I've been readind Date's Database In Depth. While discussing the Signifiicance of 5NF, he means joins that are all one-to-one. I'm new to Dbs and SQL. I wans't sure what he means by one-to-one. The content is:

Does he mean that only one attribute, a key, is involved in the "recomposition" join?

He means that the recomposition join is a bijection.

In other words A JOIN B where A and B have some superkey in common (not necessarily just one attribute) and where each tuple in A corresponds to exactly one tuple in B and vice versa.

Avoid the term "one to one". It always confuses because there is no consensus on precisely what it means.

Date is not quite right. I had to look up the passage you quoted to be sure. It's on page 146. Here's why he's not right.

Every relation R{A} satisfies the join dependency *{{A},{}} whether {} is a key or not.

Trivially therefore: R JOIN DEE = R

Perhaps this isn't very important but I don't see any sound reason to discount {} as a valid (but not terribly useful) decomposition. So it seems that a 5NF relation can have at least one nonloss decomposition that does not include a superkey.

Thanks for the replies. I'll check out the link to bijection.
I thought * { {A}, {} } was considered a trivial JD, and Date's definition, on p. 146 of 5NF, says that every nontrivial JD of a 5NF relvar must consist of a superkeys.

You are right! I made a silly mistake. I wish I'd seen your wise reply before I rashly posted my thoughts onto the TTM mailing list, which is what I just did.