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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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