# Symmetric inverse semigroup

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In [abstract algebra](/source/Abstract_algebra), the [set](/source/Set_(mathematics)) of all [partial bijections](/source/Partial_bijection) on a set *X* (a.k.a. one-to-one partial transformations) forms an [inverse semigroup](/source/Inverse_semigroup), called the **symmetric inverse semigroup**[1] (actually a [monoid](/source/Monoid)) on *X*. The conventional notation for the symmetric inverse semigroup on a set *X* is I X {\displaystyle {\mathcal {I}}_{X}} [2] or I S X {\displaystyle {\mathcal {IS}}_{X}} .[3] In general I X {\displaystyle {\mathcal {I}}_{X}} is not [commutative](/source/Commutative_semigroup).

Details about the origin of the symmetric inverse semigroup are available in the discussion on the [origins of the inverse semigroup](/source/Inverse_semigroup#Origins).

## 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**n* and its elements are called **charts** or **partial symmetries**.[4] The notion of chart generalizes the notion of [permutation](/source/Permutation). A (famous) example of (sets of) charts are the hypomorphic mapping sets from the [reconstruction conjecture](/source/Reconstruction_conjecture) in [graph theory](/source/Graph_theory).[5]

The [cycle notation](/source/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](/source/Partial_function#In_category_theory); the notation thus extended is called *path notation*.[5]

## See also

- [Symmetric group](/source/Symmetric_group)

## Notes

1. **[^](#cite_ref-Grillet1995_1-0)** Grillet, Pierre A. (1995). [*Semigroups: An Introduction to the Structure Theory*](https://books.google.com/books?id=yM544W1N2UUC&pg=PA228). CRC Press. p. 228. [ISBN](/source/ISBN_(identifier)) [978-0-8247-9662-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8247-9662-4).

1. **[^](#cite_ref-2)** [Hollings 2014](#CITEREFHollings2014), p. 252

1. **[^](#cite_ref-3)** [Ganyushkin & Mazorchuk 2008](#CITEREFGanyushkinMazorchuk2008), p. v

1. **[^](#cite_ref-4)** [Lipscomb 1997](#CITEREFLipscomb1997), p. 1

1. ^ [***a***](#cite_ref-Lipscomb97_xiii_5-0) [***b***](#cite_ref-Lipscomb97_xiii_5-1) [Lipscomb 1997](#CITEREFLipscomb1997), p. xiii

## References

- Lipscomb, S. (1997). *Symmetric Inverse Semigroups*. AMS Mathematical Surveys and Monographs. American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [0-8218-0627-0](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0627-0).

- Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). *Classical Finite Transformation Semigroups: An Introduction*. Springer. [doi](/source/Doi_(identifier)):[10.1007/978-1-84800-281-4](https://doi.org/10.1007%2F978-1-84800-281-4). [ISBN](/source/ISBN_(identifier)) [978-1-84800-281-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84800-281-4).

- Hollings, Christopher (2014). *Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups*. American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [978-1-4704-1493-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-1493-1).

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