izomorfia

Izomorfia, or isomorphism, is a fundamental concept in mathematics describing a bijective, structure-preserving correspondence between two objects of the same kind. When such a map exists between two structures, they are considered essentially the same from the perspective of the structure being studied.

Formally, let A and B be structures in a given language (with operations, relations, and constants). A map f: A -> B is an isomorphism if it is bijective and preserves the structure: for every operation symbol o of arity n, and all a1, ..., an in A, applying o in A and then f equals applying the corresponding operation in B to the images, i.e., f(o_A(a1, ..., an)) = o_B(f(a1), ..., f(an)); and for every relation symbol R of arity m, R_A(a1, ..., am) holds if and only if R_B(f(a1), ..., f(am)) holds. In short, the structure looks the same after re-labeling via f.

Examples include:

- Groups: a bijection f: G -> H is a group isomorphism if f(xy) = f(x)f(y).

- Vector spaces: a linear bijection between vector spaces is a vector space isomorphism; finite-dimensional spaces over the same field are isomorphic if and only if their dimensions match.

- Graphs: an isomorphism is a bijection between vertex sets that preserves adjacency.

Key properties: isomorphisms preserve all structural features; “isomorphic” is an equivalence relation on structures of a fixed signature; the set of all automorphisms (isomorphisms from a structure to itself) forms its automorphism group.

Applications include classifying objects up to isomorphism, studying invariants, and formalizing sameness across different models. The concept is central across algebra, geometry, combinatorics, and logic.