Symmetric inverse semigroup

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 ${\displaystyle {\mathcal {I}}_{X}}$ or ${\displaystyle {\mathcal {IS}}_{X}}$. In general ${\displaystyle {\mathcal {I}}_{X}}$ is not commutative.

Details about the origin of the symmetric inverse semigroup are available in the discussion on the origins of the inverse semigroup.