[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\$$.