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On additive idempotent \(k\)-regular semirings. (English) Zbl 1005.16044

Let \((S,+,\cdot)\) be an additively commutative and idempotent semiring. \(S\) is called \(k\)-regular if for every \(a\in S\) there exists some \(x\in S\) such that \(a+axa=axa\), and completely \(k\)-regular if additionally \(ax+ax^2a=ax^2a=xa+ax^2a\) holds. A \(k\)-unity of \(S\) is some \(e\in S\) such that \(a+ae=ae\) and \(a+ea=ea\). Note that every identity is such a \(k\)-unity but also every element which is absorbing with respect to both operations. Finally, \(S\) is a \(k\)-semifield if \(|S|>1\), and there is some \(k\)-unity in \(S\) such that for every \(a\not=0\) in \(S\) (if there is an absorbing zero \(0\)) there exists some \(x\in S\) satisfying \(e+xa=xa\) and \(e+ax=ax\). Hence every additively idempotent semifield is a \(k\)-semifield, and every \(k\)-semifield is \(k\)-regular, but not conversely. Moreover, a \(k\)-semifield has no zero divisors. By a carefull discussion of equivalence relations which correspond to Green’s relations in semigroup theory the authors prove that \(S\) is a completely \(k\)-regular semiring iff \(S\) is a union of \(k\)-semifields. (Also submitted to MR).

MSC:

16Y60 Semirings
12K10 Semifields
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