Weak \(C\)-sets are not \(C\)-sets in regular semigroups (Q5931527)

A weak \(C\)-set \(P\) of a regular semigroup \(S\) is a set of idempotents that meets each \(\mathcal R\)- and \(\mathcal L\)-class of \(S\), all products of two members of \(P\) are idempotent, and \(P\) satisfies the condition that \(qPq\) is a subset of \(P\) for all \(q\) in \(P\). The paper gives an example of a completely simple semigroup that has exactly two weak \(C\)-sets neither of which is a \(C\)-set, that is neither has the additional property that each \(x\) in \(S\) has an inverse \(x^*\) such that both \(xP^1x^*\) and \(x^*P^1x\) are contained in \(P\).