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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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