isomorphes

Isomorphes, in mathematics, refer to structures that are related by an isomorphism. An isomorphism is a bijective map that preserves the defining operations and relations of the structures involved. When two objects are connected by such a map, they are considered structurally the same for the purposes of the structure being studied.

Formally, an isomorphism f: A → B is a function that is both bijective and structure-preserving. What “preserving the structure” means depends on the context. In group theory, f(a·a') = f(a)·f(a') for all elements a, a' in A. In linear algebra, a linear isomorphism preserves addition and scalar multiplication. In ring theory, f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). For graphs, f permutes vertices so that adjacency is preserved.

Isomorphisms are an equivalence relation: every structure is isomorphic to itself (reflexive), if A is isomorphic to B then B to A (symmetric), and if A is isomorphic to B and B to C, then A is isomorphic to C (transitive). This partitions structures into isomorphism classes, illustrating that many invariants—such as dimension in vector spaces or order in finite groups—classify objects up to isomorphism. An automorphism is an isomorphism from a structure to itself, capturing its internal symmetries.

Examples illustrate the idea: the additive group of integers Z is isomorphic to 2Z; finite-dimensional vector spaces over the same field are all isomorphic if they share the same dimension; graphs are isomorphic when their vertex-edge structure matches under some relabeling.

The concept of isomorphy is central across mathematics and underpins the idea that isomorphic objects share all essential properties, even if their presentations differ.