beta(*) RELATION ON LATTICES

Abstract

In this paper, we generalize beta(*) 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 is an element of L are beta(*) equivalent, a beta(*)b, if and only if for each t is an element of L such that a V t = 1 then b V t = 1 and for each k is an element of L such that b V k = 1 then a V k = 1, this is equivalent to a V b << 1/a and a V b << 1/b. We show that the beta(*) relation is an equivalence relation. Then, we examine beta(*) relation on weakly supplemented lattices. Finally, we show that L is weakly supplemented if and only if for every x is an element of L, x is equivalent to a weak supplement in L.