John, my understanding is that an empty reaction network is one in which only the trivial relation \$$x \le x\$$ holds for all \$$x\$$. Under that definition, my argument comes out even worse: I was working with a completely unrelated relation to begin with. We could imagine that one to be an empty reaction network except that reagents may (discretely) _evaporate_.

Thinking through my argument, it's only _accidentally_ correct for empty reaction networks. It's still true that we need minimal/maximal elements of \$$\mathbb{N}\$$ to cover the forgotten reagents, since an infinity of \$$S\$$-complexes collapse onto the same \$$T\$$-complex. But the only element whose inverse image we need to worry about for computing \$$g(x)\$$ is \$$x\$$ itself.

(**EDIT:** Wait, I'm even doubting this now -- because no two of the collapsing infinity of \$$S\$$-complexes are related, we don't have a unique join.)

Also, I'm a little confused by this homomorphism:
$f( a_1 s_1 + \cdots + a_n s_n) = a_1 s_{\phi(1)} + \cdots + a_n s_{\phi(n)}$
If \$$\phi = \\{1 \mapsto 1, 2 \mapsto 1\\}\$$, then \$$f(a_1 s_1 + a_2 s_2) = a_1 s_1 + a_2 a_1 = (a_1 + a_2) s_1\$$, meaning this function doesn't _forget_ reagents, it just treats them as _interchangeable_. (But this does nicely disprove my hypothesis from [comment #1](https://forum.azimuthproject.org/discussion/comment/18335/#Comment_18335)!)