On the Digraphic Decomposition of Stable Quasi-Idempotents within Finite Partial Transformation Semigroups

semigroup Pn on the set Xn+1 = {0,1,2,...,n}. The primary focus is on the interplay between the powers of transformations, equivalence relations, and their cyclic and quasi-idempotent structures. A key observation is that for any transformation α ∈ Pn, repeated application stabilizes, leading to a periodic behavior characterized by the (m,r)-path cycle—a notation capturing both cyclic and linear components of α. To further analyze α, the concept of orbits is introduced, defined as equivalence classes under the relation x ∼ y if xαm = yαr. These orbits provide a framework for understanding the dynamics of α. The study also examines specific elements like idempotents (ε2 = ε) and quasi-idempotents (ξ2= ξ, ξ4 = ξ2), offering classifications based on the sizes of their cyclic portions within their orbits. A notable result is that stable quasi-idempotents generate the ideal Pn\Sn, where Sn denotes the symmetric group. This work contributes a digraphic characterization of Pn, advancing the understanding of its algebraic structure. The findings have potential applications in semigroup theory, automata, and computational mathematics, particularly in analyzing transformation systems with finite domains.