Published January 29, 2024
| Version v1
Journal
Open
PRODUCT OF QUASI-IDEMPOTENTS IN FINITE SEMIGROUP OF PARTIAL ORDER-PRESERVING TRANSFORMATIONS
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Description
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 \neq α 2 = α 4 . In this article, we study quasi-idempotent ele
ments in the semigroup of partial order-preserving transforma
tions and show that semigroup POn is quasi-idempotent gener
ated. Furthermore, an upper bound for quasi-idempotent rank
of POn is obtained to be d 5n 2 −4 e . Where d xe denotes the least
positive integer m such that x ≤ m ≤x+1.
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