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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).

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