## Friday, May 24, 2013

### Product orders

Let $\mathbb{N}_k$ represent an elementary total order with $k$ elements. Every divisibility relation is the product of total orders. The divisibility lattice of 24 is $\mathbb{N}_2 \times \mathbb{N}_4$.

The prime signature of a number describes the divisibility lattice of a number up to isomorphism by describing it as a product of total orders. The number of divisors of a number is the size of this ordering relation and the number of prime factors of a number counting repetition turns it into a graded lattice.