An extension to "A subsemigroup of the rook monoid"

Abstract

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