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April 29

Join vs Coarsest common refinement

In https://www.jstor.org/stable/2958591 (Here's a pdf http://www.ma.huji.ac.il/~raumann/pdf/Agreeing%20to%20Disagree.pdf), footnote 4 seems to say that the join of two partitions = the coarsest common refinement of those partitions. But it almost seems like the opposite? What's wrong with my reasoning?

reasoning

According to Join and meet, an element J is the join of A and B (all in the set S) if

$A\leq J{\text{ and }}B\leq J$
For any $w\in S,$ if $A\leq w{\text{ and }}B\leq w,$ then $J\leq w$

Partition_of_a_set#Refinement_of_partitions says the following are the same: "A is finer than B" "A ≤ B" "B is coarser than A",

https://math.stackexchange.com/questions/303834/common-knowledge-and-concept-of-coarsening-partition defines the coarsest common refinement as "the coarsest R finer than both P and Q"

So let P and Q be two partitions of S, and R be the coarsest common refinement of A and B.
By definition R ≤ P and R ≤ Q, but also by definition R is the coarsest such element, so any W that is also ≤ P and ≤ Q will be ≤ R.

And so W ≤ R almost contradicts the 2nd requirement of join (R ≤ W) (?)

AltoStev (talk) 14:46, 29 April 2022 (UTC)[reply]

Edit: Oh wait, R ≤ P and R ≤ Q is also almost the opposite of A ≤ J and B ≤ J. Same question though, what's wrong with my reasoning (which concludes that the join and coarsest common refinement are not the same) AltoStev (talk) 14:52, 29 April 2022 (UTC)[reply]

For any partial order (in this case even a lattice) there is another partial order on the same set of elements which is the converse relation. So one is free to decide in any given application which direction to call "up" and which "down". Apparently, for whatever reason, the author preferred to think of "more refined" as higher, making the partition {{1},{2},{3},{4}} the top and {{1,2,3,4}} the bottom of the lattice of partitions of {1,2,3,4}. This seems to be the common convention in the field,^[1] which may all derive from Aumann's 1976 paper. --Lambiam 18:48, 29 April 2022 (UTC)[reply]