Symmetric inverse semigroup

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 ${\displaystyle {\mathcal {I}}_{X}}$[2] or ${\displaystyle {\mathcal {IS}}_{X}}$[3] In general ${\displaystyle {\mathcal {I}}_{X}}$ 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-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]

Notes

1. ^ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.
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.
• Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. 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.