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\( \beta_* \) RELATION ON LATTICES
  • Hasan OKTEN
Hasan OKTEN

Corresponding Author:[email protected]

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Abstract

In this paper, we generalize \( \beta^* \) relation on submodules of a module (see \cite{nebiyev}) to elements of a complete modular lattice. Let \( L \) be a complete modular lattice. We say \( a,b \in L \) are \( \beta_* \ equivalent \), \( a \beta_* b \), if and only if for each \( t \in L \) such that \( a \vee t = 1 \) then \( b \vee t = 1 \) and for each \( k \in L \) such that \( b \vee k = 1 \) then \( a \vee k = 1 \), this is equivalent to \( a \vee b \ll 1/a \) and \( a \vee b \ll 1/b \). We show that the \( \beta_* \) relation is an equivalence relation. Then, we examine \( \beta_* \) relation on weak supplemented lattices. Finally, we show that \( L \) is weakly supplemented if and only if for every \( x \in L \), \( x \) is equivalent to a weak supplement in \( L \).

: 06C05, 16D10


: \( \beta_* \)-relation, weakly supplemented lattice, complemented lattice, amply supplemented lattice, hollow lattice.