$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:

$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":

$$R_X (x, y) \cdot R_Y (x + 1, y) = R_Y (x, y) \cdot R_X (x, y + 1).$$

Here we think of $R_X,R_Y$ as "advancing" along the $x,y$ directions respectively on the grid $\mathbb{Z}^2$.

*The geometric idea is the following:*

Start at the point $(x,y)$. The LHS corresponds to moving one unit to the right along the $x$-axis, and then moving one unit upward along the $y$-axis. In the RHS we first advance along the $y$-direction, then the $x$-direction.

In particular, this implies that the natural extension of more than two products is path-independent.

**Question:** In what contexts has this structure been observed?

I am familiar with the concept of WZ pairs, which is a special case. Are there other contexts?

For simplicity I chose to work here with $\mathbb{Z}^2$, but there is a natural extension of this structure to $\mathbb{Z}^d$, where we have $d$ families $R_i:\mathbb{Z}^d \to M$, which "commute pairwise" in this sense.

*Motivation:* We reached this notion in the context of studying polynomial continued fractions.

In our case $M=M_2(\mathbb{Z})$ is the space of $2 \times 2$ integer matrices, and $R_X (x, y),R_Y (x, y)$ both depend **polynomially** on $(x,y)$. (see page 11 here).