In 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: 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 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, 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 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:
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.
Let
\R+
\R
log:\R+\to\R
log(xy)=logx+logy
x,y\in\R+,
\exp:\R\to\R+
\exp(x+y)=(\expx)(\expy)
x,y\in\R,
The identities
log\expx=x
\explogy=y
log
\exp
log
log
The
log
Consider the group
(\Z6,+),
\left(\Z2 x \Z3,+\right),
These structures are isomorphic under addition, under the following scheme:or in general
(a,b)\mapsto(3a+4b)\mod6.
For example,
(1,1)+(1,0)=(0,1),
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
\Zm
\Zn
(\Zmn,+)
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\toY
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
\scriptstyle\sqsubseteq,
f:X\toY
If
X=Y,
In algebra, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example:
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)
f(v)
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 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.
In category theory, given a category C, an isomorphism is a morphism
f:a\tob
g:b\toa,
fg=1b
gf=1a.
Two categories and are isomorphic if there exist functors
F:C\toD
G:D\toC
FG=1D
GF=1C
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).
See also: Equality (mathematics).
Although, there are cases where isomorphic objects can be considered as 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 setsare ; 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\}
\{1,2,3\}
A\mapsto1,B\mapsto2,C\mapsto3,
A\mapsto3,B\mapsto2,C\mapsto1,
Also, integers and even numbers are isomorphic as ordered sets and abelian group (for addition), cannot be considered as equal sets, since one is a proper subset of the other.
On the other hand, when sets (or other mathematical objects) are definied only by their properties, without considering the nature of their elements, one consider often them as equal. This is generally the case with solutions of universal properties.
For example, the rational numbers are usually defined as equivalence classes of pairs of integers, although nobody think 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. It results that 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 transfered to the other through the isomorphism. For example the real numbers that are obtained by dividing two integers (inside the real numbers) form the smallest subfield of the real numbers. There is thus a unique isomorphism from the rational numbers (as defined as equivalence classes of pairs) to the quotients of two real numbers that are integers. This allows identifying these two sorts of rational numbers.
A,B,C