An isomorphism is a bijection from one set of a mathematical object to the set of another mathematical object such that the structures defined upon these sets in these objects, such as orderings and operations, are preserved.
For example, if one object consists of a set X with an ordering <= and the other object consists of a set Y with an ordering [=, then it must hold for the function f : X -> Y that
- f(u) [= f(v) iff u <= v.
Such an isomorphism is called an order isomorphism.
Or, if on these sets the binary operations * and @ are defined, respectively, then it must hold that
- f(u) @ f(v) = f(u * v).
When the objects in questions are groups, such an isomorphism is called a group isomorphism.