A Study on Complemented Lattices and Boolean Function

Abstract

In this thesis the nature of Complemented lattice and Boolean function is studied. Lattice theory is a part of Mathematics. In Modern algebra, Abstract algebra and Boolean function are Lattice theory play important role. A non empty set P together with a binary relation R is said to form a partially ordered set or a posel if the following conditions hold: Reflexivity Anti-symmetry Transitivity A poset (L≤) is said to form a lattice if for every a, b ϵ L if Sup {a,b} and Inf { a,b}exist in L. A lattice is said to be complemented lattice if every element has complement. In this thesis we give several results on complemented lattice, Boolean function and Boolean algebra which will certainly extend and generalize many results in lattice theory. The thesis contains four chapters. Chapter one: We have discussed the basic definition of set, partially order set, relations, functions etc. Chapter two: We have discussed lattice, sublattice, convex sublattice, complemented lattice, ideal, Filter, Prime ideal, Principle ideal and Principle Dual ideal. We have proved that two bounded lattices A and B are complemented if and only if A x B is complemented. In this chapter we have also discussed the definition of upper bound, least upper bound, lower bound, greatest lower bound and relatively complemented lattice, and established relation among them. We also studied some other properties of these concepts and we have showed that two lattices A and B are relatively complemented if and only if the cross product of two lattices A and B is relatively complemented. Chapter three: We have discussed Boolean algebra, Boolean lattice and Boolean function. Let (A, ˄, ˅, ', 0, 1) be a Boolean algebra. Expressions involving members of A and the operations ˄, ˅ and complementation are called Boolean expressions or Boolean polynomials. For example, x ˅ y', x, x ˄ 0 are etc. all Boolean expressions. Any function specifying these Boolean expressions is called a Boolean function. Thus if/(x,y) =X ˄ y then f is the Boolean function and x ˄ y is the Boolean expressions (or value of the function f). Since it is normally the functional value (and not the function) that we are interested in, we call these expressions the Boolean function. We will denote least and greatest elements of a Boolean algebra by 0 and 1 respectively. In fact, most of the times we will confine ourselves to Boolean algebras that contain only these two elements. We also discus in this chapter Disjunctive Normal form (DN form), Conjunctive Normal form (CN form), Length and Cover. A Boolean function is said to be in DN form in n variables x1, x2, …… xn if it can be written as join of terms of the type f1(x1) ˄ f2 (x2)A ........ ˄ fn (xn ) where fi(xi ) = xi, or x’i for all i =1,2,3,…… ,n and no two terms are same. Also 1 and 0 are said to be in DN form. We also prove them theorem: 'Every Boolean function can be put in DN form'. Here we give several results on DN form, CN form, Homomorphisom, Iso-morphisom and Indomorphisom. Chapter four : In this chapter we have studied series combination, parallel combination, don't care condition and Bridge circuits. By a switch we mean a contact or a device in an electric circuit which lets the current to flow through the circuit. The switch can assume two states 'closed 'or 'open '(ON or OFF). In the first case the current flows and in the second the current does not flow. We will use a,b,c, . . ..... .....x,y,z........etc. to denote switches in a circuit. Two switches a,b are said to be connected 'in series' if the current can pass only when both are in closed state and current does not flow if any one or both are open. Two switches a, b, are said to be connected 'in parallel' if current flows when any one or both are closed and current does not pass when both are open. In this chapter we also solve some circuit problems.