> Question: forgive my ignorance, but why is it called "preorder" and not just "order"? What is the significance of the "pre"?
The original concept of **totally ordered set** or **order**, still dominant today, obeys a bunch of rules:
1. **reflexivity**: \\(x \le x\\)
2. **transitivity**: \\(x \le y\\) and \\(y \le z\\) imply \\(x \le z\\)
3. **antisymmetry**: if \\(x \le y\\) and \\(y \le x\\) then \\(x = y\\)
4. **trichotomy**: for all \\(x,y\\) we either have \\(x\le y\\) or \\(y \le x\\).
The real numbers with the usual \\(\le\\) obeys all these. Then people discovered many situations where rule 4 does not apply. If only rules 1-3 hold they called it a **partially ordered set** or **poset**. Then people discovered many situations where rule 3 does not hold either! If only rules 1-2 hold they called it a **preordered set** or **preorder**.
Category theory teaches us that preorders are the fundamental thing: see [Lecture 3](https://forum.azimuthproject.org/discussion/1812/lecture-3-chapter-1-posets). But we backed our way into this concept, so it has an awkward name. Fong and Spivak try to remedy this by calling them posets, but that's gonna confuse everyone even more! If they wanted to save the day they should have made up a beautiful brand new term.