Symmetric inverse semigroup
In abstract algebra, the set of all partial bijections on a set X forms an inverse semigroup, called the symmetric inverse semigroup on X. The conventional notation for the symmetric inverse semigroup on a set X is or. 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, the inverse semigroup of one-to-one partial transformations is denoted by Cn and its elements are called charts or partial symmetries. The notion of chart generalizes the notion of permutation. A example 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 ends when it reaches the "undefined" element; the notation thus extended is called path notation.
