Division lattice

The division lattice is an infinite complete bounded distributive lattice whose elements are the natural numbers ordered by divisibility. Its least element is 1, which divides all natural numbers, while its greatest element is 0, which is divisible by all natural numbers. The meet operation is greatest common divisor while the join operation is least common multiple.

The non-negative integers partially ordered by divisibility.

The prime numbers are precisely the atoms of the division lattice, namely those natural numbers divisible only by themselves and 1.

For any square-free number n, its divisors form a Boolean algebra that is a sublattice of the division lattice. The elements of this sublattice are representable as the subsets of the set of prime factors of n. The converse also holds, namely that every sublattice of the division lattice that forms a Boolean algebra is isomorphic to the lattice of divisors of a square-free number.

References

• Davey, B. A.; Priestley, H. A. (2002), Introduction to Lattices and Order, Cambridge University Press, ISBN 978-0-521-78451-1