Study on Modular Lattice and Boolean Algebra

Abstract

This thesis studies extensively the nature of modular lattices and Boolean algebras. The modular lattices have been study by several authors including Abbott [2] , Birkhoff [ 3] and Rutherford [19 ]. A poset is said to form a lattice if for every a, b ϵ L, a ˅ b and a ˄ b exists in L, where ˅ , , ˄ are two binary operation . A lattice L is called modular lattice ifforall a,b,c ϵ L with a ≥b, a, ˄ (b ˅ c) = [b ˅ (a ˄ c)]. In this thesis we give several results on modular lattices which certainly extend and generalized many result in lattice theory. In chapter one we discuss ideals , complete lattices , relatively complemented lattices and other results on lattices which are basic to this thesis . If every interval in a lattice is complemented the lattice is said to be relatively complemented. Chapter two discusses Embeddings, Kernels and dual homomorphisms. If L, M be two lattices, a one-one homomorphism θ : L→ M is called an embedding mapping . Also in that case we say L is embedded in M. We prove that the definition of dual meet homomorphism and dual join homomorphism are equivalent. In chapter three we discuss on modular lattices and distributive lattices Distributive lattices have been studied by sever author including Cignoli [ 4 ] , Cornish [ 5 ] , Cornish and Hicman [ 6 ] and Evans [ 7 ] Nieminen [15 ], [16] . Hence we prove a lattice L is distributive if and only if (a ˅ b) ˄ (b ˅ c) ˄ (c ˅ a)=(a ˄ b) ˅ (b ˄ c) ˅ (c ˄ a) ˅ a, b, c ϵ L In chapter four we discuss Boolean algebras and Boolean functions Previously Boolean algebras, Disjunctive Normal forms and Conjunctive Normal forms have studied by Abbott [ 1 ] . Here we extend several result on Boolean Algebras and also find the DN form of the function whose CN form is f =(x ˅ y ˅ z) ˄ (x ˅ y ˅ z') ˄ (x ˅ y' ˅ z) ˄ (x ˅ y' ˅ z') ˄ (x' ˅ y ˅ z).