__NOTOC__ In [[abstract algebra]], the [[Set (mathematics)|set]] of all [[partial bijection]]s on a set ''X'' ({{aka}} one-to-one partial transformations) forms an [[inverse semigroup]], called the '''symmetric inverse semigroup'''<ref name="Grillet1995">{{cite book|first=Pierre A. |last=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> (actually a [[monoid]]) on ''X''. The conventional notation for the symmetric inverse semigroup on a set ''X'' is <math>\mathcal{I}_X</math><ref>{{harvnb|Hollings|2014|p=252}}</ref> or <math>\mathcal{IS}_X</math>.<ref>{{harvnb|Ganyushkin|Mazorchuk|2008|p=v}}</ref> In general <math>\mathcal{I}_X</math> is not [[Commutative semigroup|commutative]].
Details about the origin of the symmetric inverse semigroup are available in the discussion on the [[Inverse_semigroup#Origins|origins of the inverse semigroup]].
==Finite symmetric inverse semigroups== When ''X'' is a finite set {1, ..., ''n''}, the inverse semigroup of one-to-one partial transformations is denoted by ''C''<sub>''n''</sub> and its elements are called '''charts''' or '''partial symmetries'''.<ref>{{harvnb|Lipscomb|1997|p=1}}</ref> The notion of chart generalizes the notion of [[permutation]]. A (famous) example of (sets of) charts are the hypomorphic mapping sets from the [[reconstruction conjecture]] in [[graph theory]].<ref name=Lipscomb97_xiii>{{harvnb|Lipscomb|1997|p=xiii}}</ref>
The [[cycle notation]] of classical, group-based permutations generalizes to symmetric inverse semigroups by the addition of a notion called a ''path'', which (unlike a cycle) ends when it reaches the [[Partial_function#In_category_theory|"undefined" element]]; the notation thus extended is called ''path notation''.<ref name=Lipscomb97_xiii/>
==See also== *[[Symmetric group]]
==Notes== {{Reflist|2}}
==References== {{refbegin}} *{{cite book |first=S. |last=Lipscomb |title=Symmetric Inverse Semigroups |publisher=American Mathematical Society |series=AMS Mathematical Surveys and Monographs |date=1997 |isbn=0-8218-0627-0 }} *{{cite book |first1=Olexandr |last1=Ganyushkin |first2=Volodymyr |last2=Mazorchuk |title=Classical Finite Transformation Semigroups: An Introduction |year=2008 |publisher=Springer |isbn=978-1-84800-281-4 |doi=10.1007/978-1-84800-281-4}} *{{cite book |first=Christopher |last=Hollings |title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups |year=2014 |publisher=American Mathematical Society |isbn=978-1-4704-1493-1}} {{refend}}
[[Category:Semigroup theory]] [[Category:Algebraic structures]]
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