For some poset $(X,<)$ with a global max $\top$ and min $\bot$ with $\bot < \top$, and a subset $A = \varnothing$, we'd have $\sup \varnothing = \bot$ and $\inf \varnothing = \top$, right? So $\sup \varnothing < \inf \varnothing$.
This just seemed initially counter-intuitive so I want to make sure I'm interpreting this correctly.
Correct, $\sup \varnothing = \bot$ and $\inf \varnothing = \top$. The reason for this seemingly counterintuitive behavior is to make properties like $\sup (A \cup B) = \sup \{ \sup A, \sup B \}$ work out, without having to worry about $\varnothing$ as a special case. I.e., $\sup \varnothing = \bot$ because $\bot$ is the neutral element w.r.t. $\sup$, very much the same reason why $\sum_{i \in \varnothing} a_i = 0$ and $\prod_{i \in \varnothing} a_i = 1$.