Structural Analysis of Quasi-Idempotents in Full Contraction Semigroups via Directed Graphs

Abstract

This study introduces a digraph-theoretic framework for analyzing quasi-idempotent elements within the semigroup of full contraction mappings on a finite totally ordered set. In the classical algebraic context, a transformation is called quasi-idempotent if it is not equal to its own square, while its square is equal to its fourth power. Expanding upon prior algebraic frameworks involving categories such as symmetric, asymmetric, and stationary block structures, we propose an alternative representation by modeling each transformation as a functional directed graph. In this perspective, the elements of the underlying set correspond to the graph's nodes, while directed arcs encode the functional behavior of the transformation. We demonstrate that quasi-idempotency aligns with distinct graphical configurations. Crucially, we show how the defining contraction property, which requires that the distance between the images of any two points never exceeds the distance between the original points, strictly limits the topology of the associated functional digraph: it forces all cycles to have length at most two, confines non-trivial strongly connected components to disjoint two-cycles, and bounds the diameter of the image. Fundamental theoretical outcomes are reframed using graph-theoretic terminology, providing enhanced visualization and structural interpretation of transformation behavior.