linguistic:voice_change, information:classification
Generalization to partial functions

The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f: A′ → B′, where A′ is a subset of A and B′ is a subset of B.

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.