A subsemigroup of the rook monoid

TL;DR

This paper introduces a specific subsemigroup of the rook monoid, explores its algebraic properties, and provides explicit formulas and combinatorial interpretations for its elements and structure.

Contribution

It defines a new subsemigroup of the rook monoid, characterizes its elements, and develops formulas for products, roots, and substructures, with combinatorial and graphical insights.

Findings

All elements are idempotents or nilpotents

Explicit formulas for element products and roots

Characterization of subsemigroups and combinatorial structures

Abstract

We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that $S_n$ consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, and compute nilpotency indexes. Furthermore, we give a necessary and sufficient condition for the $j$th root of a nonzero element to exist in $S_n$, show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of $S_n$; and, using rook $n$-diagrams, graphically interpret many of our results.