isomorfia

Isomorfia, known in English as isomorphism, is a fundamental concept in mathematics that describes a structural equivalence between two objects of the same kind. At the heart of isomorfia is the idea that two objects are essentially the same for the purposes of the theory being studied, even if they are not identical as concrete objects.

Concretely, an isomorfia consists of a bijective map between two structures that preserves the operations, relations, and basic properties that define the structure. The map is called an isomorphism, and its inverse is also a structure-preserving map. If such a map exists, the two objects are said to be isomorphic. Isomorfia is an equivalence relation: it is reflexive, symmetric, and transitive, which means the class of all objects can be partitioned into isomorphism classes.

In practice, isomorfia is used across many areas. In linear algebra, any two n-dimensional vector spaces over the same field are isomorphic. In group theory, two groups are isomorphic if there is a bijective group homomorphism between them, so isomorphic groups have the same algebraic structure. In graph theory, two graphs are isomorphic if a bijection between their vertex sets preserves adjacency. Isomorfia also extends to rings, topological spaces, and many other mathematical objects, with the isomorphism preserving the essential structure of each type.

In category theory, an isomorphism is a morphism that has an inverse, formalizing the notion of “the same up to isomorphism.” The term isomorfia appears in several languages as the standard label for this concept, reflecting its central role in modern mathematics.