A Study on Standard n-Ideals of a Lattice

Abstract

This thesis studies the nature of standard n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor. For a fixed element n of a lattice L, a convex sublattice containing n is called an n-ideals. If L has an 'O', then replacing n by 0, an n-ideal becomes an ideal. Moreover, if L has 1, an n-ideal becomes a filter by replacing n by 1. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. So, any result involving n-ideals will give a generalization of the results on ideals and filters with 0 and 1 respectively in a lattice. In this thesis we give a series of results on n-ideals of a lattice which certainly extend and generalize many works in lattice theory. Chapter-1, discusses n-ideals, finitely generated n-ideals and other results on n-ideals of a lattice which are basic to this thesis. We have shown that, a lattice L is modular (distributive) if and only if in (L), the lattice of n-ideals is modular (distributive). In chapter-2, we have discussed lattices and elements with special properties. Here we have proved the coincidence of standard and neutral elements in a wide class of lattices including modular lattices, weakly modular lattices as well as relatively complemented lattices. In modular lattices and relatively complemented lattices the proves of the results are trivial but in weakly modular lattices this prove is not so simple. In this chapter, we have proved the following results: