f-Supplemented Lattices

Abstract

The main purpose of this thesis is to generalize some known results about F-supplemented modules to lattices. Let L be a complete modular lattice with smallest element 0 and greatest element 1. A homomorphic image of an f-small element under a lattice homomorphism need not be f-small unlike the module case. For compactly generated compact lattices f-supplement elements are compact. For compactly generated lattices, f-radical is the join of all f-small elements. Moreover for compact lattices, f-radical itself is an f-small element. Let L be a compactly generated compact lattice. If L is f-supplemented, then the quotient sublattice 1/ rad_f (L) of L is semiatomic. A compact lattice L is f-supplemented if and only if every maximal element m of L with f ≤ m has an f-supplement in L. A join of f-supplemented lattices containing f is f-supplemented. Let L be a compact lattice and f ≤ a be an element of L. If a/0 is f-supplemented and 1/a has no maximal element, then L is f-supplemented. If a lattice L is amply f-supplemented, then the quotient sublattice 1/a is amply (f ∨ a)-supplemented for every element a of L and the sublattice a/0 is amply f-supplemented for every f-supplement element a of L. L is amply f-supplemented if and only if for every element a of L, there is an element x ≤ a such that the sublattice x/0 is f-supplemented and the inequality x ≤ a is f-cosmall in L.