Cleavability in semigroups (Q1576301)

The notion of cleavability was originally introduced in topology: a space \(X\) is cleavable (pointwise cleavable) over a class \(\mathcal P\) of topological spaces if for every \(A \subset X\) (every \(x\in X\)) there exist \(Y\in\mathcal P\) and a continuous mapping \(f:X\to Y\) such that \(f^\gets f(A)=A\) (\(f^\gets f(x)=x\)). By replacing ``topological space'' by ``semigroup'' and ``continuous mapping'' by ``semigroup homomorphism'' the authors define cleavability of semigroups and prove two results of the following sort: If a semigroup \(S\) is (pointwise) cleavable over a class \(\mathcal P\) of semigroups, then \(S\) is so in \(\mathcal P\).