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 the semigroup of full order-preserving mapping on Xn. A transformation α in On is called quasi-idempotent if α=α2=α4. We characterise quasi-idempotent in On and show that the semigroup On is quasi-idempotent generated. Moreover, we obtained an upper bound forquasi-idempotents rank of On, that is, we showed that the cardinality of a minimum quasi-idempotents generating set for On is less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes the least positive integerm such that x⩽m