It is five years since Tatiana Gateva-Ivanova introduced me to the combinatorial Yang–Baxter equation. I may be slow; I have just understood why this is a good thing for a mathematician to study, aside from its applications in physics (whatever they may be).

Let me start with an analogy. A *binary operation* on a set *A* is a function *b* from *X*×*X* to *X*, where *X* is a set. Now given a binary operation, we can define a function *f _{x}* (left translation) on

*X*, for any

*x*∈

*X*, by

*f _{x}*(

*y*) =

*b*(

*x,y*).

Then the associative law is equivalent to the assertion that

If *b*(*x,y*) = *z*, then *f _{x}f_{y}* =

*f*,

_{z}in other words, left translation gives a homomorphism to a transformation semigroup.

Now let *r* be a map from *X*×*X* to itself. Considering *X*×*X*×X, let *r*^{12} denote the map which acts on the first two coordinates as *r* and fixes the third, and *r*^{23} be the map fixing the first coordinate and acting on the second and third as *r*. The *combinatorial Yang–Baxter equation* asserts:

*r*^{12}*r*^{23}*r*^{12} = *r*^{23}*r*^{12}*r*^{23}.

Now, for each *x*∈*X*, let *f _{x}* be the map on

*X*given by

*r*(*x,y*)=(*f _{x}*(

*y*),?).

Then, if *r* satisfies the CYB, we have:

If *r*(*x,y*) = (*u,v*), then *f _{x}f_{y}* =

*f*.

_{u}f_{v}In other words, there is a sense in which solutions to the CYB are “2-dimensional semigroups”.

So it is OK to study them!

The proof is simple. Let *r*(*x,y*) = (*u,v*). Then *r*^{23}*r*^{12}*r*^{23} maps

(*x,y,z*) → (*x,f _{y}*(

*z*),?) → (

*f*(

_{x}*f*(

_{y}*z*)),?,?) → (

*f*(

_{x}*f*(

_{y}*z*)),?,?),

while *r*^{12}*r*^{23}*r*^{12} maps

(*x,y,z*) → (*u,v,z*) → (*u,f _{v}*(

*z*),?) → (

*f*(

_{u}*f*(

_{v}*z*)),?,?).