Connected compact Hausdorff riegs

Question

Does a nontrivial connected compact Hausdorff topological rieg exist? All three binary operations—addition, multiplication, and exponentiation—are required to be continuous.

Current result

Compact rieg no-go proves that every compact Hausdorff topological rieg with cancellative addition is trivial. Connectedness is not needed for that result. Thus a positive answer must have noncancellative addition; the general existence question remains open here.

The original motivation compares connected topological algebra with profinite algebra. A path-connected topological rig is contractible: a path $p$ from $0$ to $1$ gives $H(t,x)=p(t)x$. This observation concerns path-connectedness and does not by itself settle the connected compact case.

Next step

Choose a compact connected noncancellative topological semiring and test whether it admits a jointly continuous exponentiation satisfying all the rieg equations, especially $x^0=1$, $x^1=x$, and $x^{y+z}=x^yx^z$. Failure of a particular candidate is not a general nonexistence theorem.

Source

Original problem 7.0.5.