"Loops and fold! Loops are pretty obviously complecting what you're doing and how to do it. Fold is a little bit more subtle, 'cause it seems like this nice 'somebody else is taking care of it,' but it does have this implication about the order of things. This left-to-right bit."
I can see how you'd get that. I think our interpretations are related. As far as how they're related goes...
The underlying issue may be that introducing non-commutativity makes an operation more cumbersome when used with sets, and introducing non-associativity makes it even worse.
- Commutative and associative: If you apply '* or 'max pairwise to a set, you can only get one result. (It's possible to implement 'some this way, and a 'keep that returns an unordered bag.)
- Associative: If you apply 'compose or 'join pairwise to a set, you can get many results, but you can choose between them by putting the items in a list. (It's possible to implement 'keep this way.)
- Commutative: If you apply XOR pairwise to a set, you can get many results, but you can choose between them by placing the items on the leaves of a rooted binary tree.
- Neither: If you apply 'cons pairwise to a set, you can get many results, but you can choose between them by placing them on the leaves of a rooted binary tree with a distinction between left children and right children.
Both 'foldl and 'foldr are specific enough to be deterministic when they're used with non-associative, non-commutative operations like 'cons. The fact that they're so specific is why there isn't just one 'fold.
I was intentionally avoiding 'is as an example, since it has an intuitive meaning when applied to a set (IMO), which is different from applying it piecewise.
