Products of quasi-idempotents in finite symmetric inverse semigroups

Abstract

Let $X_{n}=\{1,2,\ldots ,n\}$ and let ${\mathcal {S}}_{n}$ and ${\mathcal {I}}_{n}$ be the symmetric group and symmetric inverse semigroup on $X_{n}$ respectively. In this paper, we show that the semigroup ${\mathcal {SI}}_{n}={\mathcal {I}}_{n}{\setminus } {\mathcal {S}}_{n}$, of all strictly partial one-to-one maps on $X_{n}$, is generated by quasi-idempotent elements (non-idempotent elements $\alpha$ satisfying $\alpha ^{4}=\alpha ^{2}$). Also, we give the least upper bound for the minimum length factorisation of each $\alpha \in {\mathcal {SI}}_{n}$ into a product of quasi-idempotents.