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On page 4 of Many Lives of Lattice Theory the author wrote: "Two equivalence relations on a set are said to be independent when every equivalence class of the first meets every equivalence class of the second." I fail to understand this definition, because in lattice of partitions we define meet and join of sets of equivalence classes, not individual classes.

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up vote 1 down vote accepted

An equivalence class is a set, so it makes sense to talk about the intersection with other sets.

Yes, intersection of sets of equivalence classes can also be defined, but here they are not used.

If you are actually interested in why this definition is used, you need to give more context.

Edited to add:

At the end of the paragraph cited in the question Rota explains: "Two equivalence relations are independent when the answer to either question gives no in- formation on the possible answer to the other question." (The question is: In which equivalence class of the equivalence relations does my object lie?)

So he gives the definition and the motivation. Your problem seems to be that in other contexts one is interested in other definitions which is true, but so what?

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Regarding set intersection a property that one is usually interested in is disjointness. Is what Rota saying: "every equivalence class intersects (that is not disjoint) with the other"? – Tegiri Nenashi Oct 7 '11 at 23:01
Yes, every equivalence class of the first equivalence relation has an element in common with every equivalence class of the second equivalence relation. There are no disjoint classes between the two equivalence relations. – Phira Oct 7 '11 at 23:23
My confusion was about Rota usage of "meet" as relation (as opposed to function) in the paper about lattices. I'm simply not aware of "meet" being synonymous with "have at least one element in common" anywhere in math literature. If you exhibit one, I would happily mark your entry as answered. – Tegiri Nenashi Oct 8 '11 at 0:12
The paper of Rota is the example because he explains it in the paragraph following the sentence. But the google search ("A meets B" "not disjoint" sets) provides other examples. – Phira Oct 8 '11 at 0:15
Cool. I was misguided into checking up antonyms of "disjoint" at, and not surprisingly, found "join"(not meet!) – Tegiri Nenashi Oct 8 '11 at 0:23

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