# 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 ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$, one often writes ${\displaystyle {\mathcal {A}}\cong {\mathcal {B}}}$.[1][2]

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.[3]

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).

## References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.
2. Weisstein, Eric W. "Isomorphism". mathworld.wolfram.com. Retrieved 2020-10-12.
3. "Definition of ISOMORPHISM". www.merriam-webster.com. Retrieved 2020-10-12.