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Cofinitely Supplemented Modular Lattices

Research paper by Rafail Alizade, Sultan Eylem Toksoy

Indexed on: 12 Aug '11Published on: 12 Aug '11Published in: Arabian Journal for Science and Engineering



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.