Graph-Theoretic Characterizations of Quasi-Idempotents in Full Order-Preserving Transformation Semigroup

Abstract

This paper presents a digraph-theoretic extension of the characterization of quasi-idempotent in the semigroup On of full order-preserving transformations on a finite chain. Building on earlier results . that describe quasi-idempotent as those transformations α ∈ On satisfying α≠α^2=α^4, we provide a novel interpretation using the functional digraphs of such maps. We show that a transformation is quasi-idempotent if and only if each vertex in its associated digraph is either fixed or maps directly into a fixed point, and every non-trivial strongly connected component forms a 2-cycle. Furthermore, we prove that no directed path of the form v1 → v2 → v3 exists where all vertices are non-stationary. These findings offer a new perspective on the structure of On, bridging algebraic properties with graphical structure, and set the stage for visual and computational analysis of quasi-idempotent generation in transformation semigroups.