Discrete Mathematical Structures by Pearson

8-26 Discrete Mathematical Structures

Case II: a ≥ b and a ≥ c

Then,

(a ∧ b) ∨ (a ∧ c) = b ∨ c

Hence,

a ∧ ( b ∨ c) = (a ∧ b) ∨ (a ∨ c)

8.21 MODULAR LATTICE

A lattice (L, ∧, ∨) is said to be modular lattice if

a ∨ ( b ∧ c) = (a ∨ b) ∧ c

whenever a ≤ c.

THEOREM 8.11 In a modular lattice (L, ∧, ∨), for any a, b, c ∈ L

a ≤ c ⇒ a ∨ ( b ∧ c) = (a ∨ b) ∧ c.

Proof: Let a ≤ c, then a ∨ c = c

Also, if L is distributive, then

a ∨ ( b ∧ c) = (a ∨ b) ∧ (a ∨ c)

= (a ∨ b) ∧ c (\a ∨ c = c)

Note 8.7

1. Every distributive lattice is modular.

2. Every modular lattice is not distributive.

Example 1 Show that the lattice in the Fig. 8.25 is a non-distributive modular lattice. ...