An extension to "A subsemigroup of the rook monoid"

A recent paper studied an inverse submonoid $M_n$ of the rook monoid, by representing the nonzero elements of $M_n$ via certain triplets belonging to ${\mathbb{Z}}^3$. In this short note, we allow the triplets to belong to ${\mathbb{R}}^3$. We thus study a new inverse monoid ${\bar{M}}_n$, which is a supermonoid of $M_n$. We point out similarities and find essential differences. We show that ${\bar{M}}_n$ is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly $E^{\ast }$-unitary inverse monoid.