Editing Partial function (section)

=== 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"/>