- Finite symmetric inverse semigroups
- See also
- Notes
- References
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[1] (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is 
[2] or 
[3] 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-one partial transformations is denoted by Cn and its elements are called charts or partial symmetries.[4] 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.[5]
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.[6]
See also
- Symmetric group
Notes
2. ^Hollings 2014, p. 252
3. ^Ganyushkin and Mazorchuk 2008, p. v
4. ^Lipscomb 1997, p. 1
5. ^Lipscomb 1997, p. xiii
6. ^Lipscomb 1997, p. xiii
References
- S. Lipscomb, "Symmetric Inverse Semigroups", AMS Mathematical Surveys and Monographs (1997), {{isbn|0-8218-0627-0}}.
- {{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4}}
- {{cite book|author=Christopher 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}}
- Aphelenchoides coffeae
- Aphelenchoides fragariae
- Aphelenchoides olesistus
- Aphelenchoides oryzae
- Aphelenchoides parietinus
- Aphelenchoides pseudolesistus
- Aphelenchoides ritzemabosi
- Aphelenchus avenae
- Aphelenchus fragariae
- Aphelenchus olesistus
- Aphelenchus pseudolesistus
- Aphelinid
- Aphelinidae
- Aphelocephala
- Aphelocephala leucopsis
- Aphelocephala nigricincta
- Aphelocephala pectoralis
- AP Help 1.0
- Aphemia
- Aphenphosmphobia
- Aphephobia
- A.P. Herbert
- A P Herbert
- Apherisis
- APHE-T