An extension to "A subsemigroup of the rook monoid"

In a recent paper, we defined an inverse submonoid $M_n$ of the rook monoid and investigated its properties. That investigation was enabled by representing the nonzero elements of $M_n$ (which are $n\times n$ matrices) via certain triplets belonging to $\mathbb{Z}^3$. In this short note, we allow the aforementioned triplets to belong to $\mathbb{R}^3$. We thus study a new inverse monoid $\overline{M}_n$, which is a supermonoid of $M_n$. We prove that the elements of $\overline{M}_n$ are either idempotent or nilpotent, compute nilpotent indexes, and discuss issues pertaining to $j$th roots. We also describe the ideals of $\overline{M}_n$, determine Green's relations, show that $\overline{M}_n$ is a supersemigroup of the Brandt semigroup, and prove that $\overline{M}_n$ has infinite Sierpiński rank. While there are similarities between $M_n$ and $\overline{M}_n$, there are also essential differences. For example, $M_n$ can be generated by only three elements, all ideals of $M_n$ are principal ideals, and there exist $x\in M_n$ that do not possess a square root in $M_n$; but none of these statements is true in $\overline{M}_n$.