Dan wrote:

> I have a question regarding notation. In the equation below, is it fair to say that the sign "\\(+\\)" has two meanings?

> \[ (2 [\textrm{egg}] + [\textrm{yolk}]) + (3 [\textrm{egg}] + 3 [\textrm{pie crust}]) = 5[\textrm{egg}] + [\textrm{yolk}] + 3 [\textrm{pie crust}] \]

You can say yes, or you can say no: I believe both viewpoints can be developed in a consistent way. If you treat them as different you may need to distinguish between

\[ ([\textrm{egg}]) + ([\textrm{yolk}]) \]

and

\[ ([\textrm{egg}] + [\textrm{yolk}]) .\]

Here's how I actually think: \\(\mathbb{N}[S]\\) is the **free commutative monoid** on the set \\(S\\). There are many ways to construct this thing, but they're all canonically isomorphic so it doesn't matter which one we use.

(Sorry, that's how category theorists actually think: we are very flexible while being very precise about the ways in which we're being flexible: the words _canonically isomorphic_ summarize a lot of detail that I'm reluctant to explain right now.)

Here's one way to construct \\(\mathbb{N}[S]\\). We take elements of \\(S\\), create all possible finite sums of these elements, which are purely formal expressions of this sort:

\[ s_1 + \cdots + s_n \]

where \\(s_1, \dots, s_n \in S\\), and then we mod out by an equivalence relation which imposes the commutative monoid rules. For example, we decree that

\[ s_1 + s_2 + s_3 = s_2 + s_1 + s_3 .\]

This gives a commutative monoid \\(\mathbb{N}[S]\\). Then, we allow ourselves to abbreviate an \\(n\\)-fold sum of copies of \\(s \in S\\) as \\(n s\\), e.g.

\[ [\textrm{egg}] + [\textrm{egg}] + [\textrm{egg}] = 3[\textrm{egg}] . \]

If you don't like this, here's another equivalent to way to think about it if \\(S\\) has some finite number of elements, say \\(n\\). In this case, after we choose an ordering for \\(S\\), we can think of \\(\mathbb{N}[S]\\) as the set of \\(n\\)-tuples of natural numbers. So, for example, if

\[ S = \\{ [\textrm{egg}], [\textrm{yolk}] \\} \]

we write elements of \\(\mathbb{N}[S]\\) as pairs of natural numbers. The element \\( (2,3) \\) in here corresponds to the element

\[ 2 [\textrm{egg}] + [\textrm{yolk}] \]

in my previous, isomorphic, description of \\(\mathbb{N}[S]\\). Similarly, the equation

\[ (2,3) + (1,1) = (3,4) \]

corresponds to the equation

\[ (2 [\textrm{egg}] + 3[\textrm{yolk}] ) + ([\textrm{egg}] + [\textrm{yolk}]) =
3 [\textrm{egg}] + 4[\textrm{yolk}] .\]