I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn):
A forcing poset is an ordered triple, $(P, ≤, 1)$, where $≤$ is a preorder on $P$ that satisfies following splitting condition: For all $p ∈ P$, there are $q, r ∈ P$ such that $q, r ≤ p$ with no $s ∈ P$ such that $s ≤ q, r$
I was struck by ambiguity of the last sentence. What does $s ≤ q, r$ mean? Does this mean that "$s ≤ q$ and $s ≤ r$"? Or "$s ≤ q$ or $s ≤ r$"?
It means that $s\leq q$ and $s\leq r$. The second interpretation can be immediately ruled out as impossible, since "$q\leq q$ or $q\leq r$" is always true.
Let me also add, that I feel that this condition is extraneous. It suffices to say that unless stated otherwise, this assumption is made. But it is perfectly fine to work with quasi-orders (which will not necessarily satisfy this property) or with non-separative partial orders (which will also not satisfy this property). In fact, often it makes life easier since it allows a more natural presentation of the forcing conditions.