A partially ordered set — or poset — $P = (A, \preceq)$ is a pair consisting of a set $A$ and a partial order $\preceq$ on $A$. This partial order is a binary relation on a set $A$ such that — for all $x, y, z \in A$ — it is:
- reflexive ($x \preceq x$),
- antisymmetric ($x \preceq y$ and $y \preceq x$ imply $x = y$),
- and transitive ($x \preceq y$ and $y \preceq z$ imply $x \preceq z$).
For two elements $x, y \in P$, we say that $y$ covers $x$ if $x \prec y$ and there is no element $z \in P$ such that $x \prec z \prec y$. We write this as $x \lessdot y$.
I assumed that every poset could be described as a directed acylic graph through the cover relation between its elements. This intuitively makes sense to me, as cycles (e.g., $x \lessdot y$ and $y \lessdot x$) would be inconsistent with posets due to its antisymmetry, but $x \neq y$ given the definition of the cover relation. Thinking about cycles in orders reminded me of this great example on the link between directed acyclic graphs and genealogy, from the Handbook of Graph Theory by Jonathan L. Gross:
A “family tree” is a digraph, where the orientation is traditionally given not by arrows but by the direction down for later generations. Despite the name, a family tree is usually not a tree, since people commonly marry distant cousins, knowingly or unknowingly. However, it is always a DAG, because if there were a cycle, everyone on it would be older than everyone else on the cycle.
I did, however, stumble upon a paper titled “Cyclic Ordering through Partial Orders” by Stefan Haar. Briefly scanning through the paper shows some interesting concepts, so I am certainly curious to learn more about this!