorphic) groups. Given maps between two objects ''X'' and ''Y'', however, one asks if they are equal or not (they are both elements of the set \hom(X, Y), hence equality is the proper relationship), particularly in commutative diagrams. See also: homotopy type theory, in which isomorphisms can be treated as kinds of equality.


See also

*
Bisimulation In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if ...
* Equivalence relation *
Heap (mathematics) In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set ''H'' with a ternary operation denoted ,y,z\in H that satisfies a modified associativity property: \forall a,b,c,d,e \in H \ \ \ \ a,b,cd,e] = ,c,b.html"_;" ...
*
Isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
* Isomorphism class *
Isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
* Universal property *
Coherent isomorphism In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax ...


Notes


References


Further reading

*


External links

* * {{Authority control Morphisms Equivalence (mathematics)