Discrete Mathematical Structures by Pearson

8-20 Discrete Mathematical Structures

Proof:

(1) From the definition of join operation

b ∨ c = sup{b, c} and hence b ≤ b ∨ c.

Hence,

a ≤ b and b ≤ b ∨ c ⇒ a ≤ b ∨ c.

(2) Let a ≤ b and a ≤ c, then a is a lower bound of {b, c}.

Thus,

a ≤ b ∧ c

8.15 DUALITY

The dual lattice of any lattice (L, ∧, ∨) is obtained by interchanging the operation ∧ and ∨ and the

relation ≤ by ≥. Thus, lattices (L, ≤ ) and (L, ≥) are called duals of each other.

For example, the set of natural numbers N in the partial order relation less than or equal to,

that is, (N, ≤ ), is the dual lattice of the lattice (N, ≥).

8.16 COMPLETE LATTICE

A lattice L is said to be complete if every ...