## Distributive lattices of $$t$$-$$k$$-Archimedean semirings.(English)Zbl 1254.16042

Summary: A semiring $$S$$ in $$\mathbb{SL}^+$$ is a $$t$$-$$k$$-Archimedean semiring if for all $$a,b\in S$$, $$b\in\sqrt{Sa}\cap\sqrt{aS}$$. Here we introduce the $$t$$-$$k$$-Archimedean semirings and characterize the semirings which are distributive lattice (chain) of $$t$$-$$k$$-Archimedean semirings. A semiring $$S$$ is a distributive lattice of $$t$$-$$k$$-Archimedean semirings if and only if $$\sqrt B$$ is a $$k$$-ideal, and $$S$$ is a chain of $$t$$-$$k$$-Archimedean semirings if and only if $$\sqrt B$$ is a completely prime $$k$$-ideal, for every $$k$$-bi-ideal $$B$$ of $$S$$.

### MSC:

 16Y60 Semirings 16D25 Ideals in associative algebras
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