Semilattices of \({\mathcal L}^{**}\)-simple semigroups (Q1393036)

Green's \(\mathcal L\)-relation on a semigroup \(S\) was extended by John Fountain in studying abundant semigroups by defining \(a{\mathcal L}^*b\) if and only if \(a\) and \(b\) in \(S\) are \(\mathcal L\)-related in some oversemigroup of \(S\). In this paper, the author further extends the \({\mathcal L}^*\)-relation to a right congruence relation \({\mathcal L}^{**}\) such that \({\mathcal L}\subseteq{\mathcal L}^*\subseteq{\mathcal L}^{**}\) on semigroups. Semigroups which are semilattices of \({\mathcal L}^{**}\)-simple monoids are particularly studied. The author proves that a semigroup \(S\) is a strong semilattice of \({\mathcal L}^{**}\)-simple monoids if and only if the idempotents of \(S\) are central and each \({\mathcal L}^{**}\)-class contains an idempotent. Some other related results are also obtained.