Distributive Lattices of λ-simple Semirings

In this paper, we study the decomposition of semirings with a semilattice additive reduct. For, we introduce the notion of principal l left k-radicals Λ(a) = {x ∈ S | a → l ∞ x} induced by the transitive l l closure → l ∞ of the relation →l which induce the equivalence relation λ. l l Again non-transitivity of →l yields an expanding family {→l n} of binary relations which associate subsets Λn(a) for all a ∈ S, which again induces an equivalence relation λn. We also define λ(λn)-simple semirings, and characterize the semirings which are distributive lattices of λ(λn)-simple semirings.