## 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