An extension to "A subsemigroup of the rook monoid"

TL;DR

This paper extends the study of a specific inverse submonoid of the rook monoid by generalizing from integer triplets to real triplets, revealing new algebraic properties and differences.

Contribution

It introduces a new inverse monoid 0verline;M_n0 by allowing triplets in 0efefef^3, expanding the algebraic framework of the original monoid.

Findings

0verline;M_n is noncommutative and periodic

It is a fundamental and completely semisimple inverse monoid

It is strongly E^*-unitary and exhibits similarities and differences with M_n

Abstract

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 $\overline{M}_n$, which is a supermonoid of $M_n$. We point out similarities and find essential differences. We show that $\overline{M}_n$ is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly $E^*$-unitary inverse monoid.