Lattices:

Let L be a non-empty set closed under two binary operations called meet and join, denoted by ∧ and ∨. Then L is called a lattice if the following axioms hold where a, b, c are elements in L:

1) Commutative Law: -
(a) a ∧ b = b ∧ a           (b) a ∨ b = b ∨ a

2) Associative Law:-
(a) (a ∧ b)∧ c = a ∧(b∧ c)           (b) (a ∨ b) ∨ c = a ∨ (b ∨ c)

3) Absorption Law: -
(a) a ∧ ( a ∨ b) = a           (b) a ∨ ( a ∧ b) = a

Duality:

The dual of any statement in a lattice (L,∧ ,∨ ) is defined to be a statement that is obtained by interchanging ∧ an ∨.

For example, the dual of a ∧ (b ∨ a) = a ∨ a is           a ∨ (b ∧ a )= a ∧ a

Bounded Lattices:

A lattice L is called a bounded lattice if it has greatest element 1 and a least element 0.

Example:

1. The power set P(S) of the set S under the operations of intersection and union is a bounded lattice since ∅ is the least element of P(S) and the set S is the greatest element of P(S).
2. The set of +ve integer I₊ under the usual order of ≤ is not a bounded lattice since it has a least element 1 but the greatest element does not exist.

Properties of Bounded Lattices:

If L is a bounded lattice, then for any element a ∈ L, we have the following identities:

1. a ∨ 1 = 1
2. a ∧1= a
3. a ∨0=a
4. a ∧0=0

Theorem: Prove that every finite lattice L = {a₁,a₂,a₃....a_n} is bounded.

Proof: We have given the finite lattice:

          L = {a₁,a₂,a₃....a_n}

Thus, the greatest element of Lattices L is a₁∨ a₂∨ a_{3∨....∨an.}

Also, the least element of lattice L is a₁∧ a₂∧a₃∧....∧a_n.

Since, the greatest and least elements exist for every finite lattice. Hence, L is bounded.

Sub-Lattices:

Consider a non-empty subset L₁ of a lattice L. Then L₁ is called a sub-lattice of L if L₁ itself is a lattice i.e., the operation of L i.e., a ∨ b ∈ L₁ and a ∧ b ∈ L₁ whenever a ∈ L₁ and b ∈ L₁.

Example: Consider the lattice of all +ve integers I₊ under the operation of divisibility. The lattice D_n of all divisors of n > 1 is a sub-lattice of I₊.

Determine all the sub-lattices of D₃₀ that contain at least four elements, D₃₀={1,2,3,5,6,10,15,30}.

Solution: The sub-lattices of D₃₀ that contain at least four elements are as follows:

1. {1, 2, 6, 30}          2. {1, 2, 3, 30}
3. {1, 5, 15, 30}          4. {1, 3, 6, 30}
5. {1, 5, 10, 30}          6. {1, 3, 15, 30}
7. {2, 6, 10, 30}

Isomorphic Lattices:

Two lattices L₁ and L₂ are called isomorphic lattices if there is a bijection from L₁ to L₂ i.e., f: L₁⟶ L₂, such that f (a ∧ b) =f(a)∧ f(b) and f (a ∨ b) = f (a) ∨ f (b)

Example: Determine whether the lattices shown in fig are isomorphic.

Solution: The lattices shown in fig are isomorphic. Consider the mapping f = {(a, 1), (b, 2), (c, 3), (d, 4)}.For example f (b ∧ c) = f (a) = 1. Also, we have f (b) ∧ f(c) = 2 ∧ 3 = 1

Distributive Lattice:

A lattice L is called distributive lattice if for any elements a, b and c of L,it satisfies following distributive properties:

1. a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
2. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

If the lattice L does not satisfies the above properties, it is called a non-distributive lattice.

Example:

1. The power set P (S) of the set S under the operation of intersection and union is a distributive function. Since,
                       a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)
             and, also a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪c) for any sets a, b and c of P(S).
2. The lattice shown in fig II is a distributive. Since, it satisfies the distributive properties for all ordered triples which are taken from 1, 2, 3, and 4.

Complements and complemented lattices:

Let L be a bounded lattice with lower bound o and upper bound I. Let a be an element if L. An element x in L is called a complement of a if a ∨ x = I and a ∧ x = 0

A lattice L is said to be complemented if L is bounded and every element in L has a complement.

Example: Determine the complement of a and c in fig:

Solution: The complement of a is d. Since, a ∨ d = 1 and a ∧ d = 0

The complement of c does not exist. Since, there does not exist any element c such that c ∨ c'=1 and c ∧ c'= 0.

Modular Lattice:

A lattice (L, ∧,∨) is called a modular lattice if a ∨ (b ∧ c) = (a ∨ b) ∧ c whenever a ≤ c.

Direct Product of Lattices:

Let (L₁ ∨₁ ∧₁)and (L₂ ∨₂ ∧₂) be two lattices. Then (L, ∧,∨) is the direct product of lattices, where L = L₁ x L₂ in which the binary operation ∨(join) and ∧(meet) on L are such that for any (a₁,b₁)and (a₂,b₂) in L.

              (a₁,b₁)∨( a₂,b₂ )=(a₁ ∨₁ a₂,b₁ ∨₂ b₂)
and       (a₁,b₁) ∧ ( a₂,b₂ )=(a₁ ∧₁ a₂,b₁ ∧₂ b₂).

Example: Consider a lattice (L, ≤) as shown in fig. where L = {1, 2}. Determine the lattices (L², ≤), where L²=L x L.

Solution: The lattice (L², ≤) is shown in fig: