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

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

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

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

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

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

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.