Quasi-idempotents in finite semigroup of full order-preserving transformations

Let Xn be the finite set {1,2,3· · ·, n} and On defined by On={α∈Tn:(∀x, y∈Xn), x⩽y→xα⩽yα}be semigroup of full order-preserving mapping on Xn. A transformation α in is called quasi-idempotent if α=α2=α4. We characterise show that generated. Moreover, we obtained an upper bound forquasi-idempotents rank On, is, showed cardinality a minimum quasi-idempotents generating for less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes least positive integerm such x⩽m<x+ 1.