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, compute nilpotency indexes, determine Green’s relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth 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 \(S_n\) (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results.