KUET Institutional Repository

# Study on Modular Lattice and Boolean Algebra

 dc.contributor.advisor Rahman, Prof. Dr. Md. Bazlar dc.contributor.author Islam, Md. Rashidul dc.date.accessioned 2018-08-13T04:13:33Z dc.date.available 2018-08-13T04:13:33Z dc.date.copyright 2008 dc.date.issued 2008-04 dc.identifier.other ID 0051507 dc.identifier.uri http://hdl.handle.net/20.500.12228/351 dc.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 Philosophy in Mathematics, April 2008. en_US dc.description Cataloged from PDF Version of Thesis. dc.description Includes bibliographical references (pages 85-86). dc.description.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. en_US 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). dc.description.statementofresponsibility Md. Rashidul Islam dc.format.extent 86 pages dc.language.iso en_US en_US dc.publisher Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh. en_US dc.subject Modular Lattice en_US dc.subject Boolean Algebra en_US dc.subject Distributive Lattices en_US dc.title Study on Modular Lattice and Boolean Algebra en_US dc.type Thesis en_US dcterms.rights Khulna University of Engineering & Technology (KUET) thesis/dissertation/internship reports are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. dc.description.degree Master of Philosophy in Mathematics dc.contributor.department Department of Mathematics
﻿