Perfect completely semisimple inverse semigroups (Q1204125)

A congruence \(\rho\) on a semigroup \(S\) is called perfect if \((a\rho)(b\rho)=(ab)\rho\) for all \(a,b\in S\) where \(a\rho\) is the congruence class containing \(a\), and \((a\rho)(b\rho)=\{xy:x\in a\rho,\;y\in b\rho\}\), and both sides are equal as sets. A semigroup \(S\) is called perfect if all congruences on \(S\) are perfect. In this paper the author completely determines the structure of perfect completely semisimple inverse semigroups, and describes the three types of such semigroups. In the proof, he uses the fact due to \textit{V. Fortunatov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1972, No. 3, 80-89 (1972; Zbl 0246.20051)] and the result is a generalization of \textit{H. Hamilton} and the reviewer's [J. Aust. Math. Soc., Ser. A 32, 114-128 (1982; Zbl 0486.20038)].