This is a pedantic terminology question for a paper I'm writing.
Given a partial order $P$, a subset $U\subseteq P$ is called a down-set (also lower set, order ideal and several other names) if for every $s\in U$ and every $t\le s$ we have $t\in U$.
The empty set satisfies this condition vacuously, but I'm wondering whether there's any general consensus on whether the empty set should be considered a down-set. I consulted Wikipedia and a couple of text-books, but they didn't seem to mention this edge case.
As you said, the emptyset satisfies the definition. It doesn't matter whether it satisfies it vacuously - that makes it a downset. I think the reason you're not seeing it explicitly addressed is that it isn't considered an edge case in the first place.
(Of course, there may be some results which hold only for nonempty down sets, so it could indeed form an edge case for the results you're looking at, but that doesn't affect the downset-ness of the emptyset.)