On Quasi-Nilpotents in Finite Partial Transformation Semigroups

B. Ali Ali; M.  Yahuza; A. T.  Imam

B. Ali Ali Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna-Nigeria
M. Yahuza Department of Mathematics, Federal University of Education, Zaria-Nigeria
A. T. Imam Department of Mathematics, Ahmadu Bello University, Zaria-Nigeria

Keywords: Nilpotents, Quasi-Nilpotent, Pseudo-Quasi-Nilpotent, Generating Sets, Rank

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 height of an element αα is defined as ∣imα∣∣imα∣, we obtained quasi-nilpotents rank of K(n,r)K(n,r) that is the cardinality of a minimum quasi-nilpotents 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 idempotents rank.

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