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Generalizations of supplemented lattices. (English) Zbl 1423.06025
Summary: Some generalizations of the concept of a supplemented lattice, namely a soc-supplemented-lattice, soc-amply-supplemented-lattice, soc-weak-supplemented-lattice, soc-$$\oplus$$-supplemented-lattice and completely soc-$$\oplus$$-supplemented-lattice are introduced. Various results are proved to show the relationship between these lattices. We have also proved that, if $$L$$ is a soc-$$\oplus$$-supplemented-lattice satisfying the summand intersection property (SIP), then $$L$$ is a completely soc-$$\oplus$$-supplemented-lattice.
MSC:
 06C05 Modular lattices, Desarguesian lattices 06C15 Complemented lattices, orthocomplemented lattices and posets
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References:
 [1] Takil, Mutlu Figen, Amply weak semisimple-supplemented-modules, Int. J. Pure Appl. Math., 83, 4, 613-621, (2013) [2] Tohidi, M., Soc-$$\oplus$$-supplemented modules, Int. J. Math. Sci. Appl., 2, 2, 803-812, (2012) [3] Wang, Y.; Ding, N., Generalized supplemented modules, Taiwanese J. Math., 10, 6, 1589-1601, (2006) · Zbl 1122.16003 [4] Wisbauer, R., Foundations of Module and Ring Theory, (1991), Gordon and Breach Sc. Pub.: Gordon and Breach Sc. Pub. Reading [5] Călugăreanu, G., Lattice Concepts of Module Theory, (2000), Kluwer: Kluwer Dordrecht · Zbl 0959.06001 [6] Alizade, R.; Toksoy, S. E., Cofinitely supplemented modular lattices, Arab. J. Sci. Eng., 36, 6, 919, (2011) [7] Alizade, R.; Toksoy, S. E., Cofinitely weak supplemented lattices, Indian J. Pure. Appl. Math., 40, 5, 337-346, (2009) · Zbl 1245.06016 [8] Grätzer, G., Lattice Theory: First Concepts and Distributive Lattices, (1971), W. H. Freeman and company: W. H. Freeman and company San Francisco · Zbl 0232.06001 [9] Anderson, F. W.; Fuller, K. R., Rings and Categories of Modules, (1992), Springer Verlag · Zbl 0765.16001 [10] Akalan, E.; Birkenmeier, G. F.; Tercan, A., Corrigendum to Goldie extending modules, Commun. Algebra, 41, 2005, (2013), Original article.ibid 37(2009), 663-683 and first correction in 38(2010), 4747-4748 · Zbl 1278.16005 [11] Nimbhorkar, S. K.; Shroff, Rupal, Goldie extending elements in modular lattices, Math. Bohem., 142, 2, 163-180, (2017) · Zbl 1424.06028
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