An extension to ``A subsemigroup of the rook monoid'' (Q6145717)

The paper under review is an extension to [the authors, Semigroup Forum 105, No. 1, 191--216 (2022; Zbl 1514.20219)] in which they study an inverse submonoid \(M_n\) of the rook monoid \(R_n\) and investigate its properties. Now, they introduce a new inverse monoid \(\overline{M}_n\) which contains \(M_n\) as submonoid. Moreover, they show that \(\overline{M}_n\) is a noncommutative (Proposition 1), periodic (Corollary 5), strongly \(E^*\)-unitary inverse semigroup (Theorem 4) and completely semisimple, combinatorial and fundamental inverse monoid (Corollary 6).