On the lattice of congruences on inverse semirings

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 ${\displaystyle {\rho }_{min},{\rho }_{max},{\rho }^{min}}$ and ${\displaystyle {\rho }^{max}}$ on S and showed that ${\displaystyle \rho \theta =[{\rho }_{min},{\rho }_{max}]}$ and ${\displaystyle \rho \kappa =[{\rho }^{min},{\rho }^{max}]}$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ${\displaystyle {\rho }_{max}}$ is a distributive lattice congruence and ${\displaystyle {\rho }^{max}}$ is a skew-ring congruence on S. If η (σ) is the least distributive lattice (resp. skew-ring) congruence on S then η ∩ σ is the least Clifford congruence on S.