In mathematics, 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, and this is often denoted as . The word is derived .
The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties, from the point of view of the structure considered (excluding further information such as additional structure or names of objects). Thus isomorphic structures may often be identified, that is, considered the same. One says also that the two structures are the same up to an isomorphism. For example, there are several different ways for defining the real numbers, and despite the different nature of the resulting objects (sequences of digits, subsets of the rational numbers, sequences of rational numbers, etc.), they are identified to form a unique set of the real numbers. On the other hand, a common case 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 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 , 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 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 spaces. A homeomorphism is an isomorphism of topological spaces. 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. In geometry, 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 be the multiplicative group of positive real numbers, and let be the additive group of real numbers.
The logarithm function satisfies for all so it is a group homomorphism. The exponential function satisfies for all so it too is a homomorphism.
The identities and show that and are inverses of each other. So,
are group 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 a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.
Integers modulo 6 Consider the ring of the integers from 0 to 5 with addition and multiplication modulo 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 the Chinese remainder theorem which asserts that, if and are coprime integers, the ring of the integers modulo is isomorphic to the direct product of the integers modulo and the integers modulo .
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 such that:
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 then an isomorphism from X to Y is a bijective function such that
Such an isomorphism is called an or (less commonly) an .
If then this is a relation-preserving automorphism.
Applications In algebra, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example: Linear isomorphisms between vector spaces; they are specified by invertible matrices. Group isomorphisms between groups; the classification of isomorphism classes of finite groups is an open problem. Ring isomorphisms 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, 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 to in H. See graph isomorphism.
In order theory, an isomorphism between two partially ordered sets P and Q is a bijective map from P to Q that preserves the order structure in the sense that for any elements and of P we have less than in P if and only if is less than in Q. As an example, the set {1,2,3,6} of whole numbers ordered by the is-a-factor-of relation is isomorphic to the set {O, A, B, AB} 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 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 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, given a category C, an isomorphism is a morphism that has an inverse morphism that is, and
Two categories and are isomorphic if there exist functors and which are mutually inverse to each other, that is, (the identity functor on ) and (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 classes 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 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 intuitively correspond to isomorphism classes of well-ordered sets (though there are technical set-theoretic 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 at a point , though technically denoted to emphasize the dependence on the base point, is often written lazily as simply if 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 is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of , 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
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
while another is
and no one isomorphism is intrinsically better than any other. and have both the usual order of the integers. Viewed as ordered sets, there is only one isomorphism between them, namely
Also, integers 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.