A generalization of essential supplemented lattices
Hasan Hüseyın Ökten, Figen Erılmaz, Berna Koşar · Miskolc mathematical notes/Mathematical notes · 2026
In this work, all lattices are complete modular lattices with the smallest element 0 and the greatest element 1 . Let L be a lattice. If every essential element of L has a weak supplement in L , then L is called a weakly essential supplemented (briefly, weakly e-supplemented) lattice. In this work, some properties of these lattices are investigated. The concept of weakly essential supplemented lattice is a generalization of the concept of essential supplemented lattice. Let L be a weakly e-supplemented lattice. Then 1 ∕ r ( L ) have no essential elements with distinct from 1 . Let L be a lattice, a 1 , a 2 , … , a n ∈ L and 1 = a 1 ∨ a 2 ∨ ⋯ ∨ a n . If a i ∕ 0 is weakly e-supplemented for every i = 1 , 2 , … , n , then L is also weakly e-supplemented. Let L be a weakly e-supplemented lattice and a ∈ L . Then the quotient sublattice 1 ∕ a is weakly e-supplemented. Let L be a lattice. Then L is weakly e-supplemented if and only if every essential element of L is β ∗ equivalent to a weak supplement in L .