A recent paper studied an inverse submonoid Mnβ of the rook monoid, by
representing the nonzero elements of Mnβ via certain triplets belonging to
Z3. In this short note, we allow the triplets to belong to
R3. We thus study a new inverse monoid Mnβ, which is a
supermonoid of Mnβ. We point out similarities and find essential differences.
We show that Mnβ is a noncommutative, periodic, combinatorial,
fundamental, completely semisimple, and strongly Eβ-unitary inverse monoid