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 the same up to an isomorphism.
An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical 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 p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.
The term isomorphism 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:
- 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 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.
An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical 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 p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.
The term isomorphism 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.
Contents
- 1 Examples
- 2 Applications
- 3 Category theoretic view
- be the multiplicative group of positive real numbers, and let be the additive group of real numbers.
The logarithm function satisfies for all
Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.
If X = Y, then this is a relation-preserving automorphism.
Applications
In abstract algebra, two basic isomorphisms are defined:
- Group isomorphism, an isomorphism between groups
- Ring isomorphism, an isomorphism between rings. (Isomorphisms between fields are actually ring isomorphisms)
Just as the automorphisms of an X = Y, then this is a relation-preserving automorphism.
In abstract algebra, two basic isomorphisms are defined:
- Group isomorphism, an isomorphism between groups
- Ring isomorphism, an isomorphism
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 a
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 ƒ(u) to ƒ(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 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 → b that has an inverse morphism g: b → a, that is, fg = 1b and gf = 1a. For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.
Isomorphism vs. bijective morphism
equality on the one hand and isomorphism on the other.[2] Equality is when two objects are exactly the same, and everything that's true about one object is true about the other, while an isomorphism implies everything that's true about a designated part of one object's structure is true about the other's. For example, the sets
- be the multiplicative group of positive real numbers, and let be the additive group of real numbers.
