3 problems

Connected idempotent shelf polynomial conjecture. If $S$ is idempotent, then

Let $S$ be a shelf, with its operation and connectedness defined in the paper. A shortest-cycle characterization of connected shelves. There exists a shortest cycle that covers all…

Let $G$ be the group generated by $(\tau_n)_{n\geq1}$ with relations … and $\tau_i\tau_j=\tau_j\tau_i$ whenever $|i-j|>2$. Let $sh:G\to G$ satisfy $sh(\tau_i)=\tau_{i+1}$, and defi…