In abstract algebra, the set of all partial bijections on a set X (a.k.a. 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 I\( \mathcal{I}_X\) [2] or \( {\displaystyle {\mathcal {IS}}_{X}}\).[3] In general \( \mathcal{I}_X \) 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.[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
Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.
Hollings 2014, p. 252
Ganyushkin and Mazorchuk 2008, p. v
Lipscomb 1997, p. 1
Lipscomb 1997, p. xiii
Lipscomb 1997, p. xiii
References
S. Lipscomb (1997) Symmetric Inverse Semigroups, AMS Mathematical Surveys and Monographs, ISBN 0-8218-0627-0.
Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. doi:10.1007/987-1-84800-281-4_1. ISBN 978-1-84800-281-4.
Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. ISBN 978-1-4704-1493-1.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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