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 $Lambda(a)={x in S | a stackrel{l}{longrightarrow^{infty}} x}$ induced by the transitive closure $stackrel{l}{longrightarrow^{infty}}$ of the relation $stackrel{l}{longrightarrow}$ which induce the equivalence relation $lambda$. Again non-transitivity of $stackrel{l}{longrightarrow}$ yields an expanding family {$stackrel{l}{longrightarrow^n}}$ of binary relations which associate subsets $Lambda_n(a)$ for all $a in S$, 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.