N-Ideals of a Lattice

Abstract

This thesis studies the nature of n-ideals of a lattice. The topic arose out of a study on the kernels, around a particular element n, of a skeletal congruence on a distributive 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-ideal. 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 O 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.