Product of Quasi-Idempotents in Finite Semigroup of Partial Order-Preserving Transformations

Abstract

Let Xn be the finite set {1, 2, . . . , n}, and POn = On ∪ {α : dom(α) ⊂ Xn(∀x, y ∈ Xn), x ≤ y =⇒ xα ≤ yα} be the semigroup of all partial order-preserving transformations from Xn to itself, where On = {α ∈ Tn : (∀x, y ∈ Xn)x ≤ y =⇒ xα ≤ yα} is the full order preserving transformation on Xn and Tn the semigroup of full transformations from Xn to itself. A transformation α in POn is called quasi-idempotent if α ̸= α 2 = α 4 . In this article, we study quasi-idempotent elements in the semigroup of partial order-preserving transformations and show that semigroup POn is quasi-idempotent generated. Furthermore, an upper bound for quasi-idempotent rank of POn is obtained to be ⌈ 5n−4 2 ⌉. Where ⌈x⌉ denotes the least positive integer m such that x ≤ m ≤ x + 1.