Geodesic
Distributive lattices of t-k-Archimedean semirings
Discussiones Mathematicae. General Algebra and Applications, Tome 31 (2011) no. 2, pp. 147-158

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A semiring S in ⁺ is a t-k-Archimedean semiring if for all a,b ∈ S, b ∈ √(Sa) ∩ √(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 √B is a k-ideal, and S is a chain of t-k-Archimedean semirings if and only if √B is a completely prime k-ideal, for every k-bi-ideal B of S.
Keywords: k-radical, t-k-Archimedean semiring, completely prime k-ideal, semiprimary k-ideal 
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Mondal, Tapas. Distributive lattices of t-k-Archimedean semirings. Discussiones Mathematicae. General Algebra and Applications, Tome 31 (2011) no. 2, pp. 147-158. http://geodesic.mathdoc.fr/item/DMGAA_2011_31_2_a1/