Cofinitely weak f-supplemented lattices

In this paper, weakly f-supplemented and cofinitely weak f-supplemented lattices are introduced and studied. Let a be an element of L such that a ≪_f L. If 1/a is weakly (f ∨ a)-supplemented and a/0 is weakly (f ∧ a)-supplemented, then L is weakly f-supplemented. A d-complemented lattice with f-small f-radical is weakly f-supplemented if and only if L is f-semilocal. A lattice L is cofinitely weak f-supplemented if and only if every maximal element of L has a weak f-supplement in L. If a/0 is a cofinitely weak (f ∧ a)-supplemented sublattice of a lattice L and 1/a has no maximal element, then L is cofinitely weak f-supplemented. Let L be a compactly generated lattice such that for every compact element c of L, rad_(f∧c)(c/0) = c∧rad_f(L). Then L is cofinitely weak f-supplemented if and only if L is cofinitely (f ∧ c)supplemented. Let L be a compact lattice such that for every compact element c of L, rad_(f∧c)(c/0) = c ∧ rad_f(L). Then L is weakly f-supplemented if and only if L is (f ∧ c)-supplemented.