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Congruences on additive inverse semirings. (English) Zbl 1122.16040
An additive inverse semiring [skew-ring] is an arbitrary semiring \((S,+,\cdot)\) such that \((S,+)\) is an inverse semigroup [a group]. For \(E^+(S)=\{a\in S\mid a+a=a\}\) and any congruence \(\varrho\) on \((S,+,\cdot)\) define \(\varrho_{\max}=\{(a,b)\in S^2\mid ae\varrho be\) for all \(e\in E^+(S)\}\) and \(\ker\varrho=\{a\in S\mid a\varrho e\) for some \(e\in E^+(S)\}\), and denote by \(\varrho^{\max}\) the greatest congruence on \((S,+,\cdot)\) saturating \(\ker\varrho\). A completely regular semiring \(S\) is called a generalized Clifford semiring [a Clifford semiring] if it is an additive inverse semiring such that \(E^+(S)\) is a \(k\)-ideal of \(S\) [and a distributive lattice].
Now, let \((S,+,\cdot)\) be an additive inverse semiring such that \((E^+(S),\cdot)\) is a semilattice, and \(\varrho\) a congruence on \(S\). Then the following statements are equivalent: (i) \(S/\varrho\) is a Clifford semiring; (ii) \(S/\varrho_{\max}\) is a distributive lattice and \(S/\varrho^{\max}\) is a skew-ring; (iii) \(\varrho_{\max}=\varrho\vee\nu\) and \(\varrho^{\max}=\varrho\vee\sigma\), where \(\nu\) is the least distributive lattice congruence on \(S\) and \(\sigma\) is the least skew-ring congruence on \(S\). A similar statement is proved concerning generalized Clifford semirings in (i).
MSC:
16Y60 Semirings
20M18 Inverse semigroups
08A30 Subalgebras, congruence relations
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