nLab
multi-valued function

Multi-valued functions

Idea

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.

Definitions

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.

Examples

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\}.)

Revised on September 15, 2010 20:31:34 by Toby Bartels (75.88.78.218)