A subsemigroup of the rook monoid

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.