Problems and Comments on Boolean Algebras

The notation "" for OR is bad and misleading. Just think that in the context of boolean functions, the author uses  instead of ∨.The integers modulo 2, that is Z2  0,1, have an addition where 1  1  0 while 1 ∨ 1  1. A set A is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S, it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a  b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers N,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L. This makes sense because any element of L is an upper bound for the empty set. If there were an element x in L which would not be an upper bound then we would have an element in x ∈ ∅ such that notx ≤ a. But there is no element in ∅. Similarly, every element of L is a lower bound of ∅.Thus the greatest lower bound of ∅ must be the largest element of L. A lattice is a partially ordered set where every two element subset has a least upper bound and a greatest lower bound. It is then easy to see that in a lattice every finite non-empty subset has a least upper bound and a largest lower bound. However, there might be no smallest or largest element. Lattices which have a smallest and largest element are called bounded. In a complete lattice every subset has a least upper bound and a largest lower bound. In analysis, a least upper bound is also called supremum and a largest lower bound infimum. In a lattice, we define the least upper bound of two elements a and b as their join ∨ and their largest lower bound as their meet ∧.Thus, meet and join are binary operations. Meet and join operations are commutative and associative. By the very definition we