beta(*) RELATION ON LATTICES

Nebiyev C., Okten H. H.

MISKOLC MATHEMATICAL NOTES, vol.18, no.2, pp.993-999, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 2
  • Publication Date: 2017
  • Doi Number: 10.18514/mmn.2017.1782
  • Journal Name: MISKOLC MATHEMATICAL NOTES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.993-999
  • Keywords: beta(*)-relation, weakly supplemented lattice, complemented lattice, amply supplemented lattice, hollow lattice
  • Ondokuz Mayıs University Affiliated: Yes

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.