Prove that a Distributive Lattice is Modular

In this page we will prove that a distributive lattice is modular.

Distributive Lattice: A lattice (L, ≤) is called distributive if the distributive property holds, i.e, a∨(b∧c) = (a∨b)∧(a∨c) a∧(b∨c) = (a∧b)∨(a∧c)

Modular Lattice: A lattice (L, ≤) is called modular if for all a, b, c ∈ L with a≤c we have (a∨b)∧c = a∨(b∧c).

Theorem: Show that a distributive lattice is modular.

Proof:

Suppose that (L, ≤) is a distributive lattice.

To show it is modular, let us consider a, b, c ∈ L with a≤c.

Now, using distributive property we have that

(a∨b)∧c = (a∧c)∨(b∧c)

⇒ (a∨b)∧c = a∨(b∧c). This is because, by assumption a≤c implies that a∧c=a.

This shows that (L, ≤) is modular.

Hence, it proves that each distributive lattice is modular.

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