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A semiring S is said to be a t-k-simple semiring if it has no non-trivial proper left k-ideal and no non-trivial proper right k-ideal. We introduce the notion of t-k-simple semirings and characterize the semirings in SL + , the variety of all semirings with a semilattice additive reduct, which are distributive lattices of t-k-simple subsemirings. A semiring S is a distributive lattice of t-k-simple subsemirings if and only if every k-bi-ideal in S is completely semiprime k-ideal. Also the semirings for which every k-bi-ideal is completely prime has been characterized.

We characterize the least distributive lattice congruence on the semirings S with semilattice additive reduct in different ways arising from the nonsymmetry and non-transitivity of the binary relation −→ defined by:which are the principal completely semiprime k-ideal and the principal filter of S generated by a respectively. Both the relations M and N induced by M (a) and N (a) respectively are the least distributive lattice congruence on S. Again non-transitivity of −→ yields an expanding family {−→ n } of binary relations which again associates subsets Mn(a) and Nn(a) for all a ∈ S, and induces equivalence relations Mn and Nn. Also the n-th power of the symmetric opening ←→ of −→ gives us Σn(a) = {x ∈ S | a ←→ n x} which induces the equivalence relation σn. We have characterized the semirings which are distributive lattices of Mn(Nn, σn)-simple semirings.