On multiplicative lower sets

The set of all orders of elements of the symmetric group S_n forms a multiplicative lower set:

(#{0} 
 #{1} 
 #{2} 
 #{2 3} 
 #{3 4} 
 #{4 5 6} 
 #{4 5 6} 
 #{7 10 12} 
 #{7 8 10 12 15} 
 #{8 9 12 14 15 20})

The set of lower sets over the divisibility relation forms a distributive lattice with its own union and intersection operations.