Partial function

Function spaces

For convenience, denote the set of all partial functions ${\displaystyle f:X\rightharpoonup Y}$ from a set ${\displaystyle X}$ to a set ${\displaystyle Y}$ by ${\displaystyle [X\rightharpoonup Y].}$ This set is the union of the sets of functions defined on subsets of ${\displaystyle X}$ with same codomain ${\displaystyle Y}$:

${\displaystyle [X\rightharpoonup Y]=\bigcup _{D\subseteq X}[D\to Y],}$

the latter also written as $\bigcup _{D\subseteq X}{Y}^D.$ In finite case, its cardinality is

${\displaystyle |[X\rightharpoonup Y]|=(|Y|+1{)}^{|X|},}$

because any partial function can be extended to a function by any fixed value ${\displaystyle c}$ not contained in ${\displaystyle Y,}$ so that the codomain is ${\displaystyle Y\cup \{c\},}$ an operation which is injective (unique and invertible by restriction).

Discussion and examples

The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a function since every element on the left-hand set is associated with exactly one element in the right hand set.

Natural logarithm

The natural logarithm function mapping the real numbers to themselves is a partial function, but not a (total) function, since it is undefined for non-positive real inputs. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function.

Subtraction of natural numbers

Subtraction of natural numbers (in which ${\displaystyle \mathbb{N}}$ is the non-negative integers) is a partial function:

${\displaystyle f:\mathbb{N}\times \mathbb{N}\rightharpoonup \mathbb{N}}$

${\displaystyle f(x,y)=x-y.}$

It is defined only when ${\displaystyle x\ge y.}$

Bottom element

In denotational semantics a partial function is considered as returning the bottom element when it is undefined.

In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested.

In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function.

In category theory

In category theory, when considering the operation of morphism composition in concrete categories, the composition operation ${\displaystyle \circ :\mathrm{hom}(C)\times \mathrm{hom}(C)\to \mathrm{hom}(C)}$ is a total function if and only if ${\displaystyle \mathrm{ob}(C)}$ has one element. The reason for this is that two morphisms ${\displaystyle f:X\to Y}$ and ${\displaystyle g:U\to V}$ can only be composed as ${\displaystyle g\circ f}$ if ${\displaystyle Y=U,}$ that is, the codomain of ${\displaystyle f}$ must equal the domain of ${\displaystyle g.}$

The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science."

The category of sets and partial bijections is equivalent to its dual. It is the prototypical inverse category.

In abstract algebra

Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).

The set of all partial functions (partial transformations) on a given base set, ${\displaystyle X,}$ forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on ${\displaystyle X}$), typically denoted by ${\displaystyle {\mathcal{P}\mathcal{T}}_X.}$ The set of all partial bijections on ${\displaystyle X}$ forms the symmetric inverse semigroup.

Charts and atlases for manifolds and fiber bundles

Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps.

The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined.