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Cofinitely Supplemented Modular Lattices

Abstract

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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Correspondence to Rafail Alizade.

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Alizade, R., Toksoy, S.E. Cofinitely Supplemented Modular Lattices. Arab J Sci Eng 36, 919 (2011). https://doi.org/10.1007/s13369-011-0095-z

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Keywords

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

Mathematics Subject Classification (2010)

  • 06CXX
  • 16D10