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Complements| Lattice

Complements: Let x, y be any elements of a lattice L. If x ˄ y =0 and x ˅ y = 1 then we say y is a complements of x.

Or, Let [a, b] be an interval in a lattice L. Let x ∊ [a, b] be any element. If there exist y ∊ L such that

                                    x ˄ y = a, x ˅ y = b

we say y is a complement of x relative to [a, b], or y is relative complement of x in [a, b].

Theorem: Two bounded lattices A and B are complemented if and only if A × B is complemented.

Proof: Let A and B be complemented and suppose 0, u and 0ˊ, uˊ are the universal bounds of A and B respectively.

Then (0, 0ˊ) and (u, uˊ) will be least and greatest elements of A × B.

Let (a, b) ∊ A × B be any element.

Then a ∊ A, b ∊ B and as A, B are complemented, there exists aˊ ∊ A, bˊ∊ B such that a ˄ aˊ = 0, a ˅ aˊ = u, b ˄ bˊ =0ˊ, b ˅ bˊ = uˊ.

Now

                  (a, b) ˄ (aˊ, bˊ) = (a ˄ aˊ, b ˄ bˊ) = (0, 0ˊ)

                  (a, b) ˅ (aˊ, bˊ) = (a ˅ aˊ, b ˅ bˊ) = (u, uˊ)

Shows that (aˊ, bˊ) is complement of (a, b) in A × B.

Hence A × B is complemented.

Conversely, let A × B be complemented.

Let a ∊ A, b ∊ B be any elements.

Then (a, b) ∊ A × B and thus has a complement, say (aˊ, bˊ)

Then (a, b) ˄ (aˊ, bˊ) =(0, 0ˊ), (a, b) ˅ (aˊ, bˊ) =(u, uˊ)

⟹ (a ˄ aˊ, b ˄ bˊ) = (0, 0ˊ), (a ˅ aˊ, b ˅ bˊ) = (u, uˊ)

⟹ a ˄ aˊ = 0,    a ˄ aˊ = u

      a ˅ aˊ = 0ˊ,    a ˅ aˊ = uˊ

i.e., aˊ and bˊ are complements of a and b respectively. Hence A and B are complemented.

Or, Prove that two bounded lattice A and B are complemented if and only if A × B is complemented.

Proof: First suppose A and B are complemented, we have to show that  A × B is complemented.

Suppose least elements of A and B are 0 and 0ˊ, greatest elements of A and B are 1 and 1ˊ respectively.

Then least element of A × B is (0, 0ˊ) and greatest element of A × B is (1, 1ˊ).

Now, (a, b) ∊ A × B

   ⟹ a ∊ A and b ∊ B.

Since A and B are complemented, so there exists aˊ ∊ A and bˊ ∊ B such that

a ˄ aˊ = 0, a ˅ aˊ = 1, b ˄ bˊ =0ˊ, b ˅ bˊ = 1ˊ    ——————- (i)

Clearly, (aˊ, bˊ) ∊ A × B

Now,

(a, b) ˄ (aˊ, bˊ) = (a ˄ aˊ, b ˄ bˊ) = (0, 0ˊ)  [ from (i)]

And   (a, b) ˅ (aˊ, bˊ) = (a ˅ aˊ, b ˅ bˊ) = (1, 1ˊ)    [from (i)]

This shows that, (aˊ, bˊ) is complement of (a, b) in A × B.

Hence A × B is complemented.

Conversely, suppose that, A × B is complemented, we have to show that A and B are complemented.

Let ? ∊ A and ? ∊ B then (?, ?) ∊ A × B and thus has a complement.

Say (?ˊ, ?ˊ) then

(?, ?) ˄ (?ˊ, ?ˊ) = (0, 0ˊ),  (?, ?) ˅ (?ˊ, ?ˊ) = (1, 1ˊ)

⟹ (? ˄ ?ˊ, ? ˄ ?ˊ) = (0, 0ˊ),   (? ˅ ?ˊ, ? ˅ ?ˊ) = (1, 1ˊ)

⟹ ? ˄ ?ˊ = 0 and ? ˅ ?ˊ = 1

    ? ˄ ?ˊ = 0ˊ and ? ˅ ?ˊ = 1ˊ

This implies ?ˊ and ?ˊ are complements of ? and ? in A and B. Hence A and B are complemented.

 

 

 

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