Call
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 I X {\displaystyle {\mathcal {I}}_{X}} or I S X {\displaystyle {\mathcal {IS}}_{X}}. In general I X {\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.