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” .
Description:
This thesis is submitted to the Department of Mathematics, Khulna University of Engineering & Technology in partial fulfillment of the requirements for the degree of Master of Science in Mathematics, April 2019.
Cataloged from PDF Version of Thesis.
Includes bibliographical references (pages 49-50).