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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