Isomorfie

Isomorfie, in English isomorphism, is a fundamental concept in mathematics describing when two objects have the same structure up to renaming of elements. An isomorphism is a bijective map between two structures of the same kind that preserves the defining operations or relations. If a map f from A to B satisfies f(x ∘A y) = f(x) ∘B f(y) for all elements x, y in A, then f is a structure-preserving bijection, and its inverse f^{-1} is also structure-preserving. In this case A and B are said to be isomorphic, often written A ≅ B. Isomorphism thus identifies objects that are structurally the same, even if their elements are labeled differently.

Isomorphisms form an equivalence relation on the class of objects in a given theory, and classification up to isomorphism asks how many essentially different objects exist for that theory. Examples include:

- Groups: two groups are isomorphic if their group operations correspond under a bijection. The additive group Z_n is isomorphic to itself only for the same n.

- Vector spaces: over a field F, two vector spaces are isomorphic if and only if they have the same dimension; every n-dimensional space is isomorphic to F^n.

- Graphs: in graph theory, graphs are isomorphic if there is a permutation of vertices sending edges to edges.

- Rings, modules, and algebras: analogous notions hold, with the isomorphism preserving the respective algebraic operations.

Automorphisms are isomorphisms from a structure to itself. Endomorphisms that are invertible are automorphisms. In category theory, isomorphisms are morphisms with two-sided inverses, capturing the idea that isomorphic objects are structurally identical within the theory.