## Distributive lattice decompositions of semirings with a semilattice additive reduct.(English)Zbl 1205.16039

Let $$(S,+,\cdot)$$ be a semiring such that $$(S,+)$$ is a semilattice. For $$\emptyset\neq A\subseteq S$$ define $$\overline A=\{x\in S\mid x+a_1=a_2$$ for some $$a_1,a_2\in A\}$$ and $$\sqrt A=\{x\in S\mid x^n\in\overline A$$ for some positive integer $$n\}$$. Then $$S$$ is called $$k$$-Archimedean if $$S=\sqrt{SaS}$$ for all $$a\in S$$. Moreover, for the relation $$a\sigma b\Leftrightarrow b\in\sqrt{SaS}$$ for all $$a,b\in S$$ let $$\varrho=\sigma^*$$ denote the transitive hull of $$\sigma$$ and define $$\eta=\varrho\cap\varrho^{-1}$$.
It is proved that $$\eta$$ is the least distributive lattice congruence on $$S$$. From this fact several equivalent conditions are obtained to characterize $$S$$ as a distributive lattice of $$k$$-Archimedean semirings. One of these conditions is that $$\sigma$$ is transitive.

### MSC:

 16Y60 Semirings 08A30 Subalgebras, congruence relations
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### References:

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