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.


16Y60 Semirings