Products of Nilpotents in Partial Transformation Semigroups using Digraphic Paths and Chains

In this paper, we investigate the factorization of singular partial self-maps on a finite set into products of the least number of nilpotent elements. This research demonstrates that the semigroup of such maps can be expressed within a union of nilpotent-generated sets, specifically up to the third power. Some of our key findings include the determination of the nilpotent rank and the nilpotent depth for these maps, which vary based on whether the set size is even or odd. Additionally, this study surveys the relationship between these results and Stirling numbers, leveraging the Vagner Theorem and digraphic representations. We also examine stable quasi-idempotents, which correspond to specific digraphic paths and chains, providing further insights into the structure of partialtransformation semigroups.