The structure of compact completely simple semigroups with the congruence extension property (Q1375889)

According to the author, a topological semigroup is said to have the congruence extension property (CEP) if for each closed subsemigroup \(T\) of \(S\) and each closed congruence \(\sigma\) on \(T\), \(\sigma\) has a closed extension to \(S\). The author characterizes compact (completely) simple semigroups with the CEP property. The following is the main theorem: A compact (completely) simple semigroup \(S\) has CEP if and only if \(S\) is isomorphic to the direct product \(X\times G\times Y\) where \(X\) is a left zero semigroup, \(Y\) is a right zero semigroup and \(G\) is a compact group with \(G\) itself having CEP. As a corollary, the continuous homomorphic image of a compact completely simple semigroup with CEP has also CEP.