# The varieties axiomatized by join-semilattice, self-distributivity, and Fibonacci term identities

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 $$V_{N}$$?