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Here are some very important, very basic posets:
One topic from the book I didn't discuss is how we get partitions of a set \(S\) from surjective functions \(f: S \to T\). The basic idea is simple: for each element \(t \in T\) we get one part of our partition, namely the set of all elements that map to \(t\). Here is an exercise to check that you understand this:
Daniel Cicala gave two interesting "degenerate" examples of preorders. Take your favorite set \(X\). Then:
Setting \(x \leq y\) if and only if \(x = y\) gives you a preorder on \(X\), called the discrete preorder.
Setting \(x \leq y \) for all \(x,y \in X\) gives you a preorder on \(X\), called the codiscrete preorder.
This led to some puzzles:
Puzzle 8. What simple law do Daniel's preorders obey, that does not hold for the real numbers with its usual notion of \(\leq\)?
Puzzle 9. What do you call preorders that obey this law?
For answers go here.
And here's an inexhaustible puzzle:
Puzzle 10. There are many examples of monotone maps between posets. List a few interesting ones!
You can see some examples in the comments on Lecture 3 starting here.
To see all the exercises in Chapter 1, go here.