Directed subposet of a poset containing the minimal elements The following appears naturally in a certain context:
Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, let $E_0:=M$. Then, if $|E_i|\leq 1$, set $E_{i+1}=\emptyset$. Otherwise, for each incomparable pair $x\neq y$ in $E_i$, consider the minimal elements $z$ with $x<z>y$ and put them into the set $E_{i+1}$. This defines $E_{i+1}$ out of $E_i$. Finally, set $E=E_0\cup E_1\cup E_2\cup ...$.
Questions: Is there a more conceptual definition of the subposet $E$? Does it have a universal property making somehow clear why it is defined like above? Is it a well-known construction in the theory of posets? Does it have a name?
 A: In order to have any hope of getting a universal property you have to think in terms of order-preserving functions (ie categorically) and not picking elements one at a time (graph theoretically).
The subject that would have results like this is called domain theory. It assumes that the poset already has (honest) joins of directed subsets.
The most likely kind of universal property is that $E$ is the image of the least co-closure.
A co-closure is an order-preserving function $f:X\to X$ such that $f(x)=f(f(x))\leq x$ for each $x\in X$.
In the example where you construction arose, can you construct such a function?
If so, that's your universal property.   If not, you're at sea.
Edit in response to Werner's: Look up SFP or bifinite domains, on whom the main authors are Gordon Plotkin, Mike Smyth and Achim Jung (though the word bifinite was mine).   However, you still haven't told us where the question came from.
A: After editing the above construction a little bit, the following holds: $E$ is the smallest subposet of $P$ which contains $M$ and satisfies the following property: If $x\in P\setminus E$ then there exists a greatest element $y\in E$ with the property $y<x$.
