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 left k -radicals Λ( a ) = { x ∈ S | a l −→ ∞ x } induced by the transitive closure l −→ ∞ of the relation l −→ which induce the equivalence relation λ . Again non-transitivity of l −→ yields an expanding family { l −→ n } of binary relations which associate subsets Λ n ( a ) for all a ∈ S , which again induces an equivalence relation λ n . We also define λ ( λ n )-simple semirings, and characterize the semirings which are distributive lattices of λ ( λ n )-simple semirings.