Study on Distributive Lattices and Boolean Algebras

Abstract

In this thesis we have studied the nature of distributive Lattice and Boolean Algebra. Lattice theory is branch of Mathematics. A poset (L,≤) is said to be form a Lattice if for every a,b ϵ L,a ˅b and a˄b exist in L. where ˅, ˄ are two binary operation. A letter L is called lattice, if it is distributive lattice then we have shown that a ˄ (b ˅ c) = (a ˄ b) ˅ (a ˄ c) for all a, b, c ϵ L. In this thesis we give several results on distributive Lattice, Boolean algebra and Boolean ring which are certainly extend and generalized many results in Lattice theory. The material of this thesis has been divided into five Chapters. A brief scenario of which we present as below.

Chapter one we have discussed basic definition of set, Lattice, convex sub lattice, meet semi-lattice and joint semi-lattices which are the basic to this thesis. We also prove that if A and B are two Lattices, that the product of A and B is a Lattice. In this Chapter we have also discussed the definition of ideals, bounded lattice, finite lattice, Complemented lattice and relatively complemented lattice. We have established the relations among them. Also we studied some other properties of these concepts. We have prove that two bounded Lattice are complemented if the cartesian product of the two Lattice is complemented. In Chapter two we have discussed Modular lattice, Distributive lattice. We include some characterization of modular and distributive Lattices. We have also proved a modular lattice is distributive lattice if and only if it has no sublattice isomorphic M5 . In Chapter three we discuss Pseudocomplemented lattice, Stone lattice, Stone algebra are discussed. We have proved the theorem let L be a pseudocomplernentd distributive lattice and P be a prime ideal of L. Then the following conditions are equivalent. i. P is minimal. ii. x ∈ P implies x * ∉ P iii. x ∈ P implies x * * ∉ P iv. P ∩ D(L)= φ In Chapter four Boolean algebra has discussed here. Since Boolean Lattice, Boolean subalgebra have been studied by several authors including Cornish [9] and A. Monteiro [33] We have established the relation among them. Also we have studied some other properties of this concept. We also proved that in a Boolean algebra, the following result are holds (a')' = a (a ˄ b)' =a' v b' [De Morgan's Law] (a v b)' =a' ˄ b' [De Morgan's Law] (a ≤ b)  a ' ≥ b' a ≤ b )  a' ˄ b= 0  a' v b' = u In Chapter five Boolean ring, Disjunctive Normal form, Conjunctive Normal form are expressed here. We also have showed every Boolean ring with unity is a Boolean algebra. Last section in this chapter we should try to discussed the switching circuit system. The simplest example of such switch being on ordinary ON-OFF. These are two basic way in which switches are generally interconnected. These are referred to as in series and parallel. We have also explained with figure the circuit represented by the Boolean function f = a ˄ (b v c).