On free two-generated unary Clifford semigroup (Q1897913)

It has become standard to treat completely regular semigroups as `unary semigroups', by equipping them with the additional unary operation \(x\to x^{-1}\), where \(x^{-1}\) is the inverse of \(x\) in the maximal subgroup to which it belongs. The free completely regular semigroup \(F{\mathcal C}{\mathcal R}(X)\) on a set \(X\) may then be constructed in the usual manner. For instance \(F{\mathcal C}{\mathcal R}(\{x\})\) is just the free group on \(\{x\}\). Notice that this group is generated as a semigroup by \(\{x,x^{-1}\}\). The author's main result is that for \(|X|>1\), \(F{\mathcal C}{\mathcal R}(X)\) is not finitely generated as a semigroup.