Editing Isomorphism (section)

==Applications==
In [[algebra]], isomorphisms are defined for all [[algebraic structure]]s. Some are more specifically studied; for example:
* [[Linear isomorphism]]s between [[vector space]]s; they are specified by [[invertible matrices]].
* [[Group isomorphism]]s between [[group (mathematics)|groups]]; the classification of [[isomorphism class]]es of [[finite group]]s is an open problem.
* [[Ring isomorphism]] between [[ring (mathematics)|rings]]. 
* Field isomorphisms are the same as ring isomorphism between [[field (mathematics)|fields]]; their study, and more specifically the study of [[field automorphism]]s is an important part of [[Galois theory]].

Just as the [[automorphism]]s of an [[algebraic structure]] form a [[group (mathematics)|group]], the isomorphisms between two algebras sharing a common structure form a [[heap (mathematics)|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 [[algebra]]ic 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 (graph theory)|vertex]] ''u'' to vertex ''v'' in ''G'' if and only if there is an edge from <math>f(u)</math> to <math>f(v)</math> in ''H''. See [[graph isomorphism]].

In [[order theory]], an isomorphism between two partially ordered sets ''P'' and ''Q'' is a [[bijective]] map <math>f</math> from ''P'' to ''Q'' that preserves the order structure in the sense that for any elements <math>x</math> and <math>y</math> of ''P'' we have <math>x</math> less than <math>y</math> in ''P'' if and only if <math>f(x)</math> is less than <math>f(y)</math> 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 [[ABO blood group system|blood types]] ordered by the ''can-donate-to'' relation. See [[order isomorphism]].

In mathematical analysis, an isomorphism between two [[Hilbert space]]s 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.