Generalization of Distributive Lattice with Pseudo Complementation

Abstract

Lattice theory is an important part of Mathematics. Distributive lattices with Pseudo complementation have played many roles in development of lattice theory. Historically, lattice theory started with Boolean distributive lattices: as a result, the theory of distributive lattices is the most extensive and most satisfying chapter in the history of lattice theory. Distributive lattices 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 the most powerful variety. Throughout this thesis we will be concerned with another large variety known as the class of distributive Pseudo complemented lattice have been studied by several authors [1],[2],[3],[4],[5],[6]. On the other hand extended the notion of Pseudo complementation for meet semi lattices. There are two concepts that we should be able to distinguish: a lattice L, , , in which every element has a Pseudo complement and an algebra, L, , , ,0,1 where L, , ,0,1 is a bounded lattice and where, for every a L, the element a* is a Pseudo complement of a. We shall call the former a Pseudo complemented lattice and the later a lattice with Pseudo complementation (as an operation). The realization of special role of distributive 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 distributive lattices. In order to review, we include definitions, examples, solved problems and proof of some theorems. This work is divided into four chapters. Chapter-one is a prelude to the main text of the thesis, related to poset and various types of lattices, such as sublattice, ideal of lattice, bounded lattice, complete lattice. In chapter two we have discussed “Modular and distributive lattice” and this chapter is the concept of this work. Here we study the definition and examples of modular and distributive lattice. Some important theorem like “A modular lattice L is distributive if it has no sublattice isomorphic diagonal lattice M5 ”. Every modular lattice is distributive but converse is not true. The next chapter we discuse “Prime ideal of a lattice”, “Minimal prime ideal” and “Minimal prime n-ideal”. Chapter four dealt with the Distributive lattices with Pseudo complementation. This is the main part of my work. In this chapter we have discussed some definitions and some important theorems like “Any complete lattice that satisfies the Join Infinite Distributive (JID) identity is a Pseudo complemented distributive lattice.”