Two-level partial order ($P\bigsqcup C,p<c\text{ for all }p\in P,c\in C$)

Take disjoint antichains $P,C$ and place every element of $P$ below every element of $C$. Together with equality these comparisons define a partial order: a strict step always moves from $P$ to $C$, so there cannot be a strict two-step path or a reverse comparison.

If both levels have at least two elements, a subset $S$ has an upper bound exactly when it contains at most one element of $C$. When it contains one such element $c$, that element is its maximum and its only upper bound. A singleton $\{p\}\subseteq P$ has maximum and least upper bound $p$. A subset of $P$ with at least two elements has all of $C$ as upper bounds and has no least upper bound, because these elements are incomparable. The empty subset has all of $P\bigsqcup C$ as upper bounds and has no least upper bound. These cases prove the complete classification.