Editing Partial function (section)

== Discussion and examples ==
The first diagram at the top of the article represents a partial function that is {{em|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 ===

Consider the [[natural logarithm]] function mapping the [[real number]]s to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. 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 <math>\mathbb{N}</math> is the non-negative [[integers]]) is a partial function:
: <math>f : \N \times \N \rightharpoonup \N</math>
: <math>f(x,y) = x - y.</math>

It is defined only when <math>x \geq y.</math>

=== 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 <math>\circ \;:\; \hom(C) \times \hom(C) \to \hom(C)</math> is a total function if and only if <math>\operatorname{ob}(C)</math> has one element. The reason for this is that two morphisms <math>f : X \to Y</math> and <math>g : U \to V</math> can only be composed as <math>g \circ f</math> if <math>Y = U,</math> that is, the codomain of <math>f</math> must equal the domain of <math>g.</math>

The category of sets and partial functions is [[Equivalence of categories|equivalent]] to but not [[Isomorphism of categories|isomorphic]] with the category of [[pointed set]]s and point-preserving maps.<ref name="KoslowskiMelton2001">{{cite book|editor=Jürgen Koslowski and Austin Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author=Lutz Schröder|chapter=Categories: a free tour}}</ref> 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]]."<ref name="KoblitzZilber2009">{{cite book|author1=Neal Koblitz|author2=B. Zilber|author3=Yu. I. Manin|title=A Course in Mathematical Logic for Mathematicians|year=2009|publisher=Springer Science & Business Media|isbn=978-1-4419-0615-1|page=290}}</ref>

The category of sets and partial bijections is equivalent to its [[Opposite category|dual]].<ref name="Borceux1994">{{cite book|author=Francis Borceux|title=Handbook of Categorical Algebra: Volume 2, Categories and Structures|url=https://books.google.com/books?id=5i2v9q0m5XAC&pg=PA289|year=1994|publisher=Cambridge University Press|isbn=978-0-521-44179-7|page=289}}</ref> It is the prototypical [[inverse category]].<ref name="Grandis2012">{{cite book|author=Marco Grandis|title=Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups|url=https://books.google.com/books?id=TWqhelao4KsC&pg=PA55|year=2012|publisher=World Scientific|isbn=978-981-4407-06-9|page=55}}</ref>

=== In abstract algebra ===

[[Partial algebra]] generalizes the notion of [[universal algebra]] to partial [[Operation (mathematics)|operations]]. An example would be a [[Field (mathematics)|field]], in which the multiplicative inversion is the only proper partial operation (because [[division by zero]] is not defined).<ref name="RosenbergSabidussi1993">{{cite book|editor1=Ivo G. Rosenberg |editor2=Gert Sabidussi|title=Algebras and Orders|year=1993|publisher=Springer Science & Business Media|isbn=978-0-7923-2143-9|author=Peter Burmeister|chapter=Partial algebras – an introductory survey}}</ref>

The set of all partial functions (partial [[Transformation (function)|transformation]]s) on a given base set, <math>X,</math> forms a [[regular semigroup]] called the semigroup of all partial transformations (or the partial transformation semigroup on <math>X</math>), typically denoted by <math>\mathcal{PT}_X.</math><ref name="CliffordPreston1967">{{cite book|author1=Alfred Hoblitzelle Clifford|author2=G. B. Preston|title=The Algebraic Theory of Semigroups. Volume II|url=https://books.google.com/books?id=756KAwAAQBAJ&pg=PR12|year=1967|publisher=American Mathematical Soc.|isbn=978-0-8218-0272-4|page=xii}}</ref><ref name="Higgins1992">{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press, Incorporated|isbn=978-0-19-853577-5|page=4}}</ref><ref name="GanyushkinMazorchuk2008">{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|pages=[https://archive.org/details/classicalfinitet00gany_719/page/n26 16] and 24}}</ref> The set of all partial bijections on <math>X</math> forms the [[symmetric inverse semigroup]].<ref name="CliffordPreston1967"/><ref name="Higgins1992"/>

=== Charts and atlases for manifolds and fiber bundles ===

Charts in the [[Atlas (topology)|atlases]] which specify the structure of [[manifold]]s and [[fiber bundle]]s 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 [[Atlas (topology)#Transition maps|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.