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

For $N\geq 1$, let the variety $V_{N}$ consist of all algebras $(X,*,\vee,1)$ that satisfy the following identities:

  1. $x*(y*z)=(x*y)*(x*z)$

  2. $x*1=1,1*x=x$

  3. $(X,\vee)$ is a join-semilattice with greatest element $1$.

  4. $t_{N}(x,y)\vee t_{N+1}(x,y)=1$.

  5. $y\leq x*y$

  6. $x*(y\vee z)=(x*y)\vee(x*z)$

Observe that $V_{N}$ is a subvariety of $V_{N+1}$ for all $N$.

Is the equational theory for $V_{N}$ decidable? Is $V_{N}$ generated by its finite members? Does the algebra on one generator which is free for $V_{N}$ generate the variety $V_{N}$?


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