$\newcommand{\M}{\mathcal{M}}$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).

  • 1
    $\begingroup$ It looks like a kind of path independence, as you observe yourself. Not sure if there's more to it than that. $\endgroup$
    – R.P.
    Sep 5 at 11:16
  • 1
    $\begingroup$ One way to approach this would be to consider the free monoid with presentation given by generators indexed by two copies of $\mathbf{Z}^2$ and relators given by these relations. This is the "universal" solution to this problem. $\endgroup$
    – YCor
    Sep 5 at 22:54

1 Answer 1


This is an infinite commutative diagram on $M$ (viewed as a category with a single object $\bullet$).

$\require{AMScd}$ \begin{CD} \vdots @. \vdots @. \vdots\\ @VVR_y(0,2)V @VVR_y(1,2)V @VVR_y(2,2)V\\ \bullet @<R_x(0,2)<< \bullet @<R_x(1,2)<< \bullet @<R_x(2,2)<< \dots\\ @VVR_y(0,1)V @VVR_y(1,1)V @VVR_y(2,1)V\\ \bullet @<R_x(0,1)<< \bullet @<R_x(1,1)<< \bullet @<R_x(2,1)<< \dots\\ @VVR_y(0,0)V @VVR_y(1,0)V @VVR_y(2,0)V\\ \bullet @<R_x(0,0)<< \bullet @<R_x(1,0)<< \bullet @<R_x(2,0)<< \dots \end{CD}

If $M$ is a group, this is also a (discrete) flat connection on the trivial $M$-bundle over the lattice ${\bf Z}^2$, or the (discrete) derivative of a gauge function from ${\bf Z}^2$ to $M$.

  • 3
    $\begingroup$ In other words it is a functor from the product of linear orders $\mathbb Z\times\mathbb Z$ (viewed as a category) to a single-object category. $\endgroup$ Sep 12 at 2:18

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