= Isomorphism
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= Isomorphic
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An isomorphism is a <morphism> with a two-sided inverse that is also a <morphism>. Two objects are isomorphic when such a map exists between them. For <groups>, <rings>, and <vector spaces>, a bijective structure-preserving map has a structure-preserving inverse; in other settings the inverse condition needs separate attention, as for a <homeomorphism> between <topological spaces>.
For example, $f(k)=2k$ is a <group isomorphism> from the additive <integers> to the additive <even integers>. It is <bijective> onto that target, satisfies $f(a+b)=f(a)+f(b)$, and has inverse $2k\mapsto k$. The two groups have the same addition pattern even though they are different <sets>. It is not a <ring isomorphism> for their inherited multiplications, since $f(ab)=2ab$ while $f(a)f(b)=4ab$.
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