$ \beta_* $ relation on lattices
Celıl Nebıyev, Hasan Hüseyın Ökten · Miskolc mathematical notes/Mathematical notes · 2017
In this paper, we generalize ˇ 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 2 L are ˇ equivalent, aˇ b, if and only if for each t 2 L such that a _ t D 1 then b _ t D 1 and for each k 2 L such that b _ k D 1 then a _ k D 1, this is equivalent to a _ b 1=a and a _ b 1=b.We show that the ˇ relation is an equivalence relation.Then, we examine ˇ relation on weakly supplemented lattices.Finally, we show that L is weakly supplemented if and only if for every x 2 L, x is equivalent to a weak supplement in L.