# 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.