Structure and Classification of Stable Quasi-Idempotents in Finite Transformation Semigroups
Abstract
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 transformation 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 defects 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.
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