Given a set $S$, I am interested in the set $U$, constructed in the following way:
A set $u$ is a member of $U$ if and only if:
- The elements of $u$ are subsets of $S$;
- No element of $u$ is a subset of another element of $u$;
- Every element of $S$ is a member of at least one element of $u$.
This is similar to the set of partitions of $S$ but is not the same. We obtain the set of partitions if we strengthen condition 3 to say that every element of $S$ is in exactly one element of $u$, rather than at least one.
As Erick Wong helpfully points out, each member of $U$ forms a Sperner family.
For example, if $S=\{A,B,C\}$ then $$ \begin{align} U = \{&\\ &\{\{A\},\{B\},\{C\}\},\\ &\{\{A,B\},\{C\}\},\\ &\{\{B,C\},\{A\}\},\\ &\{\{A,C\},\{B\}\},\\ &\{\{A,B\},\{A,C\}\},\\ &\{\{A,B\},\{B,C\}\},\\ &\{\{A,C\},\{B,C\}\},\\ &\{\{A,B,C\}\}\\ \}. \end{align} $$
My question is simply whether this construction has a name, and whether there are any areas of mathematics where it arises or plays an important role. If it appears without condition 3 I'm interested in that as well.
More specifically, this set can be given a lattice structure. For $u,v\in U$ we say that $u\subseteq v$ if every member of $u$ is either a member of $v$ or a subset of a member of $v$. What I really want to understand is the structure of this lattice, and I'm interested in anything that has been written or can be said about that.