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 ${\mathcal {I}}_{X}$ ^[2] or ${\mathcal {IS}}_{X}$ .^[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 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.^[5]

Notes

^ Grillet, Pierre A. (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.
^ Hollings 2014, p. 252
^ Ganyushkin & Mazorchuk 2008, p. v
^ Lipscomb 1997, p. 1
^ ^a ^b Lipscomb 1997, p. xiii

References

Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. American Mathematical Society. ISBN 0-8218-0627-0.
Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). Classical Finite Transformation Semigroups: An Introduction. Springer. doi:10.1007/978-1-84800-281-4. ISBN 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 978-1-4704-1493-1.