The relationship between the discrete order and the multiplication on the natural numbers leads to, among other things, the study of gaps between primes.  I would nominate a class of structures S(n), which are the sets of integers relatively prime to the nth primorial (p_1p_2...p_n) as a collection worthy of the labels beautiful and intricate.  The symmetry and self-similar nature appeal to many, and while the construction is simple, there are many simple facts remaining to be established about the S(n).  For one, the largest gap between consecutive members of S(n) seems to be unknown.  (Cf https://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update for a weak upper bound; I hope to post an improvement soon.)

Gerhard "Ask Me About System Design" Paseman, 2010.12.13