Congruence openings of additive Green's relations on a semiring.
Let \((S,+,\cdot)\) be a semiring and \(\varrho\) an equivalence relation on \(S\). Then the congruence opening \(\varrho^\circ\) of \(\varrho\) is the greatest congruence on \((S,+,\cdot)\) contained in \(\varrho\). So, for Green's relations \(\mathcal L_+\) and \(\mathcal R_+\) on \((S,+)\) one has the congruences \(\mathcal L_+^\circ\), \(\mathcal R_+^\circ\) and \(\mathcal P_+=\mathcal L_+^\circ\circ\mathcal R_+^\circ=\mathcal R_+^\circ\circ\mathcal L_+^\circ\). For an arbitrary subvariety \(\mathcal V\) of the variety \(\mathcal S\) of all semirings define \(\mathcal V_{\mathcal L_+^\circ}=\mathbb H(\{S/\mathcal L_+^\circ\mid S\in\mathcal V\})\), \(\mathcal V_{\mathcal R_+^\circ}=\mathbb H(\{S/\mathcal R_+^\circ\mid S\in\mathcal V\})\) and \(\mathcal V_{\mathcal P_+}=\mathbb H(\{S/\mathcal P_+\mid S\in\mathcal V\})\), where \(\mathbb H(\mathcal C)\) denotes the class of all homomorphic images of semirings from a class \(\mathcal C\subseteq\mathcal S\). It is shown that \(\mathcal V_{\mathcal L_+^\circ}\), \(\mathcal V_{\mathcal R_+^\circ}\) and \(\mathcal V_{\mathcal P_+}=\mathcal V_{\mathcal L_+^\circ}\cap\mathcal V_{\mathcal R_+^\circ}\) are subvarieties of \(\mathcal S\). Moreover, the mappings \(\mathcal V\mapsto\mathcal V_{\mathcal L_+^\circ}\) and \(\mathcal V\mapsto\mathcal V_{\mathcal R_+^\circ}\) are opening operators on the lattice \(\mathbb L(\mathcal S)\) of all subvarieties of \(\mathcal S\). Now, for the subvariety \(\mathcal I^+\subseteq\mathcal S\) of all additively idempotent semirings define \(\mathcal V^{\mathcal L_+^\circ}=\{S\in\mathcal I^+\mid S/\mathcal L_+^\circ\in\mathcal V\}\) and \(\mathcal V^{\mathcal R_+^\circ}=\{S\in\mathcal I^+\mid S/\mathcal R_+^\circ\in\mathcal V\}\). It is proved that both classes are subvarieties of \(\mathcal I^+\) such that \(\mathcal V=\mathcal V_{\mathcal L_+^\circ}\vee\mathcal V_{\mathcal R_+^\circ}=\mathcal V^{\mathcal L_+^\circ}\cap\mathcal V^{\mathcal R_+^\circ}\) within the lattice \(\mathbb L(\mathcal I^+)\) of all subvarieties of \(\mathcal I^+\). Moreover, the mappings \(\mathcal V\mapsto\mathcal V^{\mathcal L_+^\circ}\) and \(\mathcal V\mapsto\mathcal V^{\mathcal R_+^\circ}\) are closure operators on this lattice, and these closure operators (as well as the above mentioned opening operators) yield an order embedding of \(\mathbb L(\mathcal I^+)\) into the direct product \(\{\mathcal V^{\mathcal L_+^\circ}\mid\mathcal V\in\mathbb L(\mathcal I^+)\}\times\{\mathcal V^{\mathcal R_+^\circ}\mid\mathcal V\in\mathbb L(\mathcal I^+)\}\) (resp. \(\{\mathcal V_{\mathcal L_+^\circ}\mid\mathcal V\in\mathbb L(\mathcal I^+)\}\times\{\mathcal V_{\mathcal R_+^\circ}\mid\mathcal V\in\mathbb L(\mathcal I^+)\}\)).
scientific article; zbMATH DE number 5945584
scientific article