Symmetric inverse semigroup

In abstract algebra, the set $\mathcal{I}_X$ of all partial one-one transformations on a set X forms an inverse semigroup, called the symmetric inverse semigroup (or monoid) on X. In general $\mathcal{I}_X$ is not commutative. More details 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-one partial transformations is denoted by C_n and its elements are called charts. The notion of chart generalizes the notion of permutation.

References

S. Lipscomb, "Symmetric Inverse Semigroups", AMS Mathematical Surveys and Monographs (1997), ISBN 0821806270.