duaLabThe connected compact Hausdorff problem
The former open problem "Is there a nontrivial connected compact Hausdorff rieg?" has an affirmative answer, recorded in SUMIRE:2609.00007/2.
Take $[0,1]$ with $x+y=\max(x,y)$, $xy=\min(x,y)$ and $x^y=\max(x,1-y)$. The rieg laws follow from distributivity of $\max$ and $\min$ and from the De Morgan laws for $y\mapsto 1-y$; for example $x^{y+z}=\max(x,\min(1-y,1-z))=x^yx^z$. All operations are continuous, so this is a nontrivial connected compact Hausdorff topological rieg. Its addition is idempotent, consistent with SUMIRE:2609.00007/9: a compact Hausdorff rieg with cancellative addition is trivial.
The example and its proof were produced with AI and have not been checked by a human.