[Dan](https://forum.azimuthproject.org/discussion/comment/18160/#Comment_18160), kind of, yes -- but there is no defined way to add "egg" and "yolk" together while staying in the set \\(S\\) of our base reagents. It might be more fair to say that your \\(+_1\\) is a mere syntactic way to hold two kinds of reagents at once. In other words, we can represent \\(2[\text{egg}] + [\text{yolk}]\\) more syntactically as \\((2, 1)\\), where \\([\text{yolk}]\\) and \\([\text{egg}]\\) are projections from \\(\mathbb{N}[S]\\) to \\(S\\).

This is almost exactly the same as how we write \\(4\hat{x} + 5\hat{y}\\) in linear algebra: it's a convenient notation for the integer-valued vector \\((4, 5)\\) while reminding us of our basis. The only concern against treating \\(+_1\\) as the same thing as \\(+_2\\) is that \\(\text{yolk} +_1 \text{yolk}\\) doesn't make sense, in an irritatingly technical sense. But we can do the obvious lifting of single components \\(\text{yolk}\\) to complete tuples \\(1[\text{yolk}] + 0[\text{egg}] + 0[\text{pie crust}] + \cdots\\), so it's a point that usually goes without mention. With this convention, both \\(+_1\\) and \\(+_2\\) are the same.

**EDIT:** Ah, another way to look at it. The "addition" in \\(1 + i\\) isn't really something that's actionable, is it? There's no simplification you can perform. It's a fully-reduced entity in its own right. Similarly, \\(\mathbb{C}\\) is known to have a representation in terms of \\(\mathbb{R}^2\\), via the \\(\Re\\) and \\(\Im\\) projections: \\(\Re(1 + i) = 1\\), and \\(\Im(1 + i) = 1\\).

This is almost exactly the same as how we write \\(4\hat{x} + 5\hat{y}\\) in linear algebra: it's a convenient notation for the integer-valued vector \\((4, 5)\\) while reminding us of our basis. The only concern against treating \\(+_1\\) as the same thing as \\(+_2\\) is that \\(\text{yolk} +_1 \text{yolk}\\) doesn't make sense, in an irritatingly technical sense. But we can do the obvious lifting of single components \\(\text{yolk}\\) to complete tuples \\(1[\text{yolk}] + 0[\text{egg}] + 0[\text{pie crust}] + \cdots\\), so it's a point that usually goes without mention. With this convention, both \\(+_1\\) and \\(+_2\\) are the same.

**EDIT:** Ah, another way to look at it. The "addition" in \\(1 + i\\) isn't really something that's actionable, is it? There's no simplification you can perform. It's a fully-reduced entity in its own right. Similarly, \\(\mathbb{C}\\) is known to have a representation in terms of \\(\mathbb{R}^2\\), via the \\(\Re\\) and \\(\Im\\) projections: \\(\Re(1 + i) = 1\\), and \\(\Im(1 + i) = 1\\).