9.30 (1984)

(Well-known problem). A finite set of reductions $u_i \to v_i$ of words on a finite alphabet $\Sigma = \Sigma^{-1}$ is called a group set of reductions if $\text{length}(u_i) > \text{length}(v_i)$ or $\text{length}(u_i) = \text{length}(v_i)$ and $u_i > v_i$ in the lexicographical ordering, and every word in $\Sigma$ can be reduced to the unique reduced form which does not depend on the sequence of reductions. Do there exist group sets of reductions satisfying the condition $\text{length}(v_i) \leqslant 1$ for all $i$, which are different from 1) sets of trivial reductions $x^{-\varepsilon} x^\varepsilon \to 1$, $\varepsilon = \pm 1$, 2) multiplication tables $xy \to z$ of finite groups, and 3) their finite unions?