Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$.

We say that an algebra $(X,*)$ is $N$-uniformly partially permutative if it satisfies the identities

$x*(y*z)=(x*y)*(x*z)$ and

$t_{N}(x,y)=t_{N+2}(x,y)$

$t_{N+1}(x,y)=t_{N+3}(x,y)$

The collection $V_{N}$ of all $N$-uniformly partially permutative algebras is a variety, so whenever I come across any variety, I have to ask myself the following sorts of questions.

Are the free algebras in $V_{N}$ on finitely many generators finite? I conjecture that the answer is no for 2 or more generators and for large enough $N$. For 1 generator, I am a bit more skeptical about the truth value of this conjecture.

In the equational theory of the variety of $N$-uniformly partially permutative algebras decidable?

Is the variety $V_{N}$ generated by its finite members? Can the free algebra in $V_{N}$ on one generator be embedded into the inverse limits of finite algebras in $V_{N}$?

Do the algebras in $V_{N}$ generated by a single element satisfy any identities not satified by all algebras in $V_{N}$?