Confluence question/idea

Stream: Equational

[Pace Nielsen]

A lot of the implications were resolved by taking a linear map $x\diamond y=ax+by$, where the carrier set is a commutative ring. I've found it somewhat surprising that by generalizing to non-commutative rings, no new counter-examples have yet arisen.

This led me to wonder how much further we could push, and if anything new would fall out. One type of generalization would be to taken the carrier set to be a (multiplicatively written) group (or even just a semigroup) with two unary operations $f,g$, and take $x\diamond y =f(x)g(y)$.

For concreteness, consider Equation 1323. This becomes $x=f(y)g(f(f(f(y)g(y))g(x))g(y))$. When working in a group we can invert the left factor $f(y)$ to the other side, and that gives us a reduction rule. I'm wondering if the confluence methods can apply in the case. (Also, does it matter if we work in semigroups, groups, or abelian groups? Does that affect things?)

[Zoltan A. Kocsis (Z.A.K.)]

I considered a computational variation / special case of the noncommutative case when I tried to attack 1112->8. That also didn't lead to new results.

Specifically, instead of f(x)g(y) I considered the case where the RHS is a word in $x,y,x^{-1},y^{-1}$ and constants from a group.

This is analogous to $ax+by$ where the $a,b$ are integers.

I even did a larger scale computational search trying fairly long words on a large number of parametrized infinite groups (Baumslag-Solitar groups, a family of counterexamples from a degree of satisfiability article, etc.). I found lots of examples satisfying 1112; without exception, they satisfied 8 as well.

Of course there might be an easy reason why this cannot work. Since greedy methods seem to be able to settle this, I didn't feel the need to try and prove a general impossibility result in this direction. And of course, it could work for other equations.

(please don't fault me for not writing up my negative findings; I know it's useful, but I have very little time to dedicate to ETP now and prioritized other contributions)

[Terence Tao]

Thanks for taking the time to come back and report this! We are beginning to get some mileage out of noncommutative constructions in other contexts (using translation-invariant operations on infinite regular trees), so the method does seem to have some potential, but it has to be somewhat targeted I think.