  We introduce two generalisations of the full symmetric inverse semigroup
${\mathcal{I}}_X$ and its dual semigroup ${\mathcal{I}^{\ast}}_X$ -- inverse
semigroups ${\mathcal{PI}^{\ast}}_X$ and ${\overline{\mathcal{PI}^{\ast}}}_X$.
Both of them have the same carrier and contain $\IS_X$. Binary operations on
${\mathcal{PI}^{\ast}}_X$ and ${\overline{\mathcal{PI}^{\ast}}}_X$ are
reminiscent of the multiplication in ${\mathcal{I}}_X$. We use a convenient
geometric way to realise elements from these two semigroups. This enables us to
study efficiently their inner properties and to compare them with the
corresponding properties of $\IS_X$ and ${\mathcal{I}^{\ast}}_X$.
