Birkhoff's Representation Theorem - The Partial Order of Join-irreducibles

The Partial Order of Join-irreducibles

In a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements. Equivalently, x is join-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any two smaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 is join-irreducible. An element x is join-prime if, whenever x ≤ y ∨ z, either x ≤ y or x ≤ z. In the same lattice, 4 is join-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4.

In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible is not join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y ∨ z. But then x ≤ y ∨ z, and x is not less than or equal to either y or z, showing that it is not join-prime.

There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but in a distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x ≤ y ∨ z. This inequality is equivalent to the statement that x = x ∧ (y ∨ z), and by the distributive law x = (x ∧ y) ∨ (x ∧ z). But since x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x ∧ y (equivalently x ≤ y) or x = x ∧ z (equivalently x ≤ z).

The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhoff's theorem states that the lattice itself can be recovered from the lower sets of this partial order.