Elements of distributive lattices

Every element in a distributive lattice such as the total ordering on the natural numbers can be represented as a lower set of join irreducible elements. This approach allows us to represent all natural numbers as sets:

#{}
#{#{}}
#{#{} #{#{}}}
#{#{} #{#{}} #{#{} #{#{}}}}

This also can be used to represent all natural numbers as sets of prime powers with respect to the divisibility relation:

#{}
#{2}
#{3}
#{2 4}
#{5}
#{2 3 6}
#{7}
#{2 4 8}

We can also use this to effectively represent abstract simplical complexes as sets of lower sets of the set of lower sets of an antichain.