Semilattice decompositions of semigroups (Q5917773)

Let \(S\) be a semigroup. An ideal \(I\) of \(S\) is called completely semiprime if \(a^2\in I\) implies \(a\in I\). Using this concept the authors give a new characterization of the least semilattice congruence \(\eta\) on \(S\). If \(I(a)\) denotes the least completely semiprime ideal of \(S\) containing \(a\in S\) then \(\eta\) is given by: \(a\eta b\) iff \(I(a)=I(b)\). This congruence was described by \textit{M. Petrich} [Math. Z. 85, 68-82 (1964; Zbl 0124.25801)], by means of completely prime ideals (that is, \(ab\in I\) implies \(a\in I\) or \(b\in I\)). Using completely semiprime left-(right-)ideals of \(S\) the authors define equivalences \(\lambda\), \(\rho\), and \(\tau=\lambda\cap\rho\) on \(S\) which generalize Green's relations \(\mathcal L\), \(\mathcal R\), and \(\mathcal H\), respectively. Conditions for \(S\) are given to be a semilattice of \(\lambda\)-(\(\rho\)-,\(\tau\)-)simple subsemigroups (i.e., on which \(\lambda\) (\(\rho\), \(\tau\)) is the universal relation). In particular, the case is investigated, when \(S /\eta\) (\(S/\lambda\), \(S/\rho\), \(S/\tau\)) is a chain. Finally, relations \(\eta_n\), \(\lambda_n\), \(\rho_n\), \(\tau_n\) such that \({\mathcal J}\subseteq\eta_n\subseteq\eta\), \({\mathcal L}\subseteq\lambda_n\subseteq\lambda\), \({\mathcal R}\subseteq\rho_n\subseteq\rho\), \({\mathcal H}\subseteq\tau_n\subseteq\tau\) (\(n\in\mathbb{N}\)) are introduced and semilattices and chains of \(\eta_n\)-(\(\lambda_n\)-,\(\rho_n\)-,\(\tau_n\)-)simple semigroups are described.