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.
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.