Symmetric inverse semigroup

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 (actually a monoid) on "X". The conventional notation for the symmetric inverse semigroup on a set "X" is formula_1 or formula_2. In general formula_1 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".