The ideal extension and congruence extension properties for compact semigroups. (Q1882672)

We say that a topological semigroup \(S\) has the extension congruence property if for every closed subsemigroup \(T\) of \(S\) and for every closed congruence \(\rho\) on \(T\) there exists a congruence \(\sigma\) on \(S\) such that the restriction of \(\sigma\) to \(T\) is equal to \(\rho\). We say that \(S\) has the extension ideal property if for every closed subsemigroup \(T\) of \(S\) and for every closed ideal \(I\) of \(T\) there exists an ideal \(J\) of \(S\) with \(I=J\cap T\). The paper contains proofs of two theorems. A compact semigroup \(S\) has the extension ideal property if and only if \(S\) is a semilattice of compact semigroups satisfying certain technical conditions (formulated in the paper). A compact semigroup \(S\) with the extension congruence property has the extension ideal property. An example of a compact semigroup with the extension ideal property and without the extension congruence property is presented.