Editing Bijection (section)

==Generalization to partial functions==
The notion of one-to-one correspondence generalizes to [[partial functions]], where they are called ''partial bijections'', although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a [[total function]], i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the [[symmetric inverse semigroup]].<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|date=16 July 2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}</ref>

Another way of defining the same notion is to say that a partial bijection from ''A'' to ''B'' is any relation 
''R'' (which turns out to be a partial function) with the property that ''R'' is the [[Graph of a function|graph of]] a bijection ''f'':''{{prime|A}}''→''{{prime|B}}'', where ''{{prime|A}}'' is a [[subset]] of ''A'' and ''{{prime|B}}'' is a subset of ''B''.<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>

When the partial bijection is on the same set, it is sometimes called a ''one-to-one partial [[transformation (function)|transformation]]''.<ref name="Grillet1995">{{cite book|author=Pierre A. Grillet|title=Semigroups: An Introduction to the Structure Theory|url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA228|year=1995|publisher=CRC Press|isbn=978-0-8247-9662-4|page=228}}</ref> An example is the [[Möbius transformation]] simply defined on the complex plane, rather than its completion to the extended complex plane.<ref name="Campbell2007">{{cite book|editor=C.M. Campbell |editor2=M.R. Quick |editor3=E.F. Robertson |editor4=G.C. Smith|title=Groups St Andrews 2005 Volume 2|year=2007|publisher=Cambridge University Press|isbn=978-0-521-69470-4|page=367|chapter=Groups and semigroups: connections and contrasts |author=John Meakin}} [http://www.math.unl.edu/~jmeakin2/groups%20and%20semigroups.pdf preprint] citing {{Cite journal
 | doi = 10.1006/jabr.1997.7242
| title = The Möbius Inverse Monoid
| journal = Journal of Algebra
| volume = 200
| issue = 2
| pages = 428–438
| year = 1998
| last1 = Lawson | first1 = M. V. 
| doi-access = free
}}</ref>