Geodesic
On the lattice of congruences on inverse semirings
Discussiones Mathematicae. General Algebra and Applications, Tome 28 (2008) no. 2, pp. 193-208

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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 ρ_min, ρ_max, ρ^min and ρ^max on S and showed that ρθ = [ρ_min,ρ_max] and ρκ = [ρ^min,ρ^max]. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ρ_max is a distributive lattice congruence and ρ^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.
Keywords: inverse semirings, trace, kernel, Clifford congruence, least Clifford congruence 
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Bhuniya, Anwesha; Bhuniya, Anjan. On the lattice of congruences on inverse semirings. Discussiones Mathematicae. General Algebra and Applications, Tome 28 (2008) no. 2, pp. 193-208. http://geodesic.mathdoc.fr/item/DMGAA_2008_28_2_a4/