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!