Recently I have been asking a couple of questions about complete posets [1], [2]. To me, this is interesting as a decategorification of the notion of category as indicated in this answer on MO. In this corresponcence the study of powersets correspond to the study of presheaves, that play a key role in category theory. On the other hand these question proved to be slightly interesting also from people that come from different fields.
The question of today is the following.
Is there a representation theorem of an atomi(sti)c preorder with sups?
Recall that I am using the following notation. This notation has proven to be a bit misleading in this question.
Def. In a preorder with sups $\mathbb{P}$ an atom (1) is an element $p$ such that $$\text{if } p \leq \bigvee_{i \in I} a_i \text{ then } p \leq a_j \text{ for some } j \in I. $$
Def. A subset $A$ of a preorder with sups $\mathbb{P}$ is (join-)dense if for each element $p$ there is a family of $(a_i)$ in A such that $$p = \bigvee a_i. $$
Def. A complete poset is atomic if the set of its atoms is dense.
Choosing, as a study case, partial orders, one can prove that:
Powersets are precisely complete atomic posets in my sense.
