  Starting from the symmetric group $S_n$, we construct two fiat
$2$-categories. One of them can be viewed as the fiat "extension" of the
natural $2$-category associated with the symmetric inverse semigroup
(considered as an ordered semigroup with respect to the natural order). This
$2$-category provides a fiat categorification for the integral semigroup
algebra of the symmetric inverse semigroup. The other $2$-category can be
viewed as the fiat "extension" of the $2$-category associated with the maximal
factorizable subsemigroup of the dual symmetric inverse semigroup (again,
considered as an ordered semigroup with respect to the natural order). This
$2$-category provides a fiat categorification for the integral semigroup
algebra of the maximal factorizable subsemigroup of the dual symmetric inverse
semigroup.
