# Is the variety of algebras that satisfy the Yang-Baxter equation generated by its finite members?

Suppose that $$f,g:X^{2}\rightarrow X$$, and $$T:X^{2}\rightarrow X^{2}$$ is the function where $$T(x,y)=(f(x,y),g(x,y))$$. Then $$(X,f,g)$$ is said to satisfy the Yang-Baxter equation if $$(T\times 1_{X})\circ(1_{X}\times T)\circ(T\times 1_{X})=(1_{X}\times T)\circ(T\times 1_{X})\circ(1_{X}\times T)$$. In other words, $$(X,f,g)$$ satisfies the Yang-Baxter equation precisely when it satisfies the following identities:

1. $$f(f(x,y),f(g(x,y),z))=f(x,f(y,z))$$
2. $$g(f(x,y),f(g(x,y),z))=f(g(x,f(y,z)),g(y,z))$$
3. $$g(g(x,y),z)=g(g(x,f(y,z)),g(y,z))$$

The collection of all algebras that satisfy the Yang-Baxter equations is a variety. Is this variety generated by its finite members?

Does there exist an inverse system $$((X_{n})_{n},(\phi_{m,n})_{m,n})$$ of finite algebras that satisfy the Yang-Baxter identity along with a sequence $$(e_{n})_{n}\in\varprojlim X_{n}$$ where each $$X_{n}$$ is generated by $$e_{n}$$ and where $$(e_{n})_{n}$$ generates a free subalgebra of $$\varprojlim X_{n}$$?

The motivation of this question comes from the fact that the Yang-Baxter equation is a generalization of self-distributivity, and if $$T(x,y)=(x*y,x)$$, then $$T$$ satisfies the Yang-Baxter equation precisely when $$T$$ is self-distributive. The variety of self-distributive algebras is generated by its finite members (in particular, the multigenic Laver tables) and the free self-distributive algebra on one generator embeds into an inverse limit of finite self-distributive algebras known as the classical Laver tables.

• welcome back!.... – BigM Jan 3 at 19:02
• @BigM. Thanks. I have been at the cryptography and theoretical computer science SE sites. – Joseph Van Name Jan 3 at 19:05
• If not, there is an equation satisfied by the finite members which is not a logical consequence of these equations. The only result I recall like this is Freese's result involving (modular, I think) lattices. You might consider that result. Gerhard "Memory Generated By Finite Thoughts" Paseman, 2019.01.03. – Gerhard Paseman Jan 4 at 0:05
• In the case of self-distributivity, I can come up with identities rarely satisfied by finite algebras. For example, all racks and quandles satisfy the identity $(x*x)*y=x*y$. All reasonably small classical Laver tables (the only known counterexamples can only be proven using strong large cardinal hypotheses) and their generalizations satisfy the identity $(x^{n}*x)*y=y$ where $x^{n}*x=x*(x*...(x*x)$. – Joseph Van Name Jan 4 at 2:40