# Basic questions about varieties of uniformly partially permutative algebras

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

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

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

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

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.

1. 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.

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

3. 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 $$V_{N}$$?

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

• In other words $t_n(x,y)=x^{F_{n-1}}y^{F_{n-2}}$ ? – Max Alekseyev Mar 22 at 4:51
• The operation $*$ is not commutative nor associative, so $t_{5}(x,y)=((xy)x)(xy)$. The word algebra is used in a universal algebraic sense, and not a ring theoretic sense. – Joseph Van Name Mar 22 at 11:59
• Is there any reason why this question is downvoted? – Joseph Van Name Mar 23 at 16:57
• I doubt that the downvoters have much relevant knowledge about self-distributivity, the Fibonacci terms, or any topic relevant to this question. – Joseph Van Name Mar 23 at 17:15