symmetric inverse semigroup

Let X be a set. A partial map on X is an application defined from a subset of X into X. We denote by 𝔉(X) the set of partial map on X. Given α𝔉(X), we denote by dom(α) and ran(α) respectively the domain and the range of α, i.e.


We define the compositionMathworldPlanetmath of two partial map α,β𝔉(X) as the partial map αβ𝔉(X) with domain


defined by the common rule


It is easily verified that the 𝔉(X) with the composition is a semigroup.

A partial map α𝔉(X) is said bijective when it is bijective as a map α:ran(α)dom(α). It can be proved that the subset (X)𝔉(X) of the partial bijective maps on X is an inverse semigroup (with the composition ), that is called symmetric inverse semigroup on X. Note that the symmetric groupMathworldPlanetmathPlanetmath on X is a subgroupMathworldPlanetmathPlanetmath of (X).

Title symmetric inverse semigroup
Canonical name SymmetricInverseSemigroup
Date of creation 2013-03-22 16:11:14
Last modified on 2013-03-22 16:11:14
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 6
Author Mazzu (14365)
Entry type Definition
Classification msc 20M18
Defines partial map
Defines composition of partial maps
Defines symmetric inverse semigroup