β*RELATION ON LATTICES.

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 = 1 then b⋁t = 1 and for each k ∊ L such that b⋁k D1 then a⋁k D1, 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. [ABSTRACT FROM AUTHOR]