# Lattices As Algebraic System

**Lattices as Algebraic System**

**Definition.** A Lattice is an algebraic system (L, ∨ , ∧ ) with two binary operations ∨ and ∧ , called join and meet respectively, on a non-empty set L which satisfy the following axioms for a, b, c ∈ L :

**1. Commutative Law :**

a ∨ b = b ∨ a and a ∧ b = b ∧ a .

**2. Associative Law :**

(a ∨ b) ∨ c = a ∨ (b ∨ c)

and

(a ∧ b)∧ c = a ∧ (b ∧ c)

**3. Absorption Law :**

(i) a ∨ (a ∧ b) = a

(ii) a ∧ (a ∨ b) = a

We note that Idempotent Law follows from axiom 3 above. In fact,

a ∨ a = a ∨ [a ∧ (a ∨ b)] using 3(ii) = a using 3(i)

The proof of a ∧ a = a follows by principle of duality.