Poset of idempotents 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 $A = \cup_{e \in I} eAe$.
 A: The poset of idempotents is equipped with a relative complement operation: $e \le f$ if and only if $ef = fe = e$, and then we compute that
$$(f - e)^2 = f - e$$
hence $f - e$ is also an idempotent. This operation, which I"ll write as $e \Rightarrow f$,  satisfies the following axioms:


*

*$e \Rightarrow f$ is order-preserving in the second variable and order-reversing in the first,

*$(0 \Rightarrow f) = f$,

*$((e \Rightarrow f) \Rightarrow f) = e$,

*$\inf(e, e \Rightarrow f) = 0$. 


In particular, for fixed $f$ it defines an order-reversing isomorphism from the interval $[0, f]$ to itself. Consequently, any poset with a minimal element $0$ and an element $e$ such that the interval $[0, e]$ is not isomorphic to its opposite cannot admit a relative complement and so cannot be the poset of idempotents of a ring. A poset with this property of minimal size is the poset $\{ 0 \lt a \lt b, c \lt 1 \}$. 
Another minimal counterexample is the total order $3 = \{ 0 \lt 1 \lt 2 \}$, which admits a unique map $e \Rightarrow f$ satisfying the first, second, and third axioms above, and this map does not satisfy the fourth axiom. 

Another thing you can say about the idempotents in a rng is that they are equipped with an interesting relation, namely commutativity. If $e, f$ are commuting idempotents, then $ef$ gives their inf and $e + f - ef$ gives their sup. More generally, any commuting subposet of the idempotents is a lattice.
