Properties Of Lattices

The proof of (ii) is analogous to the proof of part (i).
L4 : (i) Since a ∧ b ≤ the definition of least upper bound of a ∧ b and a. Therefore, by the definition of least upper bound
a ∧ (a ∧ b) ≤ a................. (8)
On the other hand, by the definition of LUB, we have
a ≤ a ∨ (a ∧ b)..................... (9)
The expression (8) and (9) yields
a ∨ (a ∧ b) = a.
(ii) Since a ≤ a ∨ b and a ≤ a, it follows that a is a lower bound of a ∨ b and a.
Therefore, by the definition of GLB,
a ≤ a ∧ (a ∨ b).................. (10)
Also, by the definition of GLB, we have
a ∧ (a ∨ b) ≤ a........................ (11)
Then (10) and (11) imply
a ∧ (a ∨ b) = a
and the proof is completed.

In view of L3, we can write a ∨ (b ∨ c) and (a ∨ b) ∨ c as a ∨ b ∨ c.
Thus, we can express
LUB ({a1, a2,….an) as a1 ∨ a2 ∨……∨ an
GLB ({a1, a2,….an) as a1 ∧ a2 ∧……∧ an

Remark: Using commutativity and absorption property, part (ii) of previous Theorem can be proved as follows :

Let a ∧ b = a. We note that
b ∨ (a ∧ b ) = b ∨ a = a ∨ b (Commutativity)
But
b ∨ ( a ∧ b) = b (Absorption property)    ,  Hence
a∨ b = b   and so by part (i), a ≤ b. Hence a ∧ b = a if and only if a ≤ b.

Theorem: Let (L, ≤) be a lattice. Then for any a, b, c ∈ L, the following properties hold :
1. (Isotonicity) : If a ≤ b, then
(i) a ∨ c ≤ b ∨ c
(ii) a ∧ c ≤ b ∧ c

This property is called “Isotonicity”.
2. a ≤ c and b ≤ c if and only if a ∨ b ≤ c
3. c ≤ a and c ≤ b if and only if c ≤ a Ù b
4. If a ≤ b and c ≤ d, then
(i) a ∨ c ≤ b ∨ d
(ii) a ∧ c ≤ b ∧ d.
Proof : 1 (i). We know that a ∨ b = b if and only if a ≤ b.
Therefore, to show that a ∨ c ≤ b ∨ c, we shall show that
(a ∨ c) ∨ (b ∨ c) = b ∨ c.
We note that
(a ∨ c) ∨ (b ∨ c) = [(a ∨ c) ∨ b] ∨ c

= a ∨ (c ∨ b) ∨ c
= a ∨ (b ∨ c) ∨ c
= (a ∨ b) ∨ (b ∨ c)
= b ∨ c (Qa ∨ b = b and c ∨ c = c)
The part 1 (ii) can be proved similarly.

2. If a ≤ c, then 1(i) implies a ∨ b ≤ c ∨ b    But
b ≤ c <-> b ∨ c = c <-> c ∨ b = c (commutativity)
Hence a ≤ c and b ≤ c if and only if a ∨ b ≤ c

3. If c ≤ a, then 1(ii) implies
c ∧ b ≤ a ∧ b    But
c ≤ b ∧ c ∧ b = c
Hence c ≤ a and c ≤ b if and only if c ≤ a Ù b.
4 (i) We note that 1(i) implies that
if a ≤ b, then a ∨ c ≤ b ∨ c = c ∨ b
if c ≤ d, then c ∨ b ≤ d ∨ b = b ∨ d
Hence, by transitivity
a ∨ c ≤ b ∨ d
(ii) We note that 1(ii) implies that
if a ≤ b, then a ∧ c ≤ b ∧ c = c ∧ b             
if c ≤ d, then c ∧ b ≤ d ∧ b = b ∧ d.
Therefore transitivity implies
a ∧ c ≤ b ∧ d.