Cofinitely weak f-supplemented lattices

Abstract

In this paper, weakly [Formula: see text]-supplemented and cofinitely weak [Formula: see text]-supplemented lattices are introduced and studied. Let [Formula: see text] be an element of [Formula: see text] such that [Formula: see text]. If [Formula: see text] is weakly [Formula: see text]-supplemented and [Formula: see text] is weakly [Formula: see text]-supplemented, then [Formula: see text] is weakly [Formula: see text]-supplemented. A [Formula: see text]-complemented lattice with [Formula: see text]-small [Formula: see text]-radical is weakly [Formula: see text]-supplemented if and only if [Formula: see text] is [Formula: see text]-semilocal. A lattice [Formula: see text] is cofinitely weak [Formula: see text]-supplemented if and only if every maximal element of [Formula: see text] has a weak [Formula: see text]-supplement in [Formula: see text]. If [Formula: see text] is a cofinitely weak [Formula: see text]-supplemented sublattice of a lattice [Formula: see text] and [Formula: see text] has no maximal element, then [Formula: see text] is cofinitely weak [Formula: see text]-supplemented. Let [Formula: see text] be a compactly generated lattice such that for every compact element [Formula: see text] of [Formula: see text], [Formula: see text]. Then [Formula: see text] is cofinitely weak [Formula: see text]-supplemented if and only if [Formula: see text] is cofinitely [Formula: see text]-supplemented. Let [Formula: see text] be a compact lattice such that for every compact element [Formula: see text] of [Formula: see text], [Formula: see text]. Then [Formula: see text] is weakly [Formula: see text]-supplemented if and only if [Formula: see text] is [Formula: see text]-supplemented.