Β* Relation on Lattices

Abstract

In this paper, we generalize β* relation on submodules of a module (see [1]) to elements of a complete modular lattice. Let L be a complete modular lattice. We say a,b ∈ L are β* equivalent, aβ*b if and only if for each t ∈ L such that a ∨ t D 1 then b ∨ t = 1 and for each k ∈ L such that b ∨k = 1 then a ∨k = 1 this is equivalent to a∨b ≪1/a and a∨b ≪1/b We show that the β* relation is an equivalence relation. Then, we examine β* relation on weakly supplemented lattices. Finally, we show that L is weakly supplemented if and only if for every x ∈ L, x is equivalent to a weak supplement in L. © 2017 Miskolc University Press.