Isomorphism

A bijective structure-preserving map between two mathematical objects whose inverse is also structure-preserving; it identifies the objects as the same up to the specified structure.

A bijective mapping between two mathematical objects that preserves the relevant structure (operations, relations, or distinguished elements) and whose inverse likewise preserves that structure; it identifies the objects as the same up to the structure in question.

An isomorphism is a bijective homomorphism between two algebraic structures whose inverse map is also a homomorphism; it exhibits the two structures as structurally identical within the chosen signature.

A homomorphism that is bijective and whose inverse is also a homomorphism; it establishes an equivalence of algebraic structures so that the two objects have the same algebraic form.