On the lattice of congruences on inverse semirings

Anwesha Bhuniya; Anjan Kumar Bhuniya

Discussiones Mathematicae - General Algebra and Applications, Tome 28 (2008), p. 193-208 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $ρ_{m i n}, ρ_{m a x}, ρ^{m i n}$ and $ρ^{m a x}$ on S and showed that $ρ θ = [ρ_{m i n}, ρ_{m a x}]$ and $ρ κ = [ρ^{m i n}, ρ^{m a x}]$ . Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_{m a x}$ is a distributive lattice congruence and $ρ^{m a x}$ 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.

Publié le : 2008-01-01

Zbl 1196.16039

EUDML-ID : urn:eudml:doc:276932

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Anwesha Bhuniya; Anjan Kumar Bhuniya. On the lattice of congruences on inverse semirings. Discussiones Mathematicae - General Algebra and Applications, Tome 28 (2008) pp. 193-208. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1143/

Bibliographie

[000] [1] R. Feigenbaum, Kernels of orthodox semigroup homomorphisms, J. Austral. Math. Soc. 22 (A) (1976), 234-245. | Zbl 0343.20033

[001] [2] R. Feigenbaum, Regular semigroup congruences, Semigroup Forum 17 (4) (1979), 373-377. | Zbl 0433.20043

[002] [3] D.G. Green, The lattice of congruences on an inverse semigroup, Pacific J. Math. 57 (1975), 141-152. | Zbl 0279.20051

[003] [4] M.P. Grillet, Semirings with a completely simple additive Semigroup, J. Austral. Math. Soc. 20 (A) (1975), 257-267. | Zbl 0316.16039

[004] [5] J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford 1995.

[005] [6] S.K. Maity, Congruences on additive inverse semirings, Southeast Asian Bull. Math. 30 (2006). | Zbl 1122.16040

[006] [7] F. Pastijn and M. Petrich, Congruences on regular semigroups, Trans. Amer. Math. Soc. 295 (2) (1986), 607-633. | Zbl 0599.20095

[007] [8] M. Petrich, Congruences on Inverse Semigroups, Journal of Algebra 55 (1978), 231-256. | Zbl 0401.20054

[008] [9] M. Petrich, Inverse Semigroups, John Wiley & Sons 1984.

[009] [0] M. Petrich and N.R. Reilly, A network of congruences on an inverse semigroup, Trans. Amer. Math. Soc. 270 (1) (1982), 309-325. | Zbl 0484.20026

[010] [1] N.R. Reilly and H.E. Scheiblich, Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349-360. | Zbl 0159.02503

[011] [2] H.E. Scheiblich, Kernels of inverse semigroup homomorphisms, J. Austral. Math. Soc. 18 (1974), 289-292. | Zbl 0294.20061

[012] [3] M.K. Sen, S. Ghosh and P. Mukhopadhyay, Congruences on inverse semirings, pp. 391-400 in: Algebras and Combinatorics (Hong Kong, 1997), Springer, Singapore 1999. | Zbl 0980.16041

[013] [4] M.K. Sen, S.K. Maity and K.P. Shum, On completely regular semirings, Bull. Cal. Math. Soc. 98 (4) (2006), 319-328. | Zbl 1115.16026

[014] [5] M.K. Sen, S.K. Maity and K.P. Shum, Clifford semirings and generalized Clifford semirings, Taiwanese Journal of Mathematics 9 (3) (2005), 433-444. | Zbl 1091.16028