= Two-level partial order
{title2=$P\sqcup C,\quad p<c\text{ for all }p\in P,c\in C$}
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 $P\sqcup C$ as <upper bounds> and has no <least upper bound>. These cases prove the complete classification.
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