A Boolean algebra may also be defined as a partially ordered set with certain additional

Chapter 8, Problem 29

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A Boolean algebra may also be defined as a partially ordered set with certain additional properties. Let (B, d) be a partially ordered set. For any x, y [ B, we define the least upper bound of x and y as an element z such that x d z, y d z, and if there is any element z* with x d z* and y d z*, then z d z*. The greatest lower bound of x and y is an element w such that w d x, w d y, and if there is any element w* with w* d x and w* d y, then w* d w. A lattice is a partially ordered set in which every two elements x and y have a least upper bound, denoted by x + y, and a greatest lower bound, denoted by x # y. a. Prove that in any lattice (i) x # y = x if and only if x d y (ii) x + y = y if and only if x d y b. Prove that in any lattice (i) x + y = y + x (ii) x # y = y # x (iii) (x + y) + z = x + (y + z) (iv) (x # y) # z = x # (y # z) c. A lattice L is complemented if there exists a least element 0 and a greatest element 1, and for every x [ L there exists x [ L such that x + x = 1 and x # x = 0. Prove that in a complemented lattice L, x + 0 = x and x # 1 = x for all x [ L. d. A lattice L is distributive if x + (y # z) = (x + y) # (x + z) and x # (y + z) = (x # y) + (x # z) for every x, y, z [ L. By parts (b) and (c), a complemented, distributive lattice is a Boolean algebra. Which of the following Hasse diagrams of partially ordered sets do not represent Boolean algebras? Why? (Hint: In a Boolean algebra, the complement of an element is unique.)