Order preserving and a-decreasing partial transformation semigroup

Let Xn = {1, 2, ..., n} and let ∅ ≠ A ⊆ Xn. The set POn(A)∖{1n} = {α ∈ POn∖{1n}: ∀x ∈ A, ∀y ∈ dom(α) y ≤ x ⇒ yα ≤ x} is defined as the set of partial order preserving and A-decreasing transformations on Xn. This study investigates the structural properties of the semigroup formed by these partial transformations. The structure of this semigroup is examined in detail in terms of the relations and equivalence classes that constitute its fundamental components. In particular, the study focuses on the structural properties of the extended Green's relations L*, R*, and D* within the semigroup POn(A)∖{1n}. It is shown that each L* and each R* class in POn(A)∖{1n} contains at least one dempotent element. The left and right zero divisors of the semigroup are determined. Furthermore, a necessary and sufficient condition is established for an element in this semigroup to be nilpotent.