Maybe it's to avoid Aaronson #9. Quoted directly from his blog: But when it comes to something like P≠NP, to “motivate” your result is to insult your readers’ intelligence.
I am not assuming any of those things. If an operation is distributive and associative, then isn't it automatically closed? This is what forms rings from abelian groups.
The operations $+$ and $\cdot$ are intended to be less abstract. Yes, $+$ need not be closed over $G\cup H$, but is it required $\cdot$ is closed over it?
In your first post of this thread, it seems as though you have misunderstood my question. Obviously, there is only one non-isomorphic monoid...
Suppose that $(G,+)$ and $(H,+)$ are both monoids and that the operation $\cdot$ is closed, associative, and distributive over $+$ in $G$ and $H$. My question then is whether or not $(G\cup H,\cdot)$ is necessarily a monoid. I have evidence to suggest that it might, though I cannot prove it.
Not to mention:
$$\varphi =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$
$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}$$
$$e=2+\frac{1}{1+\frac{1}{2+\frac{1}{1+...}}}=[2;1,2,1,1,4,1,1,6,1,1,8,...]$$
Fraction:$\frac{187465}{6744582}$
Continued fraction: $[0,35,1,44,111,2,2,12,4]$
Unit fraction expansion: $\frac{1}{36}+\frac{1}{58395}+\frac{1}{9724688047}+\frac{1}{283708672824669334580}$