Given a (finite) poset $(P,\leq)$, we can construct the poset $(\mathcal{D}, \subseteq)$ isomorphic to $P$ simply by letting $\mathcal{D}$ be the collection of all "descendant sets" i.e. $\mathtt{D}(x)=\{y\in P: y\leq x\}$ for each $x\in P$. This is such a straightforward observation that I'm assuming it has been mentioned somewhere – but where? A more interesting question is: are there known connections between the existence of joins in $P$ and properties of $\mathcal{D}$?
To give a little bit more motivation, I've been working with directed acyclic graphs – DAGs – which naturally pop up in connection to (mathematical formalizations of) the study of evolutionary histories of, say, different species. For historical reasons, this history is represented with graphs (the DAGs considered originally were just rooted trees). However, every DAG $G=(V,E)$ can be associated to the poset $P_G=(V,\preceq_G)$, where $u\preceq_G v$ if and only if there is a directed path for $v$ to $u$ in $G$. In principle, no interesting information is lost in this conversion between $G$ and $P_G$ –– the only thing of $G$ that is not recoverable from $P_G$ are so-called shortcut edges $(u,v)$, for which there exists a directed path from $u$ to $v$ that avoids the edge $(u,v)$. In particular, since I often use least common ancestors of the DAGs, the corrsponding posets often have some or all joins well-defined. They are not always join-semilattices (although sometimes). Still, I have a nagging feeling that since I don't have a solid background in poset theory, I'm missing some obvious references and connections that could, in a general sense, be helpful in this line of research.
