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Tyrrell_McAllister comments on Probability Space & Aumann Agreement - Less Wrong
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Under the usages familiar to me, the common coarsening is the join, not the meet. That's how "join" is used on the Wikipedia page for set partitions. Using "meet" to mean "common refinement" is the usage that makes sense to me in the context of the proof in the OP. [ETA: I've been corrected on this point; see below.]
Of course, what you call "meet" or "join" depends on which way you decide to direct the partial order on partitions. Unfortunately, it looks like both possibilities are floating around as conventions.
See for example on Wikipedia: Common knowledge (logic)
The idea is that the partitions define what each agent is able to discern, so no refinement of what a given agent can discern is possible (unless you perform additional communication). Aumann's agreement theorem is about a condition for when the agents already agree, without any additional discussion between them.
Hmm. Then I am in a state of confusion much like Psy-Kosh's. These opposing convention aren't helping, but, at any rate, I evidently need to study this more closely.
It was confusing for me too, which is why I gave an imperative definition: first form the union of I and J, then merge any overlapping elements. Did that not help?
It should have. The fault is certainly mine. I skimmed your definition too lightly because you were defining a technical term ("meet") in a context (partitions) where I was already familiar with the term, but I hadn't suspected that it had any other usages than the one I knew.
The term "meet" would correspond to considering a coarser partition as "less" than a finer partition, which is natural enough if you see partitions as representing "precision of knowledge". The coarser partition is able to discern less. Greatest lower bound is usually called "meet".
It's always called that, but the greatest lower bound and the least upper bound switch places if you switch the direction of the partial order. And there's a lot of literature on set partitions in which finer partitions are lower in the poset. (That's the convention used in the Wikipedia page on set partitions.)
The justification for taking the meet to be a refinement is that refinements correspond to intersections of partition elements, and intersections are meets in the poset of sets. So the terminology carries over from the poset of sets to the poset of set partitions in a way that appeals to the mathematician's aesthetic.
But I can see the justification for the opposite convention when you're talking about precision of knowledge.