Matthew Doty answered Puzzles 1-4 correctly! But Puzzle 4 is far from finished: there are zillions of interesting posets - interesting to different people for different reasons. So, I urge people to come up with more examples.

Some comments on Matthew's remarks:

> > Puzzle 3: what do mathematicians usually call the thing that Fong and Spivak call a poset?

> They are usually called [preordered sets](https://en.wikipedia.org/wiki/Category_of_preordered_sets).

Yes! And category theorists often call them simply **preorders**. This could be confusing: we're calling a set \\(S\\) with a preorder \\(\le\\) simply a "preorder". But in practice it works well, and it's nice and short, so I'm likely to do this.

In summary:

**Definition.** Given a set \\(S\\), a **preorder** on \\(S\\) is a binary relation \\(\le\\) that is:

1. **reflexive**: \\(x \le x\\) for all \\(x \in S\\),

2. **transitive** \\(x \le y\\) and \\(y \le z\\) imply \\(x \le z\\) for all \\(x,y,z \in S\\).

Fong and Spivak, unlike everyone else on the planet, call a set with a preorder a **poset**. Category theorists call a set with a preorder simply a **preorder**.

**Definition.** A preorder is called a **partial order** if it's also **antisymmetric**: if \\(x \le y\\) and \\(y \le x\\) then \\(x = y\\) for all \\(x,y\\). Everyone except Fong and Spivak call a set with a partial order a **partially ordered set** or **poset**. They call it a **skeletal poset**.

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

**Puzzle 5.** Why is this property called "trichotomy"?

Some comments on Matthew's remarks:

> > Puzzle 3: what do mathematicians usually call the thing that Fong and Spivak call a poset?

> They are usually called [preordered sets](https://en.wikipedia.org/wiki/Category_of_preordered_sets).

Yes! And category theorists often call them simply **preorders**. This could be confusing: we're calling a set \\(S\\) with a preorder \\(\le\\) simply a "preorder". But in practice it works well, and it's nice and short, so I'm likely to do this.

In summary:

**Definition.** Given a set \\(S\\), a **preorder** on \\(S\\) is a binary relation \\(\le\\) that is:

1. **reflexive**: \\(x \le x\\) for all \\(x \in S\\),

2. **transitive** \\(x \le y\\) and \\(y \le z\\) imply \\(x \le z\\) for all \\(x,y,z \in S\\).

Fong and Spivak, unlike everyone else on the planet, call a set with a preorder a **poset**. Category theorists call a set with a preorder simply a **preorder**.

**Definition.** A preorder is called a **partial order** if it's also **antisymmetric**: if \\(x \le y\\) and \\(y \le x\\) then \\(x = y\\) for all \\(x,y\\). Everyone except Fong and Spivak call a set with a partial order a **partially ordered set** or **poset**. They call it a **skeletal poset**.

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

**Puzzle 5.** Why is this property called "trichotomy"?