Definition:Isomorphism (Abstract Algebra)

Definition
An isomorphism is a homomorphism which is a bijection.

That is, it is a mapping which is both a monomorphism and an epimorphism.

An algebraic structure $\left({S, \circ}\right)$ is isomorphic to another algebraic structure $\left({T, *}\right)$ iff there exists an isomorphism from $\left({S, \circ}\right)$ to $\left({T, *}\right)$, and we can write $S \cong T$ (although notation may vary).

Category Theory
Let $\mathcal C$ be a category, and let $X,Y$ be objects of $\mathcal C$.

A morphism $f : X \to Y$ is an isomorphism if there exists a morphism $g : Y \to X$ such that


 * $g \circ f = \operatorname{id}_X$, and $f \circ g = \operatorname{id}_Y$

where $\operatorname{id}_Z$ denotes the identity morphism on an object $Z$ of $\mathcal C$.

Isomorphism on an Ordered Structure
An isomorphism from an ordered structure $\left({S, \circ, \preceq}\right)$ to another $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:


 * An isomorphism, i.e. a bijective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$;
 * An order isomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.

Isomorphic Copy
Let $\phi: S \to T$ be an isomorphism.

Let $x \in S$.

Then $\phi \left({x}\right) \in T$ is known as the isomorphic copy of $x$ (under $\phi$).

Also see

 * Automorphism

Linguistic Note
The word isomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.

Ordered Structure definition

 * : $\S 15$