Symmetric inverse semigroup

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of all <a href="/facts/Partial_bijection/j4gTuTmW">partial bijections</a> on a set X (a.k.a. one-to-one partial transformations) forms an <a href="/facts/Inverse_semigroup/gz4ezaRq">inverse semigroup</a>, called the symmetric inverse semigroup (actually a <a href="/facts/Monoid/6AHve7p8">monoid</a>) on X. The conventional notation for the symmetric inverse semigroup on a set X is 
 
 
 
 
 
 
 I
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {I}}_{X}}
 
 or 
 
 
 
 
 
 
 I
 S
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {IS}}_{X}}
 
. In general 
 
 
 
 
 
 
 I
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {I}}_{X}}
 
 is not <a href="/facts/Commutative_semigroup/UPwpNRVX">commutative</a>.
Details about the origin of the symmetric inverse semigroup are available in the discussion on the <a href="/facts/Inverse_semigroup/gz4ezaRq">origins of the inverse semigroup</a>.

Symmetric inverse semigroup open-in-new

Symmetric inverse semigroup