# Permutative Yang-Baxter monoids

Suppose that $$f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$$ are mappings such that $$T(x,y)=(f(x,y),g(x,y))$$. An element $$1\in X$$ is said to be an identity if $$T(1,x)=(x,1),T(x,1)=(1,x)$$. The mapping $$T$$ is said to satisfy the Yang-Baxter equation if $$(T\times 1_{X})\circ(1_{X}\times T)\circ(T\times 1_{X}) =(1_{X}\times T)\circ(T\times 1_{X})\circ(1_{X}\times T).$$

The mapping $$T$$ is said to be permutative if $$T$$ satisfies the Yang-Baxter equation, $$T$$ has an identity $$1$$ and where for all $$x,y\in X$$, there is some $$n$$ where $$T^{n}(x,y)=(1,r)$$ for some $$r\in X$$. Observe that if $$(X,T)$$ is permutative, then the identity $$1$$ is unique.

The motivation behind the notion of a permutative Yang-Baxter monoid along with examples of such Yang-Baxter monoids come from the very large cardinals and some of the notions behind these very large cardinals extend to the Yang-Baxter monoids. Let $$\mathcal{E}_{\lambda}$$ be the set of all elementary embeddings $$j:V_{\lambda}\rightarrow V_{\lambda}$$. Then $$\mathcal{E}_{\lambda}$$ can be endowed with an operation $$*$$ defined by $$j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$$. If $$\gamma$$ is a limit ordinal with $$\gamma<\lambda$$, then define an equivalence relation $$\equiv^{\gamma}$$ on $$\mathcal{E}_{\lambda}$$ by letting $$j\equiv^{\gamma}k$$ if and only if $$j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$$ for each $$x\in V_{\gamma}$$. Then $$\equiv^{\gamma}$$ is a congruence on $$(\mathcal{E}_{\lambda},*)$$. If $$T:(\mathcal{E}_{\lambda}/\equiv^{\gamma})^{2}\rightarrow(\mathcal{E}_{\lambda}/\equiv^{\gamma})^{2}$$ is the mapping where $$T([j]_{\gamma},[k]_{\gamma})=([j*k]_{\gamma},[j]_{\gamma})$$, then $$(\mathcal{E}_{\lambda}/\equiv^{\gamma},T)$$ is a permutative Yang-Baxter monoid. Are there any other natural examples of permutative Yang-Baxter monoids besides the ones that arise from self-distributive algebras that somewhat resemble the algebras of elementary embeddings?

Since the permutative Yang-Baxter monoids resemble the algebras of elementary embeddings, I want to know how well set theoretic ideas such as the notion of a critical point translate to notions concerning permutative Yang-Baxter monoids.

Is there a characterization of all the permutative Yang-Baxter monoids on one generator? Is every finitely generated permutative Yang-Baxter monoid necessarily finite? Do the finite permutative Yang-Baxter monoids generate the variety of all algebras $$(X,T)$$ that satisfy the Yang-Baxter equation?

Observe that there is a unique binary operation $$\circ$$ on $$X$$ where for all $$x,y$$, there is an $$n$$ where $$T^{n}(x,y)=(1,x\circ y)$$. The algebra $$(X,\circ,1)$$ is a monoid that satisfies the identities

1. $$x\circ y=f(x,y)\circ g(x,y)$$, (the operation $$\circ$$ is uniquely determined by this condition and the fact that $$x\circ 1=1\circ x=1$$)

2. $$g(g(x,y),z)=g(x,y\circ z)$$,

3. $$f(x,y\circ z)=f(x,y)\circ f(g(x,y),z)$$,

4. $$f(x,f(y,z))=f(x\circ y,z)$$, and

5. $$g(x\circ y,z)=g(x,f(y,z))\circ g(y,z)$$.

Since it is not every day that you get an associative operation along with 5 strange looking additional identities for free, so there may be some good mathematics behind the permutative Yang-Baxter monoids (or maybe not).

Feel free to add or remove tags. I had trouble picking good tags for this question.