**Or, Suppose L is a lattice, What do you mean by I(L), prove that (I(L), ⊆) is a lattice.**

**Proof**: Let I(L) is a set of all ideals of a lattice L.

Then clearly, L ∊ I(L)

∴ I(L) is non empty.

Now, let us first, show that (I(L), ⊆) is a poset, clearly ⊆ is reflexive in I(L) , because A ⊆ A, ∀ A.

Next, ∀ A, B ∊ I(L)

A ⊆ B and B ⊆ C ⟹ A = B

∴ ⊆ is antisymetric in I(L)

Last of all, ∀ A, B , C∊ I(L)

A ⊆ B and B ⊆ C ⟹ A ⊆ C

∴ ⊆ is transitive in I(L)

Hence I(L) is a poset.

**Now we have to show that I(L) is a lattice.**

∀ A, B ∊ I(L), let us consider A ˄ B = A ∩ B

We know, intersection of two ideals is always an ideal.

∴ A ˄ B I(L)

Let us consider ∀ A, B ∊ I(L)

X ={x ∊ L | x ≤ a ˅ b, a ∊ A, b ∊ B}

Clearly ∀ a ∊ A

a ≤ a ˅ b, ∀ b ∊ B

∴ a ∊ X [by definition]
∴ A ⊆ X and so X is non empty.

Similarly we can show, B ⊆ X

∴ A ∪ B ⊆ X

**Let us show that X is an ideal of L**

Let ∀ p, q ∊ X

p ∊ X ⟹ there exists a1 ∊ A, b1 ∊ B such that p ≤ a1 ˅ b1

q ∊ X ⟹ there exists a2 ∊ A, b2 ∊ B such that q ≤ a2 ˅ b2

Now, p ˅ q ≤ (a1 ˅ b1) ˅ (a2 ˅ b2)

= (a1 ˅ a2) ˅ (b1 ˅ b2)

Since a1, a2 ∊ A and A is an ideal, so

a1 ˅ a2 ∊ A, similarly (b1 ˅ b2)∊ B

∴ p ˅ q ∊ X

Now, ∀ p ∊ X and ∀ l ∊ L

p ∊ X ⟹ there exists a1 ∊ A and there exists b1 ∊ B such that a1 ˅ b1 ——–(i)

Again, clearly p ˄ l ≤ p ————–(ii) [by property of meet]
From (i) and (ii) we get,

p ˄ l ≤ a1 ˅ b1; for some a1 ∊ A, b1 ∊ B

∴ p ˄ l ∊ X

Hence X is an ideal of L

∴ X ∊ I(L)

If C is any ideal of L such that

A ⊆ C and B ⊆ C

Then ∀ a ∊ A and ∀ b ∊ B, we get,

a ∊ C and b ∊ C

⟹ X ⊆ C

Hence X is the smallest ideal of L containing A ∪ B

Therefore, X = sup {A, B} = A ˅ B

∴ A ˅ B ∊ I(L), ∀ A ∊ I(L) and ∀ B ∊ I(L)

Therefore, I(L) is a lattice which is called ideal lattice. **(Proved)**

Home / Lattices and Boolean Algebras / The set of all ideals of a lattice L forms a lattice under ⊆ relation.

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