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 Lambda(a) = {x is an element of S vertical bar a ->(l infinity) x} induced by the transitive closure ->(l infinity )of the relation ->(l) which induce the equivalence relation lambda. Again non-transitivity of ->(l) yields an expanding family {->(ln)} of binary relations which associate subsets Lambda(n)(a) for all a is an element of S, which again induces an equivalence relation lambda(n). We also define lambda(lambda(n))-simple semirings, and characterize the semirings which are distributive lattices of lambda(lambda(n))-simple semirings.