Skew pairs of idempotents in transformation semigroups. (Q882733)

A pair of idempotents \((e,f)\) in a semigroup is called `skew' if \(ef\) is idempotent while \(fe\) is not. The authors had previously classified skew pairs into four distinct types: `strong' if \(fe=(ef)^2\), `left regular' if \(fe=fef\neq efe\), `right regular' if \(fe=efe\neq fef\), and `discrete' otherwise. In this paper it is proved that the full transformation semigroup \(T_X\) contains skew pairs that are strong, left regular, right regular or discrete if and only if \(|X|\geq 3,4,5\), or \(6\), respectively.