Discrete Mathematical Structures by Pearson

Ordered Sets and Lattices 8-21

Define f: L

1

→L

2

by

f (1) = f, f (2) = {a}, f (3) = {b}, f (6) = {a, b}

The mapping f is one-one and onto.

Also, for all a, b ∈ L

1

, f (a ∨ b) = f (a) ∨ f ( b) and f (a ∧ b) = f (a) ∧ f ( b).

THEOREM 8.6 The dual of lattice is a lattice.

Proof: Let (L, R) be a given lattice and let (L, R

−1

) be its dual where R is defined by

a R

−1

b iff b R a

Let a, b ∈ L, then LUB{a, b} exists as R is a lattice.

Then,

a ∨ b = LUB{a, b} ∈ L

Now,

a R (a ∨ b) and b R (a ∨ b)

  ⇒ (a ∨ b) R

−1

a and (a ∨ b) R

−1

b

  ⇒ a ∨ b is lower bound in (L, R

−1

)

To show that a ∨ b is GLB of {a, b} in (L, R

−1

).

Let c be any lower bound of (a, b) in (L, R

−1

).

Then, ...