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Distributive lattices

Definition: A lattice L is called a distributive lattice if

a ˄ (b ˅ c) = (a ˄ b) ˅ (a ˄ c)    ∀ a, b, c ∊ L.

Example: A chain is distributive lattice.

Solution: Let a, b, c be any three members of a chain, then any two of these are comparable.

Suppose a ≤ b, a ≥ c, b ≤ c.

then a ≤ b ≤ c ≤ a ⟹ a = b = c.

Thus a ˄ (b ˅ c) = a= (a ˄ b) ˅ (a ˄ c)

If a ≤ b, a ≥ c, b ≤ c

then c ≤ a, a ≤ b, c ≤ b

thus a ˄ (b ˅ c) = a ˄ b = a

So, (a ˄ b) ˅ (a ˄ c) = a ˅ c = a

One can check that under different cases (a ≤ b, a ≤ c; a ≤ c, a ≥ b; a ≥ b, a ≥ c) the condition of distributivity holds and thus a chain is always a distributive lattice.  (Proved)

Theorem: A lattice L is distributive if and only if

a ˅ (b ˄ c) = (a ˅ b) ˄ (a ˅ c)     ∀ a, b, c ∊ L.

Proof: Let L be distributive.

Now (a ˅ b) ˄ (a ˅ c) = [(a ˅ b) ˄a] ˅ [(a ˅ b) ˄ c]

= a ˅ [(a ˅ b) ˄ c]       [by absorption property]

= a ˅ [(a ˄ c) ˅ (b ˄ c)]

= [a ˅ (a ˄ c)] ˅ (b ˄ c)

= a ˅ (b ˄ c).

Conversely, let a, b, c ∊ L be any three elements, then

(a ˄ b) ˅ (a ˄ c) = [(a ˄ b) ˅ a] ˄ [(a ˄ b) ˅ c]

= a ˄ [(a ˄ b) ˅ c]

= a ˄ [(c ˅ a) ˄ (c ˅ b)]

= [a ˄ (a ˅ c)] ˄ (b ˅ c)

= a ˄ (c ˅ b)

= a ˄ (b ˅ c).

i.e., L is distributive. (Proved)

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