Dan #8 you are right, for Hasse diagrams at least this property always holds. Where the book talks about generative effects (1.1.) it also says this holds for any C but I don't know if there is some other context where the Join fails to keep (A ∨ B) ≤ C when A ≤ C and B ≤ C.

If there was a C such that C ≤ (A ∨ B) then there would be an element in either A or B that isn't in C violating A ≤ C and B ≤ C.

* The reason I'm not sure is because my argument hinges on the concept of an element which is not necessarily something I have access to when reasoning about categories?