## $$k$$-bi-ideals and quasi $$k$$-ideals in $$k$$-Clifford and left $$k$$-Clifford semirings.(English)Zbl 1289.16090

Summary: A semiring $$S$$ in $$\mathbb{SL}^+$$ (the variety of all semirings whose additive reduct is a semilattice) is a $$k$$-Clifford semiring (left) if and only if it is a distributive lattice of $$k$$-semifields (left). Here we characterize both the $$k$$-Clifford semirings and left $$k$$-Clifford semirings by their ideals, especially by their $$k$$-bi-ideals and quasi $$k$$-ideals. A semiring $$S\in\mathbb{SL}^+$$ is a $$k$$-Clifford semiring if and only if $$I\cap B=\overline{IBI}$$ for every $$k$$-ideal $$I$$ and every $$k$$-bi-ideal $$B$$ of $$S$$.

### MSC:

 16Y60 Semirings 16D25 Ideals in associative algebras

### Keywords:

Clifford semirings; bi-ideals; quasi ideals