Definition 6. A lattice L with0 and 1 is said to be complemented if for every a EL there exist an a' such that aUa=l and ana=0 Here, a' is also called a complement of a. Sometimes, we can relax the restrictions by defining complement of b relative to a as bUb=a, bnb=0 if b, 1 < a Example 2.< P(S), S> is complemented for any nonempty set s It is easy to verify that LU B(Al, A2)= Al U A2 and GLB(A1, A2)= A1n A2 where U and n is the operation union and intersection respectively And we know n and U on set is complemented Example 3. Given a poset 10, a, b, c, 11, R>described in Figure 2. Figure 2: Complemented Lattice We can see that a has two complements b and c n and U on set is distributive. Similarly, we have Definition 7. A lattice L is distributive if for any a, b,cEL such that 1.a∩(bUc)=(a∩b)U(anc) 2. aU(bnc=(abn(auc If a lattice is not distributive. we call it non-distributive Example 4. <P(S), C> is distributive for any nonempty set S Based on Example 2, we have that n and U on set is distributive. 4 Boolean algebra Historically, Boolean algebras were the first lattices to be studied. They were introduced by Boole in order to formalize the calculus of propositions. Actually, the theory of Boolean algebra is equivalent to the theory of a special class of ringsDefinition 6. A lattice L with 0 and 1 is said to be complemented if for every a ∈ L there exists an a ′ such that a ∪ a ′ = 1 and a ∩ a ′ = 0. Here, a ′ is also called a complement of a. Sometimes, we can relax the restrictions by defining complement of b relative to a as b ∪ b1 = a, b ∩ b1 = 0 if b, b1 ≤ a. Example 2. < P(S), ⊆> is complemented for any nonempty set S. It is easy to verify that LUB(A1, A2) = A1 ∪ A2 and GLB(A1, A2) = A1 ∩ A2 where ∪ and ∩ is the operation union and intersection respectively. And we know ∩ and ∪ on set is complemented. Example 3. Given a poset < {0, a, b, c, 1}, R > described in Figure 2. a b c 1 0 Figure 2: Complemented Lattice. We can see that a has two complements b and c. ∩ and ∪ on set is distributive. Similarly, we have Definition 7. A lattice L is distributive if for any a, b, c ∈ L such that: 1. a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c). 2. a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c). If a lattice is not distributive, we call it non-distributive. Example 4. < P(S), ⊆> is distributive for any nonempty set S. Based on Example 2, we have that ∩ and ∪ on set is distributive. 4 Boolean Algebra Historically, Boolean algebras were the first lattices to be studied. They were introduced by Boole in order to formalize the calculus of propositions. Actually, the theory of Boolean algebra is equivalent to the theory of a special class of rings. 3