An extension to “A subsemigroup of the rook monoid”

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