Study of Pseudocomplemented Lattice

Abstract

This thesis studies the nature of Pseudocomplemented lattice. We can define a lattice in two ways; (i) Set theoretically and (ii) Algebraically. Set theoretically: A poset <L; ≤ > is a lattice if for every a, b ϵ L both Sup {a,b} and Inf {a,b} exists in L. Algebraically : A nonempty set L with two binary operations ˄ and ˅ is called a lattice if ∀ a, b, C ϵ L. The following conditions hold. i. a ˄ a=a, a ˅ a= a ii. a ˄ b=b ˄ a ,a ˅ b=b ˅ a. iii. a ˄ (b ˄ c)=(a ˄ b) ˄ c, a ˅ (b ˅ c) = (a ˅ b) ˅ c, iv. a ˄ (a ˅ b)=a, a ˅ (a ˄ b)=a. In this thesis, we have studied several properties of pseudocomplemented lattices. Moreover, we give several results on pseudocomplemented lattices which certainly extend and generalize many results in lattice theory. In Chapter one, we have discussed posets, lattices and Ideals of a lattice which are explain with some examples and generalized many theorems of them. In chapter two, congruence of lattices, distributive lattices, Complemented lattices and Boolean algebra have been discussed, which are basic concept of this thesis. In chapter three we give a description of pseudocomplemented lattices. We have also studied distributive pseudocomplemented lattices and algebraic lattices. Pseudocompiemented lattices have been studied by G. Gratzer [7] and many other authors. Here we extend several results of G. Gratzer [7] to lattices. Chapter four introduces the concepts of stone lattices. Stone lattices have been studied by Gratzer [7], Katrinak [11] and many other authors. We have given a characterization of minimal prime ideals of pseudocompiemented distributive lattices. Chapter five introduces the concept of distributive and modular lattice with n-ideals. Here we include several characterizations of n-ideals. We have proved some interesting result which are generalizes several results on distributive ,modular and ideals of a lattices. Latif [20] in his thesis has introduced the concept of standard n-ideals of a lattice. We conclude this thesis with some more properties of standard and neutral n-ideals.