nLab multi-valued function

Multi-valued functions

Multi-valued functions


A multi-valued function f:ABf: A \to B is like a function from AA to BB except that there may be more than one possible value f(x)f(x) for a given element xx of AA. (Compare a partial function, where f(x)f(x) may not exist at all.)

In older literature (into the 20th century, especially in analysis), functions were often considered to be multi-valued by default, requiring one to specify a singe-valued function otherwise. As set-theoretic formalisation spread, this intuition became difficult to maintain, and the modern concept of function must be single-valued. If you want multi-valued functions, then you can get them in terms of single-valued functions as below.


With power sets

Given sets AA and BB, a multi-valued function ff from AA to BB is a function ff from AA to the power set 𝒫(B)\mathcal{P}(B) such that for each element xx in AA the subset f(x)f(x) of BB is inhabited. By uncurrying the function one gets an entire relation.

Without power sets

Given sets AA and BB, a multi-valued function ff from a AA to BB is a span

D π f A B \array { & & D \\ & \swarrow_\pi & & \searrow^f \\ A & & & & B \\ }

of single-valued functions, where π:DA\pi: D \to A is a surjection. (This condition can be dropped to define a multi-valued partial function, which is simply a span.)

We will call AA and BB the source and target of ff as usual; then we call DD the domain of ff and π:DA\pi: D \to A the projection of the domain onto the source. By abuse of notation, the multi-valued function ff is conflated with the (single-valued) function f:DBf: D \to B.

Often one can assume that the induced function DA×BD \to A \times B is an injection; in that case, a multi-valued function is the same as an entire relation. On the other hand, if you're considering all of the multi-valued functions for a given DD, then this restriction is not really appropriate.

We consider two multi-valued functions (with the same given source and target) to be equal if there is a bijection between their domains that makes the obvious diagrams commute.


  • In 19th-century analysis, one considered the square-root function, the logarithm, and so forth to be multi-valued functions of complex numbers. We now understand this in terms of Riemann surfaces; the domain DD above is a Riemann surface. (Notice that the logarithm is actually a multi-valued partial function from C\mathbf{C} to C\mathbf{C}, although it is a multi-valued total function on C{0}\mathbf{C} \setminus \{0\}.)

  • On anyon-wavefunctions as multi-valued functions on a configuration space of points: see there


See also:

Last revised on May 26, 2023 at 13:45:53. See the history of this page for a list of all contributions to it.