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Here we introduce the notion of strong quasi k-ideals of a semiring in SL⁺ and characterize the semirings that are distributive lattices of t-k-simple(t-k-Archimedean) subsemirings by their strong quasi k-ideals. A quasi k-ideal Q is strong if it is an intersection of a left k-ideal and a right k-ideal. A semiring S in SL⁺ is a distributive lattice of t-k-simple semirings if and only if every strong quasi k-ideal is a completely semiprime k-ideal of S. Again S is a distributive lattice of t-k-Archimedean...

We introduce the k-radicals of Green's relations in semirings with a semilattice additive reduct, introduce the notion of left k-regular (right k-regular) semirings and characterize these semirings by k-radicals of Green's relations. We also characterize the semirings which are distributive lattices of left k-simple subsemirings by k-radicals of Green's relations.

An element $e$ of an ordered semigroup S is called an ordered idempotent if e ≤ e². Here we characterize the subsemigroup $$generatedbythesetofallorderedidempotentsofaregularorderedsemigroupS.IfSisaregularorderedsemigroupthen$$ is also regular. If S is a regular ordered semigroup generated by its ordered idempotents then every ideal of S is generated as a subsemigroup by ordered idempotents.

Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences ${\rho }_{min},{\rho }_{max},{\rho }^{min}$ and ${\rho }^{max}$ on S and showed that $\rho \theta =[{\rho }_{min},{\rho }_{max}]$ and $\rho \kappa =[{\rho }^{min},{\rho }^{max}]$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ${\rho }_{max}$ is a distributive lattice congruence and ${\rho }^{max}$ is a skew-ring congruence on S. If η (σ) is the least distributive...