On Principal n-ideals of a Distributive nearlattice

Abstract

This thesis studies some of the nature of Principal n-ideal of a distributive near lattice. The n-ideals of a lattice have been studied by several authors including [4] and [36]. For a fixed element n of a lattice L, a convex sub lattice of L containing n is called an n-ideal of L. If a lattice L has a '0', then replacing n by 0, an n-ideal becomes an ideal and if L has a 'I', then it 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. Thus, the study on n-ideals generalized many results on lattice theory involving ideals and filters. Then many authors have extended the concept of n-ideals in a near lattice. A near lattice is a meet semilattice with the property that any two elements possessing a common upper bound, have a supremum. For a fixed element n of a near lattice S, a convex sub near lattice of S containing n is called an n-ideal of S. For two n-ideals I and J of a near lattice S, [30] has given a neat description of Iv J, while the set theoretic intersection is the infimum. Hence the set of all n-ideals of a near lattice S, denoted by in(S) is an algebraic lattice. An n-ideal generated by a finite number of elements ai, a2. ..., an is called a finitely generated n-ideal, denoted by< a1, a2., an >n, while the n-ideal generated by a single element a is called a principal n-ideal, denoted by < a >n, The set of finitely generated n-ideals and the set of principal n-ideals are denoted by Fn(S) and Pn(S) respectively. In this thesis, we devote ourselves in studding several properties of a distributive near lattice S. Replacing n by 0, Pn(S) becomes the set of all principal ideals of S which is isomorphic to S itself. Thus, all the results on Pn(S) are generalizations of the corresponding results of S. In this thesis our results generalize many results on semi-Boolean, generalized Stone, normal, relatively normal, m-normal and relatively m-normal near lattices. We have also generalized some results on annulets and a -ideals in terms of n-ideals.