On Quasi-Nilpotents in Finite Partial Transformation Semigroups

Abstract

Let XnXn​ be the finite set {1,2,3,⋯ ,n}{1,2,3,⋯,n} and PnPn​ be the partial transformation on XnXn​. A transformation αα in PnPn​ is called quasi-nilpotent if when αα is raised to some certain power it reduces to a constant map, i.e., αmαm reduces to a constant map for m≥1m≥1. We characterize quasi-nilpotents in PnPn​ and show that the semigroup PnPn​ is quasi-nilpotent generated. Moreover, if K(n,r)K(n,r) is the subsemigroup of PnPn​ consisting of all elements of height rr or less, where the height of an element αα is defined as ∣imα∣∣imα∣, we obtain the quasi-nilpotent rank of K(n,r)K(n,r), that is, the cardinality of a minimum quasi-nilpotent generating set for PnPn​, as the Stirling number of the second kind S(n+1,r+1)S(n+1,r+1), which is the same as its idempotent rank.