Every totally ordered set
is a distributive lattice with max as join and min as meet.
We will show:
We may suppose (If not, and we may switch and .)
Recall that is equivalent to . Hence implies , i.e., , so the right hand side of the equation is equal to . On the left hand side we have , so equality is established.
Note that the relation
is true in all lattices, as both and are bounded above by .
Let L be a lattice, and let x be an element of L. If x is meet-prime, then it is meet-irreducible.
Suppose x = a v b. Then x ≤ a v b. x being meet-prime, x ≤ a or x ≤ b. Without loss of generality suppose x ≤ a. Then a v b ≤ a. By definition of v, a v b ≥ a. Therefore a v b = a and x = a.
Let L be a distributive lattice, and let x be an element of L. If x is join-irreducible, then it is join-prime.
Recall that in a lattice x ≤ y ⇔ x ^ y = x.
Suppose x ≤ a v b. This is equivalent to x ^ (a v b) = x which by distributivity is in turn equivalent to (x ^ a) v (x ^ b) = x. x being meet-irreducible, x = x ^ a or x = x ^ b. This is equivalent to x ≤ a or ≤ b.