Join is the least common coarsening of two Judgements — the weakest Judgement that contains both.
Where Meet asks “what do two Judgements share?” Join asks the complementary question: “what is the least you can say that covers both?” The Join of A and B is the weakest Judgement that subsumes both A and B. It is the “or” of Judgements: to satisfy the Join of A and B is to satisfy at least one of them.
Join is forced by Order for the same reason as Meet. If Judgements can be compared, the question “what covers both of these?” must have a determinate answer.
Together with Meet, Join forces Implication. The combined structure of Order, Meet, Join, and Implication is a Heyting algebra — constructive logic.