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
I
X
or
I
S
X
In general
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.

When *X* is a finite set {1, ..., *n*}, the inverse semigroup of one-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*.