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?