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On completely regular semirings. (English) Zbl 1115.16026
Let $$(S,+,\cdot)$$ be an arbitrary semiring. An element $$a\in S$$ is called completely regular, if there exists some $$x\in S$$ such that (i) $$a=a+x+a$$, (ii) $$a+x=x+a$$, and (iii) $$a(a+x)=a+x$$. If every element $$a\in S$$ is completely regular, $$(S,+,\cdot)$$ is called completely regular. A skew-ring is a semiring $$(S,+,\cdot)$$ such that $$(S,+)$$ is a (not necessarily Abelian) group, and a semiring $$(S,+,\cdot)$$ is called a $$b$$-lattice if $$(S,\cdot)$$ is a band and $$(S,+)$$ a semilattice.
The main result is that a semiring is completely regular if and only if it is a union of skew-rings, or equivalently, it is a $$b$$-lattice of completely simple semirings.

##### MSC:
 16Y60 Semirings