Stable Quasi-Idempotents of Higher Defects in Transformation Semigroups

 This article investigates the structure of stable quasi-idempotents ξ of arbitrary defect d(ξ) ≥ 1.
 We show that, unlike transpositions, stable quasi-idempotents generate the singular transfor
mation semigroups Tn\Sn and Pn\Sn, with the inclusion Tn\Sn ⊆ Pn\Sn. These semigroups
 are significant because every finite semigroup is either a subsemigroup or an embedding of them,
 highlighting the universality of Pn. We classify stable quasi-idempotents in terms of their de
fects and path-cycle structures, establishing explicit enumerative formulas. In particular, a
 defect-1 stable quasi-idempotent of span s has rank  nCs =  n!/ (n −s)!s!. This classification clarifies the relationship between stable quasi-idempotents, idempotents, and  quasi-idempotents, and provides a framework for analyzing the subsemigroups they generate.  Our results connect classical work on transformation semigroups with new enumerative and  structural insights.