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 v b there exists an element c of b/0 such that a v c 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. © 2011 King Fahd University of Petroleum and Minerals. © 2011 Elsevier B.V. All rights reserved.