Symmetric inverse semigroup explained

In abstract algebra, the set of all partial bijections on a set X (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

l{I} _{X} or l{IS} _{X} . In general l{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, 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

References

• Book: Lipscomb, S. . Symmetric Inverse Semigroups . American Mathematical Society . AMS Mathematical Surveys and Monographs . 1997 . 0-8218-0627-0 .
• Book: Olexandr . Ganyushkin . Volodymyr . Mazorchuk . Classical Finite Transformation Semigroups: An Introduction . 2008 . Springer . 978-1-84800-281-4 . 10.1007/978-1-84800-281-4.
• Book: Hollings, Christopher . Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups . 2014 . American Mathematical Society . 978-1-4704-1493-1.

Notes and References

1. Book: Grillet, Pierre A. . Semigroups: An Introduction to the Structure Theory. 1995. CRC Press. 978-0-8247-9662-4. 228.