isomorphic

Isomorphic is an adjective used in mathematics and related fields to describe a relationship between two objects that have the same structure, up to renaming of elements. It derives from the Greek iso, meaning equal, and morphe, meaning form. In its standard use, two objects are isomorphic if there exists a bijective mapping between their underlying sets that preserves the defining operations or relations.

In algebra, two structures are isomorphic when there exists a structure-preserving bijection, known as an isomorphism. For example, groups G and H are isomorphic if there is a bijective function f: G → H that respects the group operation (f(xy) = f(x)f(y)). In linear algebra, vector spaces are isomorphic if a bijective linear map exists between them; finite-dimensional vector spaces over the same field are isomorphic if and only if their dimensions match. For graphs, two graphs are isomorphic if a bijection between their vertex sets preserves adjacency, making the graphs essentially the same up to renaming.

In topology and category theory, the idea appears in related forms. An isomorphism in a category is a morphism that has an inverse, establishing a strong notion of sameness. In topology, objects are often considered up to homeomorphism rather than isomorphism, since homeomorphism preserves the topological structure. In computer science, isomorphism describes when two data structures or programs have the same behavior or form, and the isomorphism problem asks whether a given pair is isomorphic.

See also: morphism, automorphism, homomorphism, category theory, graph isomorphism.