Representation of Quasi-Idempotents in Finite Partial Transformation Semigroups via Digraphs

The study of finite partial transformation semigroups, PnPₙPn, plays a pivotal role in algebraic structures, with applications spanning automata theory, combinatorics, and computational algebra. PnPₙPn consists of all partial mappings on the set Xn+1=0, 1, 2, …, nX₍+₁ = \0, 1, 2, , n\Xn+1=0, 1, 2, …, n, with composition as the fundamental operation. This work revisits the notions of products, decompositions, and quasi-idempotents in PnPₙPn, building on the classical results of Clifford, Preston, Howie, and subsequent advancements by Lipscomb, Ganyushkin, and Mazorchuk. We focus on stable quasi-idempotents, which unify idempotent and quasi-idempotent behaviors, and analyze their structure through digraphic representations such as (m, r) (m, r) (m, r) -path cycles, mmm-paths, and 2-chains. The order of the domain set Z (ξ) Z () Z (ξ) corresponds to the diameter of the associated digraph, linking structural properties to quasi-nilpotency indices. Our investigation emphasizes the generation of Pn∖SnPₙ SₙPn∖Sn by stable quasi-idempotents and the characterization of their compositional behaviors. We also contextualize these findings within the study of order-preserving semigroups, highlighting parallels with recent results on collapsible elements in TnTₙTn and quasi-idempotent generation in POnPOnPOn. The results provide a comprehensive framework for understanding the decomposition, stability, and generation of partial transformations, contributing to both theoretical semigroup theory and its computational applications.