Symmetric inverse semigroup

In modern 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^[1]. The notion of chart generalizes the notion of permutation.

Notes

^ Lipscomb 1997, p. 1

References

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

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[1] Lipscomb 1997, p. 1

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