Show that the elements of the lattice (N,

Question

Show that the elements of the lattice (N, <=), where N is the set of positive integers and a <= b if and only if a divides b, satisfy the distributive property.

Accepted Answer

gcd(a, lcm(b,c)) = lcm(gcd(a,b), gcd(a,c)) for all a, b, c, so (N, |) is a distributive lattice. In the divisibility order, the meet of two numbers is their gcd and the join is their lcm. The trick is to look at each prime separately. Write the exponent of a prime p in a, b, c as x, y, z. Then gcd takes the smaller exponent (min) and lcm takes the larger (max). Distributivity of the lattice becomes the elementary fact min(x, max(y, z)) = max(min(x, y), min(x, z)), which is true for all real numbers. Since it holds prime by prime, it holds for the whole numbers, so the lattice is distributive.