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Study on Distributive Lattices and Boolean Algebras

 dc.contributor.advisor Rahman, Prof. Dr. Md. Bazlar dc.contributor.author Rahman, Md. Zaidur dc.date.accessioned 2018-08-13T03:32:48Z dc.date.available 2018-08-13T03:32:48Z dc.date.copyright 2006 dc.date.issued 2006-04 dc.identifier.other ID 0051501 dc.identifier.uri http://hdl.handle.net/20.500.12228/348 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, 2006. en_US dc.description Cataloged from PDF Version of Thesis. dc.description Includes bibliographical references (pages 107-110). dc.description.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. en_US 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). dc.description.statementofresponsibility Md. Zaidur Rahman dc.format.extent 110 pages dc.language.iso en_US en_US dc.publisher Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh. en_US dc.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.subject Distributive Lattices en_US dc.subject Boolean Algebras en_US dc.subject Boolean Ring en_US dc.title Study on Distributive Lattices and Boolean Algebras en_US dc.type Thesis en_US dc.description.degree Master of Philosophy in Mathematics dc.contributor.department Department of Mathematics
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