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 ^{[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-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. 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

## 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.