Thegroup of fifthroots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.
Inmathematics, anisomorphism is a structure-preservingmapping ormorphism between twostructures of the same type that can be reversed by aninverse mapping. Two mathematical structures areisomorphic if an isomorphism exists between them, and this is often denoted as. The word is derived from Ancient Greekἴσος (isos)'equal' and μορφή (morphe)'form, 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 often be identified. Inmathematical jargon, one says that two objects are the sameup 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 avector space are isomorphic and cannot be identified.
Anautomorphism is an isomorphism from a structure to itself. An isomorphism between two structures is acanonical isomorphism (acanonical map that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of auniversal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for everyprime numberp, allfields withp elements are canonically isomorphic, with a unique isomorphism. Theisomorphism theorems provide canonical isomorphisms that are not unique.
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.
The identities and show that and areinverses of each other. So,aregroup isomorphisms that are inverse of each other.
The 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 aruler and atable of logarithms, or using aslide rule with a logarithmic scale.
Consider thering of the integers from 0 to 5 with addition and multiplicationmodulo 6. Also consider the ring of the ordered pairs where the first element is an integer modulo 2 and the second element is an integer modulo 3, with component-wise addition and multiplication modulo 2 and 3.
These rings are isomorphic under the following map:or in general
For example, which translates in the other system as
This is a special case of theChinese remainder theorem which asserts that, if and arecoprime integers, the ring of the integers modulo is isomorphic to the direct product of the integers modulo and the integers modulo.
If one object consists of a setX with abinary relation R and the other object consists of a setY with a binary relation S then an isomorphism fromX toY is a bijective function such that:[1]
For example, R is anordering ≤ and S an ordering then an isomorphism fromX toY is a bijective function such thatSuch an isomorphism is called anorder isomorphism or (less commonly) anisotone isomorphism.
If then this is a relation-preservingautomorphism.
Field isomorphisms are the same as ring isomorphism betweenfields; their study, and more specifically the study offield automorphisms is an important part ofGalois theory.
Just as theautomorphisms of analgebraic structure form agroup, the isomorphisms between two algebras sharing a common structure form aheap. Letting a particular isomorphism identify the two structures turns this heap into a group.
Ingraph theory, an isomorphism between two graphsG andH is abijective mapf from the vertices ofG to the vertices ofH that preserves the "edge structure" in the sense that there is an edge fromvertexu to vertexv inG if and only if there is an edge from to inH. Seegraph isomorphism.
Inorder theory, an isomorphism between two partially ordered setsP andQ is abijective map fromP toQ that preserves the order structure in the sense that for any elements and ofP we have less than inP if and only if is less than inQ. As an example, the set {1,2,3,6} of whole numbers ordered by theis-a-factor-of relation is isomorphic to the set {O,A,B,AB} ofblood types ordered by thecan-donate-to relation. Seeorder isomorphism.
In mathematical analysis, an isomorphism between twoHilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.
Incybernetics, thegood 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.
Incategory theory, given acategoryC, an isomorphism is a morphism that has an inverse morphism that is, and
Two categoriesC andD areisomorphic if there existfunctors and which are mutually inverse to each other, that is, (the identity functor onD) and (the identity functor onC).
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 anequivalence relation. Anequivalence class given by isomorphisms is commonly called anisomorphism class.[2]
Examples of isomorphism classes are plentiful in mathematics.
Two sets are isomorphic if there is abijection 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 afinite-dimensional vector space can be identified with the non-negative integer representing its dimension.
Ordinals intuitively correspond to isomorphism classes of well-ordered sets (though there are technical set-theoretic issues involved).
There are three isomorphism classes of the planarsubalgebras 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 amathematical structure, it is common that twosubstructures 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, allsubspaces of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc.
Inhomotopy theory, thefundamental group of aspace at a point, though technically denoted to emphasize the dependence on the base point, is often written lazily as simply if ispath connected. The reason for this is that the existence of a path between two points allows one to identifyloops at one with loops at the other; however, unless isabelian this isomorphism is non-unique. Furthermore, the classification ofcovering spaces makes strict reference to particularsubgroups of, specifically distinguishing between isomorphic butconjugate 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.
Although there are cases where isomorphic objects can be considered equal, one must distinguishequality andisomorphism.[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 setsareequal; they are merely different representations—the first anintensional one (inset builder notation), and the secondextensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets and are notequal 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
while another is
and no one isomorphism is intrinsically better than any other.[note 1]
On the other hand, when sets (or othermathematical objects) are specified 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 ofuniversal properties. For examples, the polynomial rings, and are considered as equal, since they have the same universal property.
For example, therational numbers are formally defined asequivalence 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 afield 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. Thereal 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. So, the rational numbers may be identified to the elements of a subset of the real numbers. However, in some contexts this identification is not allowed. For example, incomputer languages andtype theory, real numbers and rational numbers have different representations, and the identification must be replaced with atype conversion.
The most common notation to denote that two objectsA andB are isomorphic is, and if mapsA isomorphically toB, then one can also write. However, depending on context, some authors may also use symbols including,, or = to denote an isomorphism.
