Structural properties of the power graph of the full transformation semigroup

Abstract

We study the power graph \(\mathcal{P}(\mathcal{T}_n)\) of the full transformation semigroup \(\mathcal{T}_n\) using functional digraphs. We characterize idempotent and nilpotent transformations and examine the structural roles of idempotent, quasi-idempotent, nilpotent, and permutation elements. We prove that distinct idempotents are non-adjacent, proper quasi-idempotents are adjacent to idempotents, and the nilpotent subgraph forms a forest rooted at constant transformations. For \(n=3\), we obtain a complete computational description of \(\mathcal{P}(\mathcal{T}_3)\), consisting of ten connected components and 18 edges on 27 vertices. The results are verified using GAP, providing a basis for further investigation of power graphs of full transformation semigroups.