## 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:
## Duality:The dual of any statement in a lattice (L,∧ ,∨ ) is defined to be a statement that is obtained by interchanging ∧ an ∨.
## Bounded Lattices:A lattice L is called a bounded lattice if it has greatest element 1 and a least element 0.
- 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).
- 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: - a ∨ 1 = 1
- a ∧1= a
- a ∨0=a
- a ∧0=0
L = {a Thus, the greatest element of Lattices L is a Also, the least element of lattice L is a Since, the greatest and least elements exist for every finite lattice. Hence, L is bounded. ## Sub-Lattices:Consider a non-empty subset L
Determine all the sub-lattices of D
1. {1, 2, 6, 30} 2. {1, 2, 3, 30} ## Isomorphic Lattices:Two lattices L
## Distributive Lattice:A lattice L is called distributive lattice if for any elements a, b and c of L,it satisfies following distributive properties: - a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
- 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:- 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). - 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.
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 (L (a
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