Symmetric inverse semigroup

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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[edit]

When X is a finite set {1, ..., n}, the inverse semigroup of one-one partial transformations is denoted by Cn and its elements are called charts.[1] The notion of chart generalizes the notion of permutation.


  1. ^ Lipscomb 1997, p. 1


S. Lipscomb, "Symmetric Inverse Semigroups", AMS Mathematical Surveys and Monographs (1997), ISBN 0-8218-0627-0.