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Cofinitely Supplemented Modular Lattices


In this paper it is shown that a lattice L is a cofinitely supplemented lattice if and only if every maximal element of L has a supplement in L. If a/0 is a cofinitely supplemented sublattice and 1/a has no maximal element, then L is cofinitely supplemented. A lattice L is amply cofinitely supplemented if and only if every maximal element of L has ample supplements in L if and only if for every cofinite element a and an element b of L with \({a\vee b=1}\) there exists an element c of b/0 such that \({a\vee c=1}\) where c is the join of finite number of local elements of b/0. In particular, a compact lattice L is amply supplemented if and only if every maximal element of L has ample supplements in L.

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  1. 1

    Alizade R., Bilhan G., Smith P.F.: Modules whose maximal submodules have supplements. Commun. Algebra 29(6), 2389–2405 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2

    Alizade R., Toksoy S.E.: Cofinitely weak supplemented lattices. Indian J. Pure Appl. Math. 40(5), 337–346 (2009)

    MathSciNet  MATH  Google Scholar 

  3. 3

    Cǎlugǎreanu G.: Lattice Concepts of Module Theory. Kluwer, Dordrecht (2000)

    Google Scholar 

  4. 4

    Clark, J.; Lomp, C.; Vanaja, N.; Wisbauer, R.: Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers Mathematics. Birkhäuser, Basel (2006)

  5. 5

    Çetindil, Y.: Generalizations of Cofinitely Supplemented Modules to Lattices. M.Sc. thesis, Izmir Institute of Technology (2005)

  6. 6

    Stenström B.: Radicals and socles of lattices. Arch. Math. 20, 258–261 (1969)

    MATH  Article  Google Scholar 

  7. 7

    Wisbauer R.: Foundations of Module and Ring Theory. Gordon and Breach, Philadelphia (1991)

    MATH  Google Scholar 

  8. 8

    Zöschinger H.: Komplemente als direkte Summanden. Arch. Math. 25, 241–253 (1974)

    MATH  Article  Google Scholar 

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Correspondence to Rafail Alizade.

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Alizade, R., Toksoy, S.E. Cofinitely Supplemented Modular Lattices. Arab J Sci Eng 36, 919 (2011).

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  • Cofinite element
  • Ample supplement
  • Amply supplemented lattice
  • Cofinitely supplemented lattice
  • Amply cofinitely supplemented lattice

Mathematics Subject Classification (2010)

  • 06CXX
  • 16D10