Isomorphism
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an isomorphism is a structure-preserving mapping or morphism 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 is derived . 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 often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
are isomorphic and cannot be identified. 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 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 structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
s and categories. In the case of algebraic structures, 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 space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s. * A
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
is an isomorphism of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s. * A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
s. * A symplectomorphism is an isomorphism of symplectic manifolds. * A permutation is an automorphism of a set. * In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.
Category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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, 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 \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, i.e., \R^+ \cong \R via the isomorphism \log x. 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 A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...
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 and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
 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 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 In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
.


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, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order,
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then calle ...
, 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 \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 a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, isomorphisms are defined for all
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
s. 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s; 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 In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
. Just as the automorphisms of an
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
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, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, the Laplace transform is an isomorphism mapping hard differential equations into easier
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
ic equations. In
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, 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
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an isomorphism between two partially ordered sets ''P'' and ''Q'' is a bijective map f from ''P'' to ''Q'' that preserves the order structure in the sense that for any elements x and y of ''P'' we have x less than y in ''P'' if and only if f(x) is less than f(y) in ''Q''. As an example, the set of whole numbers ordered by the ''is-a-factor-of'' relation is isomorphic to the set 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
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 Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the effects of a system's actions (its outputs) return as inputs to that system, influencing subsequent action. It is concerned with ...
, 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 Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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. Two categories and are isomorphic if there exist
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s 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 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, the identity is an isomorphism, and 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.


Examples

Examples of isomorphism classes are plentiful in mathematics. * Two sets are isomorphic if there is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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 Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
X at a point p, though technically denoted \pi_1(X,p) to emphasize the dependence on the base point, is often written lazily as simply \pi_1(X) if 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 \pi_1(X,p) is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
s of \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

Although there are cases where isomorphic objects can be considered equal, one must distinguish and . 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\ \quad \text \quad B = \ are ; 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 \ and \ are not 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 :\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 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 \mapsto 1, \text \mapsto 2, \text \mapsto 3. 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. Also,
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s 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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s 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 number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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

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


Notes


References


Further reading

*


External links

* * {{Authority control Morphisms Equivalence (mathematics)