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?



Directed subposet of a poset containing the minimal elements
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 $y<x$.


