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\(\mathcal D\)-subvarieties of the variety of idempotent semirings. (English) Zbl 1004.16050
Let \(\mathbf I\) be the variety of all additively and multiplicatively idempotent semirings. For any \((S,+,\cdot)\) in \(\mathbf I\) denote by \(\Delta\) [\(\nabla\)] the diagonal [universal] relation on \(S\), and by \(\mathbf{Con}(S)\) the set of all congruence relations on \(S\). Moreover, let \({\mathcal D}^+\) [\({\mathcal D}^\cdot\)] be Green’s \(\mathcal D\)-relation on \((S,+)\) [\((S,\cdot)\)]. Then eleven subclasses of \(\mathbf I\) are defined with the help of these relations, e.g., \({\mathbf B}_0=\{S\in{\mathbf I}:{\mathcal D}^+\cap{\mathcal D}^\cdot\in\mathbf{Con}(S)\}\), \(\mathbf{BR}=\{S\in{\mathbf I}:{\mathcal D}^+\cap{\mathcal D}^\cdot=\nabla\}\), and \(\mathbf{GBS}_{\ell}=\{S\in{\mathbf I}:{\mathcal D}^+\cap{\mathcal D}^\cdot=\Delta\}\). In a first step it is shown that all these classes form subvarieties of \(\mathbf I\). In a second step various decompositions of these varieties with respect to Mal’cev products are proved, e.g., \({\mathbf B}_0=\mathbf{BR}\circ\mathbf{GBS}_{\ell}\). Finally, the set theoretical relationship among these varieties is discussed. (Also submitted to MR).

16Y60 Semirings
08B05 Equational logic, Mal’tsev conditions
08B15 Lattices of varieties