Quasi-idempotents in certain transformation semigroups

Abstract

Let \(P_{n}\) and \(T_{n}\) be the partial transformations semigroup and the (full) transformations semigroup on the set \(X_{n}=\{1,\ldots ,n\}\), respectively. In this paper, we first state the orbit structure of quasi-idempotents (non-idempotent element whose square is an idempotent) in \(P_{n}\). Then, for \(2\leq r\leq n-1\), we find the quasi-idempotent ranks of the subsemigroup \(PK(n,r)=\{\alpha \in P_{n}: \mathrm{h}(\alpha) \leq r\}\) of \(P_{n}\), and the subsemigroup \(K(n,r)=\{\alpha \in T_{n}: \mathrm{h}(\alpha) \leq r\}\) of \(T_{n}\), where \(\mathrm{h}(\alpha)\) denotes the cardinality of the image set of \(\alpha\).