On a new approach to the dual symmetric inverse monoid $I*_X$
We construct the \emph{inverse partition semigroup} $\mathcal{IP}_X$, isomorphic to the \emph{dual symmetric inverse monoid} $\mathcal{I}^{\ast}_X$, introduced in [6]. We give a convenient geometric illustration for elements of $\mathcal{IP}_X$. We describe all maximal subsemigroups of $\mathcal{IP}_X$ and find a generating set for $\mathcal{IP}_X$ when $X$ is finite. We prove that all the automorphisms of $\mathcal{IP}_X$ are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets $X$, we establish that, up to equivalence, there is a unique faithful effective transitive representation of $\mathcal{IP}_n$, namely to $\mathcal{IS}_{2^n-2}$. Finally, we construct an interesting $\mathcal{H}$-cross-section of $\mathcal{IP}_n$, which is reminiscent of $\mathcal{IO}_n$, the $\mathcal{H}$-cross-section of $\mathcal{IS}_n$, constructed in [4].