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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...