Isomorphism

In mathematics (particularly in abstract algebra), two mathematical structures are isomorphic when they are the same in some sense. More specifically, an isomorphism is a function between two structures that preserves the relationships between the parts. To indicate isomorphism between two structures [math]\displaystyle{ \mathcal{A} }[/math] and [math]\displaystyle{ \mathcal{B} }[/math], one often writes [math]\displaystyle{ \mathcal{A} \cong \mathcal{B} }[/math].

Using the language of category theory, this means that morphisms map to morphisms without breaking composition. An isomorphism is also a homomorphism that is one-to-one.

As an example, one can consider the operation of adding integers Z. The doubling function φ(x) = 2x maps elements of Z to elements of the even integers 2Z. Since φ(a+b) = 2(a+b) = 2a+2b = φ(a)+φ(b), adding in Z is structurally identical as adding in 2Z (which makes this an example of isomorphism).

Related pages

• Automorphism