Proof. Define $\ltimes_0$: $A \ltimes_0 B = (A \otimes I_r) (B \otimes I_q)$. Then we get $$\left\{\begin{array}{l} I_r = I_{{\operatorname{lcm}} (q, r) / q} \otimes I_{\gcd (q, r)},\\ I_q = I_{{\operatorname{lcm}} (q, r) / r} \otimes I_{\gcd (q, r) .} \end{array}\right.$$ Thus $$\begin{aligned} (A \ltimes_0 B) & = (A \otimes I_{{\operatorname{lcm}} (q, r) / q} \otimes I_{\gcd (q, r)}) (B \otimes I_{{\operatorname{lcm}} (q, r) / r} \otimes I_{\gcd (q, r)})\\ & = (A \ltimes B) \otimes I_{\gcd (q, r)} . \end{aligned}$$ Considering $(A \ltimes_0 B) \ltimes_0 C$ and $A \ltimes_0 (B \ltimes_0 C)$, $$\begin{aligned} (A \ltimes_0 B) \ltimes_0 C & = ((A \ltimes_0 B) \otimes I_t) (C \otimes I_{qs})\\ & = ((A \otimes I_r) (B \otimes I_q) \otimes I_t) (C \otimes I_{qs})\\ & = (A \otimes I_{rt}) (B \otimes I_{qt}) (C \otimes I_{qs})\\ & = (A \otimes I_{rt}) ((B \otimes I_t) (C \otimes I_s) \otimes I_q)\\ & = (A \otimes I_{rt}) ((B \ltimes_0 C) \otimes I_q)\\ & = A \ltimes_0 (B \ltimes_0 C), \end{aligned}$$ thus the operator $\ltimes_0$ is associative.

Then we consider the relation between $(A \ltimes_0 B) \ltimes_0 C$ and $(A \ltimes B) \ltimes C$. $$\begin{aligned} (A \ltimes_0 B) \ltimes_0 C & = ((A \ltimes B) \otimes I_{\gcd (q, r)}) \ltimes_0 C\\ & = (((A \ltimes B) \otimes I_{\gcd (q, r)}) \ltimes C) \otimes I_{\gcd (qs, t)}\\ & = (((A \ltimes B) \otimes I_{\gcd (q, r)}) \otimes I_{{\operatorname{lcm}} (qs, t) / (qs)}) (C \otimes I_{{\operatorname{lcm}} (qs, t)}) \otimes I_{\gcd (qs, t)}\\ & = (((A \ltimes B) \otimes I_{\gcd (q, r)}) \otimes I_t) (C \otimes I_{qs})\\ & = ((A \ltimes B) \otimes I_{\gcd (q, r) t}) (C \otimes I_{qs}) \end{aligned}$$ For convenience, we denote $I \{ n \} : = I_n$, and then we get $$\begin{aligned} ((A \ltimes B) \otimes I_{\gcd (q, r) t}) & = (A \ltimes B) \otimes I \left\{ \frac{r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}{s {\operatorname{lcm}} (q, r)} \right\} \otimes I \left\{ \frac{st {\operatorname{lcm}} (q, r) \gcd (q, r)}{r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} \right\}\\ & = (A \ltimes B) \otimes I \left\{ \frac{r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}{s {\operatorname{lcm}} (q, r)} \right\} \otimes I \left\{ \frac{sqt}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} \right\}\\ & = (A \ltimes B) \otimes I \left\{ \frac{r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}{s {\operatorname{lcm}} (q, r)} \right\} \otimes I \left\{ \gcd (q, r) \gcd \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) \right\}, \end{aligned}$$ and $$(C \otimes I_{qs}) = C \otimes I \left\{ \frac{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}{t} \right\} \otimes I \left\{ \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} \right\} .$$ it’s not hard to find that $\gcd (q, r) \gcd \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) = \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}$, $$\begin{aligned} \gcd (q, r) \gcd \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) = \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} & \Leftrightarrow tqs = \gcd (q, r) \frac{st {\operatorname{lcm}} (q, r)}{r}\\ & \Leftrightarrow tqs = \frac{sqrt}{r}, \end{aligned}$$ and that’s trivial.

Thus we get $(A \ltimes_0 B) \ltimes_0 C = ((A \ltimes B) \ltimes C) \otimes I \left\{ \gcd (q, r) \gcd \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) \right\}$.

By the same method, we consider $A \ltimes_0 (B \ltimes_0 C)$. $$\begin{aligned} A \ltimes_0 (B \ltimes_0 C) & = A \ltimes_0 ((B \ltimes C) \otimes I \{ \gcd (s, t) \})\\ & = (A \ltimes ((B \ltimes C) \otimes I \{ \gcd (s, t) \})) \otimes I \{ \gcd (q, rt) \}\\ & = \begin{aligned} &\left( A \otimes I \left\{ \frac{{\operatorname{lcm}} (q, rt)}{q} \right\} \right) \left( (B \ltimes C) \otimes I \left\{ \frac{\gcd (s, t) {\operatorname{lcm}} (q, rt)}{rt} \right\} \right)\\ &\otimes I \{ \gcd (q, rt) \} \end{aligned}\\ & = (A \otimes I \{ rt \}) ((B \ltimes C) \otimes I \{ q \gcd (s, t) \}) \end{aligned}$$ $$\begin{aligned} (A \otimes I \{ rt \}) & = A \otimes I \left\{ \frac{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)}{q} \right\} \otimes I \left\{ \frac{rtq}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} \right\} \end{aligned}$$ and $$\begin{aligned} ((B \ltimes C) \otimes I \{ q \gcd (s, t) \}) & = (B \ltimes C) \otimes I \left\{ \frac{s {\operatorname{lcm}} \left( \frac{r {\operatorname{lcm}} (s, t)}{s}, q \right)}{r {\operatorname{lcm}} (s, t)} \right\} \otimes I \left\{ \frac{qrt}{{\operatorname{lcm}} \left( \frac{r {\operatorname{lcm}} (s, t)}{s}, q \right)} \right\} \end{aligned}$$ so we get $$A \ltimes_0 (B \ltimes_0 C)= (A \ltimes (B \ltimes C)) \otimes I \left\{ \frac{qrt}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} \right\}$$.

And then we prove that $\frac{qrt}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} = \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}$. $$\begin{aligned} \frac{qrt}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} = \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} & \Leftrightarrow \frac{r}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} = \frac{s}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)}\\ & \Leftrightarrow r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) = s {\operatorname{lcm}} \left( \frac{r {\operatorname{lcm}} (s, t)}{s} \right) . \end{aligned}$$ When $q, r, s, t$ are coprime, $$\begin{aligned} \frac{s}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} & = \frac{s}{{\operatorname{lcm}} (sq, t)}\\ & = \frac{1}{qt}\\ & = \frac{r}{prst / s}\\ & = \frac{r}{{\operatorname{lcm}} \left( \frac{r {\operatorname{lcm}} (s, t)}{s}, q \right)} . \end{aligned}$$ And when they are not coprime, Let $q = \prod_{i = 1}^n p_i^{a_i}$, $r = \prod_{i = 1}^n p_i^{b_i}$, $s = \prod_{i = 1}^n p_i^{c_i}$, $t = \prod_{i = 1}^n p_i^{d_i}$, and then we get $$\begin{aligned} r {\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right) & = \prod_{i = 1}^n p_i^{b_i} {\operatorname{lcm}} \left( \prod_{i = 1}^n p_i^{\min (a_i, b_i) + c_i - d_i}, \prod_{i = 1}^n p_i^{d_i} \right)\\ & = \prod_{i = 1}^n p_i^{b_i + \min (\min (a_i, b_i) + c_i - b_i, d_i)},\\ s {\operatorname{lcm}} \left( \frac{r {\operatorname{lcm}} (s, t)}{s} \right) & = \prod_{i = 1}^n p_i^{c_i} {\operatorname{lcm}} \left( \prod_{i = 1}^n p_i^{\min (c_i, d_i) + b_i - c_i}, \prod_{i = 1}^n p_i^{a_i} \right)\\ & = \prod_{i = 1}^n p_i^{c_i + \min (\min (c_i, d_i) + b_i - c_i, a_i)} . \end{aligned}$$ It remains to prove that $$\begin{aligned} & b_i + \min (\min (a_i, b_i) + c_i - b_i, d_i) = c_i + \min (\min (c_i, d_i) + b_i - c_i, a_i)\\ \Leftrightarrow &b_i - c_i + \min (\min (a_i, b_i) + c_i - b_i, d_i) = \min (\min (c_i, d_i) + b_i - c_i, a_i) \\ \Leftrightarrow & \min (\min (a_i, b_i), d_i + b_i - c_i) = \min (\min (c_i, d_i) + b_i - c_i, a_i) \\ \Leftrightarrow & \min (\min (a_i, b_i), d_i + b_i - c_i) = \min (\min (b_i, d_i + b_i - c_i) , a_i) \\ \Leftrightarrow &\min (a_i, b_i, d_i + b_i - c_i) = \min (b_i, d_i + b_i - c_i, a_i) \end{aligned}$$ and that’s trivial.

Then by doing projection, $$\begin{aligned} A \ltimes_0 (B \ltimes_0 C) = (A \ltimes_0 B) \ltimes_0 C & \Leftrightarrow (A \ltimes_0 (B \ltimes_0 C)) \Psi = ((A \ltimes_0 B) \ltimes_0 C) \Psi\\ & \Leftrightarrow A \ltimes (B \ltimes C) = (A \ltimes B) \ltimes C, \end{aligned}$$ where $$\begin{aligned} \Psi &= \left( I \left\{ qsu - \frac{qrt}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} \right\} \otimes {\boldsymbol{1}} \left\{ \frac{qrt}{{\operatorname{lcm}} \left( q, \frac{r {\operatorname{lcm}} (s, t)}{s} \right)} \right\} \right)\\ &= \left( I \left\{ qsu - \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} \right\} \otimes {\boldsymbol{1}} \left\{ \frac{tqs}{{\operatorname{lcm}} \left( \frac{s {\operatorname{lcm}} (q, r)}{r}, t \right)} \right\} \right) \end{aligned}$$ ◻