isomorphism

Abstract algebra: Take a group G, and a group H, with a group operation *. G is isomorphic to H if there exist a map f: G->H such that:

1: f is injective
2: f is surjective
3: f(a*b)=f(a)*f(b) for all a,b in G

If f satisfies these three properties, f is called an isomorphism.

The map f: Z -> E given by f(a)=2a where Z is the integers and E is the even integers is an isomorphism.

Proof:
Showing injectivity

f(b)=f(a) => 2a=2b (from the given function) <=> a=b

Showing surjectivity
Suppose n is in E. n is an even integer hence n=2k for some integer k.

f(k)=2k=n, hence f is surjective.

Homomorphism:
f(a+b)=2(a+b)=2a+2b=f(a)+f(b)

Hence f is an isomorphism. Q. E. D.

isomorphism by qsqazxcvfrew March 29, 2018

isomorphism

In category theory an isomorphism is a morphism between two (possibly equal) objects admitting a morphism in the opposite direction such that composing the two morphisms in either order results in an identity morphism. Intuitively it is an identification of one object with another.

In the category of sets and functions isomorphisms are precisely the bijections.

isomorphism by 27182818284tropy June 26, 2025

Isomorphic JavaScript 

A JavaScript application which can be executed in both a browser and a non-browser runtime. This was the first term used to describe this concept, but "Universal JavaScript" now has wider acceptance. Mathematicians and functional programmers will scoff at you if you use "isomorphic" to describe JavaScript, so save face by using "universal" instead.

You aren't writing isomorphic JavaScript? Ha, you remind me of my old Java coworkers.

Isomorphic JavaScript by chexxor March 30, 2017