Isomorfiza

Isomorfiza is a verb used in mathematical Romanian to denote the act of establishing an isomorphism between two structures, or to transform one object into another that is structurally identical to it. The corresponding noun is isomorfie (isomorphism). In practice, to isomorfiza two objects means to exhibit a bijective map between them that preserves the relevant operations or relations.

In abstract algebra and related fields, two objects A and B are isomorphic if there exists a bijective function f: A → B that preserves the structure: for all elements x and y in A, the operation in A satisfies f(x ⋅A y) = f(x) ⋅B f(y). The inverse function f⁻¹: B → A is then also structure-preserving. If such a map exists, A and B are said to be isomorphic, and they are considered the same up to relabeling of elements.

Examples:

- In linear algebra, two vector spaces of the same finite dimension over a field are isomorphic; a linear bijection provides the isomorphism.

- In graph theory, two graphs are isomorphic if there is a bijection between their vertex sets that preserves adjacency; demonstrating this bijection is an isomorfizare of the graphs.

- In category theory, an isomorphism is a morphism with an inverse, showing that two objects are essentially the same within the category.

Isomorfiza emphasizes the constructive aspect of mathematics: to prove two structures are isomorphic, one typically provides an explicit map or a method to transform one into the other. It contrasts with weaker notions such as homomorphisms, which may not be bijective.