A study on standard n-ideal of a lattice

Abstract

Lattice theory is an important part of mathematics . Ideal lattice and n-ideal of a lattice have played many roles in development of lattice theory. Historically, lattice theory started with Boolean distributive lattices: as a result, the theory of ideal lattice and n-ideal of a lattice is the most extensive and most satisfying chapter in the history of lattice theory. Ideal lattice have provided the motivation for many results, in general lattice theory. Many conditions on lattices and on element and ideals of lattices are weakened forms of distributivity is imposed on lattices arising in various areas of mathematics, especially algebra. In lattice theory there are different classes of lattices known as variety of lattices. Class of Boolean lattice is of course most powerful variety. Throughout this thesis we will be concerned with another large variety known as the class of ideal lattice and n-ideal of a lattice have been studied by several authors. The realization of special role of ideal lattices moved to break with the traditional approach to lattice theory, which proceeds from partially ordered sets to general lattices, semi modular lattices, modular lattices and finally ideal lattices. In this thesis we give several result on ideal and n-ideal which will certainly extend and generalize many results in lattice theory. In order to review, we include definations, examples, solved problems and proof of some theorems. The thesis contains four chapter. Chapter 1 we have discussed the basic defination of relation, poset, lattice, complete lattice, convex sub lattice, complemented and relatively complemented lattice. We also proved that, Dual of a complete lattice is complete . Chapter 2 have discussed basic concept of ideal and n-ideal of lattice. Here we study the defination and examples of ideal and n-ideal. Some imprtant theorem like “If n is a neutral element of a lattice, then I (L) n is modular if and only if L is modular”. Chapter 3 we have discussed Standard element and n-ideals. We also discussed in this chapter Congruence relation. Chapter 4 deals with standard n-ideal and Principal n-ideal. This is the main part of this thesis work. In this chapter we have discussed some defination and some important theorems like “For a neutral element n and a standard n-ideal S and an n-ideal I, S I is also a standard n-ideal” .