Symmetric inverse semigroup
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In abstract algebra, the set of all partial bijections on a set X ({{aka}} one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup{{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}} (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is {{harvnb|Hollings|2014|p=252}} or .{{harvnb|Ganyushkin|Mazorchuk|2008|p=v}} In general is not commutative.
Details about the origin of the symmetric inverse semigroup are available in the discussion on the 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 Cn and its elements are called charts or partial symmetries.{{harvnb|Lipscomb|1997|p=1}} 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.{{harvnb|Lipscomb|1997|p=xiii}}
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 "undefined" element; the notation thus extended is called path notation.
See also
Notes
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References
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- {{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}}
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