Scott: here's one interesting nuance. Your original chart:

showed unordered sets (i.e., just sets) as disjoint from ordered ones. A mathematician would prefer to see a bunch of nested blobs, as in your new chart:

The point is that a preordered set is a special kind of set, a partially ordered set is a special kind of preordered set, and an ordered set is a special kind of partially ordered set.

Category theory goes much further with this line of thought. I don't want to get into it now, but each of these blobs is actually a _category_. There's a _category_ of sets, a _category_ of preordered sets, and so on.

showed unordered sets (i.e., just sets) as disjoint from ordered ones. A mathematician would prefer to see a bunch of nested blobs, as in your new chart:

The point is that a preordered set is a special kind of set, a partially ordered set is a special kind of preordered set, and an ordered set is a special kind of partially ordered set.

Category theory goes much further with this line of thought. I don't want to get into it now, but each of these blobs is actually a _category_. There's a _category_ of sets, a _category_ of preordered sets, and so on.