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 (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is ${\displaystyle {\mathcal {I}}_{X}}$ or ${\displaystyle {\mathcal {IS}}_{X}}$. 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-to-one partial transformations is denoted by C_n and its elements are called charts or partial symmetries. 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.

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.

See also

• Symmetric group

Notes

1. Grillet, Pierre A. (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 & Mazorchuk 2008, p. v
4. Lipscomb 1997, p. 1
5. Lipscomb 1997, p. xiii