Some properties of weakly g-supplemented lattices

In this work, all lattices are complete modular lattices with the greatest element 1 and the smallest element 0 . Let L be a lattice and a , b ∈ L . If a ∨ b = 1 and a ∧ b ≪ g L , then b is called a weak g-supplement of a in L . If every element of L has a weak g-supplement in L , then L is called a weakly g-supplemented lattice. In this work, some properties of these lattices are investigated. Let L be a lattice and a , b , c ∈ L . If c ≪ L , then b is a weak g-supplement of a in L if and only if b is a weak g-supplement of a ∨ c in L . Let L be a lattice and 1 = a 1 ∨ a 2 ∨ ⋯ ∨ a n with a i ∈ L ( 1 ≤ i ≤ n ) . If a i ∕ 0