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.