Congruence openings of additive Green's relations on a semiring.
type: Item
instance of: scholarly article
title: Congruence openings of additive Green's relations on a semiring.
zbMATH Open document ID: 1248.16036
author: Nada Damljanović
author: Miroslav Ćirić
author: Stojan Bogdanović
published in: Semigroup Forum
publication date: 2011-09-06T00:00:00Z
review text: 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(\mathca…
reviewed by: Udo Hebisch
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 5945584
zbMATH Keywords: additively idempotent semirings
zbMATH Keywords: semigroups
zbMATH Keywords: Green relations
zbMATH Keywords: congruence openings
zbMATH Keywords: varieties of semirings
zbMATH Keywords: congruences
zbMATH Keywords: equivalences
zbMATH Keywords: lattices of varieties
MaRDI profile type: Publication
full work available at URL: s00233-010-9255-9
OpenAlex ID: W2140701593
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DOI: 10.1007/S00233-010-9255-9
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description: scientific article; zbMATH DE number 5945584