Anindya wrote in #22:

> Second – and this is more of an itch on my part – I suspect there's a whole bundle of subtleties wrapped up in what you've called the "upper semilattice identity".

I don't think so. Well, perhaps one could manufacture subtleties, but none naturally come to mind.

> But are the indexing widgets necessarily sets?

Yes.

Of course you could order the indexing sets if you want, or equip with them with some other structure, but that other structure plays no role in the definition of join, so it has no effect on the answer, so there's really no point to bringing it in.