Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:

1. Given what I have, _is it possible_ to get what I want?
2. Given what I have, _how much will it cost_ to get what I want?
3. Given what I have, _how long will it take_ to get what I want?
4. Given what I have, _what is the set of ways_ to get what I want?

Our first approach will only tackle question 1. Given \\(y\\), we will only ask _is it possible_ to get \\(x\\). This is a yes-or-no question, unlike questions 2-4, which are more complicated. If the answer is yes we will write \\(x \le y\\).

So, for now our resources will form a "preorder", as defined in [Lecture 3](https://forum.azimuthproject.org/discussion/1812/lecture-3-chapter-1-posets/p1).

**Definition.** A **preorder** is a set \\(X\\) equipped with a relation \\(\le\\) obeying:

1. **reflexivity**: \\(x \le x\\) for all \\(x \in X\\).

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

All this makes sense. Given \\(x\\) you can get \\(x\\). And if you can get \\(x\\) from \\(y\\) and get \\(y\\) from \\(z\\) then you can get \\(x\\) from \\(z\\).

What's new is that we can also _combine_ resources. In chemistry we denote this with a plus sign: if we have a molecule of \\(\text{H}_2\text{O}\\) and a molecule of \\(\text{CO}_2\\) we say we have \\(\text{H}_2\text{O} + \text{CO}_2\\). We can use almost any symbol we want; Fong and Spivak use \\(\otimes\\) so I'll often use that. We pronounce this symbol "tensor". Don't worry about why: it's a long story, but you can live a long and happy life without knowing it.

It turns out that when you have a way to combine things, you also want a special thing that acts like "nothing". When you combine \\(x\\) with nothing, you get \\(x\\). We'll call this special thing \\(I\\).

**Definition.** A **monoid** is a set \\(X\\) equipped with:

1. a binary operation \\(\otimes : X \times X \to X\\)
2. an element \\( I \in X \\)

such that these laws hold:

1. the **associative law**: \\( (x \otimes y) \otimes z = x \otimes (y \otimes z) \\) for all \\(x,y,z \in X\\)

2. the **left and right unit laws**: \\(I \otimes x = x = x \otimes I\\) for all \\(x \in X\\).

You know lots of monoids. In mathematics, monoids rule the world! I could talk about them endlessly, but today we need to combine the monoids and preorders:

**Definition.** A **monoidal preorder** is a set \\(X\\) with a relation \\(\le\\) making it into a preorder, an operation \\(\otimes : X \times X \to X\\) and element \\(I \in X\\) making it into a monoid, and obeying:

$$ x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .$$

This last condition should make sense: if you can turn an egg into a fried egg and turn a slice of bread into a piece of toast, you can turn an egg _and_ a slice of bread into a fried egg _and_ a piece of toast!

You know lots of monoidal preorders, too! Many of your favorite number systems are monoidal preorders:

1. The set \\(\mathbb{R}\\) of real numbers with the usual \\(\le\\), the binary operation \\(+: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \\) and the element \\(0 \in \mathbb{R}\\) is a monoidal preorder.

2. Same for the set \\(\mathbb{Q}\\) of rational numbers.

3. Same for the set \\(\mathbb{Z}\\) of integers.

4. Same for the set \\(\mathbb{N}\\) of natural numbers.

_Money_ is an important resource: outside of mathematics, _money_ rules the world.
We combine money by addition, and we often use these different number systems to keep track of money. In fact it was bankers who invented negative numbers, to keep track of debts! The idea of a "negative resource" was very radical: it took mathematicians over a century to get used to it.

But sometimes we combine numbers by multiplication. Can we get monoidal preorders this way?

**Puzzle 60.** Is the set \\(\mathbb{N}\\) with the usual \\(\le\\), the binary operation \\(\cdot : \mathbb{N} \times \mathbb{N} \to \mathbb{N}\\) and the element \\(1 \in \mathbb{N}\\) a monoidal preorder?

**Puzzle 61.** Is the set \\(\mathbb{R}\\) with the usual \\(\le\\), the binary operation \\(\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\\) and the element \\(1 \in \mathbb{R}\\) a monoidal preorder?

**Puzzle 62.** One of the questions above has the answer "no". What's the least destructive way to "fix" this example and get a monoidal preorder?

**Puzzle 63.** Find more examples of monoidal preorders.

**Puzzle 64.** Are there monoids that cannot be given a relation \\(\le\\) making them into monoidal preorders?

**Puzzle 65.** A **monoidal poset** is a monoidal preorder that is also a poset, meaning

$$ x \le y \textrm{ and } y \le x \textrm{ imply } x = y $$

for all \\(x ,y \in X\\). Are there monoids that cannot be given any relation \\(\le\\) making them into monoidal posets?

**Puzzle 66.** Are there posets that cannot be given any operation \\(\otimes\\) and element \\(I\\) making them into monoidal posets?

**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Chapter_2)**