Lattices:Let L be a nonempty 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:  2) Associative Law: 3) Absorption Law:  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:
Properties of Bounded Lattices:If L is a bounded lattice, then for any element a ∈ L, we have the following identities:
Theorem: Prove that every finite lattice L = {a_{1},a_{2},a_{3}....a_{n}} is bounded. Proof: We have given the finite lattice: L = {a_{1},a_{2},a_{3}....a_{n}} Thus, the greatest element of Lattices L is a_{1}∨ a_{2}∨ a_{3∨....∨an.} Also, the least element of lattice L is a_{1}∧ a_{2}∧a_{3}∧....∧a_{n}. Since, the greatest and least elements exist for every finite lattice. Hence, L is bounded. SubLattices:Consider a nonempty subset L_{1} of a lattice L. Then L_{1} is called a sublattice of L if L_{1} itself is a lattice i.e., the operation of L i.e., a ∨ b ∈ L_{1} and a ∧ b ∈ L_{1} whenever a ∈ L_{1} and b ∈ L_{1}. 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 sublattice of I_{+}. Determine all the sublattices of D_{30} that contain at least four elements, D_{30}={1,2,3,5,6,10,15,30}. Solution: The sublattices of D_{30} that contain at least four elements are as follows: 1. {1, 2, 6, 30} 2. {1, 2, 3, 30} Isomorphic Lattices:Two lattices L_{1} and L_{2} are called isomorphic lattices if there is a bijection from L_{1} to L_{2} i.e., f: L_{1}⟶ L_{2}, 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:
If the lattice L does not satisfies the above properties, it is called a nondistributive lattice. Example:
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 (L_{1} ∨_{1} ∧_{1})and (L_{2} ∨_{2} ∧_{2}) be two lattices. Then (L, ∧,∨) is the direct product of lattices, where L = L_{1} x L_{2} in which the binary operation ∨(join) and ∧(meet) on L are such that for any (a_{1},b_{1})and (a_{2},b_{2}) in L. (a_{1},b_{1})∨( a_{2},b_{2} )=(a_{1} ∨_{1} a_{2},b_{1} ∨_{2} b_{2}) Example: Consider a lattice (L, ≤) as shown in fig. where L = {1, 2}. Determine the lattices (L^{2}, ≤), where L^{2}=L x L. Solution: The lattice (L^{2}, ≤) is shown in fig:
Next TopicBoolean Algebra
