Products of quasi-idempotents in finite symmetric inverse semigroups (Q309960)

In the paper under review, an element \(a\) of some semigroup is called \textit{quasi-idempotent} provided that \(a^{2}\) is an idempotent. The main result of the paper shows that the singular part (that is the set of non-invertible elements) of the full symmetric inverse semigroup on a finite set is generated by quasi-idempotents. Moreover, the paper also provides an upper bound for a minimum length factorization of non-invertible partial permutations into a product of quasi-idempotents. The terminology chosen in the paper should not be confused with the terminology used in associative algebras, where a quasi-idempotent is an element \(a\) such that \(a^{2}=\lambda a\) for some scalar \(\lambda\).