On Quasi-Nilpotents in Finite Partial Transformation Semigroups

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