**Dual lattice**: A non empty subset I of a lattice L is called dual ( or filter) of L if

(i) ∀ a , b ∊ I ⟹ a ˄ b ∊ I

(ii) ∀ a ∊ I , ∀ l ∊ L ⟹ a ˅ l ∊ I

*Dual ideal generated by a subset H of L is denoted by [H]*

**Principle ideal**: Let L be a lattice and a ∊ L be any element. The set (a] = {x ∊ L | x ≤ a} forms an ideal of L, is called principle ideal. *It is generated by a.*

**Principle dual ideal**: Let L be a lattice and a ∊ L be any element the set [a) = {x ∊ L | a ≤ x} forms a dual ideal of L is called the principle dual ideal. *It is generated by a.*

**Prime ideals**: An ideal A of a lattice L is called a prime ideal of L if A is properly contained in L and wherever a ˄ b ∊ A then a ∊ A or b ∊ A.

**Dual prime ideal**: A proper dual ideal I of a lattice is called a prime ideal if a ˅ b ∊ I ⟹ a ∊ I or b∊ I.

**Ideal lattice**: The set of all ideals of a lattice L is called ideal lattice of L. *It is denoted by I(L).*

# Theorem: A dual ideal is a convex sublattice.

Or, Let I and D be ideal and dual ideal of a lattice L. Then I ∩ D is a convex sublattice.

**Proof**: Let I and D be ideal and dual ideal of lattice L. So I and D are sublattice. Hence I ∩ D is a sublattice.

Now let a, b ∊ I ∩ D, where a ≤ b

Then a, b ∊ I and a, b ∊ D

⟹ [a, b] ⊆ I and [a, b] ⊆ D as I and D is convex sublattices.

Thus [a, b] ⊆ I ∩ D or that I ∩ D is convex sublattice of L.