Let $A$ be an associative algebra and consider the poset $I$ of idempotents in $A$, where as usual if $e,f \in A$ we say $e \leq f$ provided that $e = fef$. Clearly $0 \in I$ is minimal, but in general I can't say anything else about this poset.
This leads me to ask: are there any other general properties of $I$ without more assumptions on $A$? In particular, if I am given some poset with a minimal element, is it possible to construct an algebra with the prescribed poset of idempotents?
Edit: I should have said I'm working over a field (or at least a ring with no nontrivial idempotents), since otherwise idempotents in the base ring complicate things. Also: bonus points for producing examples where $A = \cup_{e \in I} eAe$.