Lecture 3 - Preorders

Okay, let's get started!

Fong and Spivak start out by explaining preorders, which is short for "preordered sets". Whenever you have a set of things and a reasonable way deciding when anything in that set is "bigger" than some other thing, or "more expensive", or "taller", or "heavier", or "better" in any well-defined sense, or... anything like that, you've got a preorder. When \(y\) is bigger than \(x\) we write \(x \le y\). (You can also write \(y \ge x\), of course.)

What do I mean by "reasonable"? We demand that the \(\le\) relation obeys these rules:

reflexivity: \(x \le x\)
transitivity \(x \le y\) and \(y \le z\) imply \(x \le z\).

Any set with a relation obeying these rules is called a preorder.

This is a fundamental concept! After all, humans are always busy trying to compare things and see what's better. So, we'll start by studying preorders.

But I can't resist revealing a secret trick that Fong and Spivak are playing on you here. Why in the world should a book on applied category theory start by discussing preorders? Why not start by discussing categories?

The answer: a preorder is a specially simple kind of category. A category, as you may have heard, has a bunch of 'objects' \(x,y,z,\dots\) and 'morphisms' between them. A morphism from \(x\) to \(y\) is written \(f : x \to y\). You can "compose" a morphism from \(f : x \to y\) with a morphism from \(g: y \to z\) and get a morphism \(gf : x \to z\). Every object \(x\) has an "identity" morphism \(1_x : x \to x\). And a few simple rules must hold. We'll get into them later.

But a category with at most one morphism from any object \(x\) to any object \(y\) is really just a preorder! If there's a morphism from \(x\) to \(y\) we simply write \(x \le y\). We don't need to give the morphism a name because there's at most one from \(x\) to \(y\).

So, the study of preorders is a baby version of category theory, where everything gets much easier! And when Fong and Spivak are teaching you about preorders, they're sneakily getting you used to categories. Then, when they introduce categories explicitly, you can always fall back on preorders as examples.

Here are some puzzles on preorders.

Puzzle 1. Show that the set of real numbers, \(\mathbb{R}\), becomes a preorder with the usual notion of \(\le\).

Of course we don't have to use the symbol \(\le\) to denote the relation in our preorder, we can use whatever symbol we want, and sometimes another symbol will be more convenient.

Puzzle 2. Show that the set of natural numbers, \(\mathbb{N}\) becomes a preorder with the relation '\(n\) divides \(m\)', often written \(n \vert m\). To be precise, \(n \vert m\) if and only if \(m = n k\) for some natural number \(k\).

Puzzle 3. For any set \(S\), we call the set of all subsets of \(S\) the power set of \(S\), and write it as \(P(S)\). Show that \(P(S)\) becomes a preorder with the relation \(X \subseteq Y\).

Sometimes instead of \(\le\), the symbol we use for the relation in our preorder is \(\ge\)! The reason is that we can "turn around" any preorder and get another preorder. This can get confusing, but it's extremely important.

Puzzle 4 Show that if some set \(S\) with some relation \(\le\) is a preorder, so is that set with the relation \(\ge\) defined so that \(x \ge y\) if and only if \(y \le x\). This is called the opposite preorder.

Puzzle 5. Show that any set becomes a preorder if we define \(x \le y\) to mean \(x = y\). This is called a discrete preorder.

Puzzle 6. Show that any set \(S\) becomes a preorder if we say that \(x \le y\) for all elements \(x,y \in S\). This is called a codiscrete preorder. Here is a very important special kind of preorder:

Definition. A preorder is called a partial order if it's also antisymmetric: that is, if two elements \(x\) and \(y\) have \(x \le y\) and \(y \le x\) then \(x=y\).

Puzzle 7. Which of the examples of preorders in Puzzles 1--6 are partial orders? (For Puzzle 4, the answer depends on what preorder we started with, but there's still a nice answer.)

The nicest or at least most familiar kind of preorder is a so-called 'linear order':

Definition. A partial order is called a linear order or total order if it also obeys trichotomy: for all elements \(x, y\) we either have \(x \le y\) or \(y \le x\).

Puzzle 8. Which of the examples of preorders in Puzzle 1-6 are linear orders? (Again, for Puzzle 4 the answer depends on the preorder we started with.)

I can't resist giving away a bit of the answer to Puzzle 8. The usual \(\le\) on the set of real numbers is a linear order, and the fact that the real numbers form a line is probably the reason for the term "linear order". You can visualize any linear order as some sort of "line". For example, the integers with their usual \(\le\) form some sort of line, though it's quite different from the real line.

Puzzle 9. What is the reason for the term "trichotomy"?

I also encourage you to list some interesting and important examples of preorders that haven't been mentioned yet. There are lots!

To read other lectures go here.